Abstract Algebra — Chapter 12 Homological algebra

Chapter 12
Homological algebra

We work in this chapter largely in an abelian category. At times, proofs of statements that hold true in arbitrary abelian categories will be given only in categories of modules over a ring. This choice, which simplifies the exposition, will be given a more rigorous justification in the course of the chapter.

12.1. Exact sequences

Though we’ve managed to suppress them to this point, exact sequences are ubiquitous in algebra. Let’s give the definitions.

Definition 12.1.1. §

Let $I$ be the set of integers in an interval in $ℝ$ . A diagram $A_{\cdot}$ in a category $𝒞$ of the form

\dots \to A_{i + 1} \to_{}^{d_{i + 1}^{A}} A_{i} \to_{}^{d_{i}^{A}} A_{i - 1} \to \dots

is a sequence, where the $A_{i}$ are defined for $i \in I$ and the morphisms $d_{i}^{A} : A_{i} \to A_{i - 1}$ are defined for $i \in I$ with $i - 1 \in I$ . We will refer to $I$ as the defining interval of the sequence $A_{\cdot}$ .

Notation 12.1.2. §

In an abelian category, if $A$ is a subobject of an object $B$ , we write $A \subseteq B$ to denote this and $B ∕ A$ for the cokernel of the inclusion morphism $A \to B$ . For $f : B \to C$ , we will let $f (A)$ denote the image of the composite of the inclusion with $f$ .

We are particularly interested in exact sequences.

Definition 12.1.3. §

We say that a diagram

A \to_{}^{f} B \to_{}^{g} C

in an abelian category $𝒞$ is exact if $g \circ f = 0$ and the induced monomorphism $im f \to \ker g$ is an isomorphism.

Definition 12.1.4. §

A sequence $A_{\cdot} = (A_{i}, d_{i}^{A})$ with defining interval $I$ in an abelian category $𝒞$ is exact, or an exact sequence, if the subdiagram

A_{i + 1} \to_{}^{d_{i}} A_{i} \to_{}^{d_{i - 1}} A_{i - 1}

is exact for each $i \in (I + 1) \cap (I - 1)$ .

That is, $A_{\cdot}$ is exact if $d_{i} \circ d_{i + 1} = 0$ and the canonical morphism $im d_{i + 1} \to \ker d_{i}$ is an isomorphism for all $i$ for which $d_{i + 1}$ and $d_{i}$ are defined: we write this more simply as the identification $im d_{i + 1} = \ker d_{i}$ for subobjects of $A_{i}$ .

Remark 12.1.5. §

One can make the same definitions of exact sequences in the category of groups, or more generally in any “semi-abelian” category, and much of the discussion that follows remains the same.

Remark 12.1.6. §

If the interval $I$ of definition of $A_{\cdot}$ has a left (resp., right) endpoint $N$ such that $A_{N} = 0$ , then one can extend $A_{\cdot}$ to the left (resp., right) by taking $A_{i} = 0$ for all $i < N$ (resp., $i > N$ ). In fact, we could do this for any sequence for which $I$ has an endpoint (without the condition $A_{N} = 0$ ), but we do not as the operation does not preserve exactness (nor do the to-be-defined morphisms between two sequences defined on different intervals extend to morphisms under this operation).

Definition 12.1.7. §

Let $𝒞$ be an abelian category.

a.: A short exact sequence in $𝒞$ is an exact sequence in $𝒞$ of the form
$0 \to A \to B \to C \to 0 .$
b.: A left short exact sequence in $𝒞$ is an exact sequence in $𝒞$ of the form
$0 \to A \to B \to C .$
c.: A right short exact sequence in $𝒞$ is an exact sequence in $𝒞$ of the form
$A \to B \to C \to 0 .$

Remark 12.1.8. §

To say that

0 \to A \to_{}^{f} B \to_{}^{g} C \to 0

is exact is to say that $f$ is a monomorphism, $im f = \ker g$ , and $g$ is an epimorphism.

Example 12.1.9. §

Multiplication-by- $n$ provides a short exact sequence of abelian groups

0 \to ℤ \to_{}^{n} ℤ \to ℤ ∕ 𝑛ℤ \to 0 .

Remark 12.1.10. §

If $f : A \to B$ is any morphism in an abelian category, then we have an exact sequence

0 \to \ker f \to A \to_{}^{f} B \to coker f \to 0 .

Note that this provides two short exact sequences

\begin{array}{rcl} 0 \to \ker f \to A \to im f \to 0 & and & 0 \to im f \to B \to coker f \to 0, \end{array}

since $coim f \to im f$ is an isomorphism. We can “splice these back together” to get the $4$ -term sequence by taking the composite $A \to im f \to B$ , which is $f$ .

Definition 12.1.11. §

A long exact sequence in an abelian category $𝒞$ is an exact sequence $A_{\cdot} = {(A_{i}, d_{i}^{A})}_{i \in ℤ}$ .

Frequently, a long exact sequence is expressed in the form

\dots \to A_{2} \to A_{1} \to A_{0} \to 0,

and we can extend it to all integers by setting $A_{i} = 0$ for all $i \leq - 1$ .

Example 12.1.12. §

The sequence

\dots \to ℚ^{2} \to_{}^{f} ℚ^{2} \to_{}^{f} ℚ^{2} \to_{}^{f} ℚ^{2} \to \dots

with $f : ℚ^{2} \to ℚ^{2}$ defined by $f (a, b) = (0, a)$ for $a, b \in ℚ$ is a long exact sequence of $ℚ$ -vector spaces. The sequence

0 \to ℚ \to_{}^{ι_{2}} ℚ^{2} \to_{}^{f} ℚ^{2} \to_{}^{f} ℚ^{2} \to \dots

of $ℚ$ -vector spaces with $ι_{2} (b) = (0, b)$ is also a long exact sequence.

We next study maps between sequences. Let us make a formal definition.

Definition 12.1.13. §

Let $A_{\cdot} = (A_{i}, d_{i}^{A})$ and $B_{\cdot} = (B_{i}, d_{i}^{B})$ be sequences in a category $𝒞$ with defining intervals $I$ and $J$ , respectively. A morphism of sequences $f_{\cdot} : A_{\cdot} \to B_{\cdot}$ in a category is a collection ${(f_{i})}_{i \in I \cap J}$ of morphisms $f_{i} : A_{i} \to B_{i}$ in $𝒞$ such that $d_{i}^{B} \circ f_{i} = f_{i - 1} \circ d_{i}^{A}$ for all $i \in I \cap J \cap (I + 1) \cap (J + 1)$ .

Remark 12.1.14. §

We can view the condition for a sequence of maps $f_{i} : A_{i} \to B_{i}$ between the terms of sequences $A_{\cdot}$ and $B_{\cdot}$ defined at all $i \in ℤ$ to be a map of sequences as saying that the diagram

A morphism of sequences. A full diagram description follows.

Diagram description: A morphism of sequences

The structural squares and triangles displayed here commute.

Objects, listed by row and column:

Row 1, from left to right: column 1: ellipsis; column 2: A subscript (i plus 1); column 3: A subscript (i); column 4: A subscript (i minus 1); column 5: ellipsis.
Row 2, from left to right: column 1: ellipsis; column 2: A subscript (i plus 1); column 3: A subscript (i); column 4: A subscript (i minus 1); column 5: ellipsis.

Arrows and lines:

An arrow from ellipsis (row 1, column 1) to A subscript (i plus 1) (row 1, column 2), without a label.
An arrow from A subscript (i plus 1) (row 1, column 2) to A subscript (i) (row 1, column 3), labelled d subscript (i plus 1) superscript (A).
An arrow from A subscript (i plus 1) (row 1, column 2) to A subscript (i plus 1) (row 2, column 2), labelled f subscript (i plus 1).
An arrow from A subscript (i) (row 1, column 3) to A subscript (i minus 1) (row 1, column 4), labelled d subscript (i) superscript (A).
An arrow from A subscript (i) (row 1, column 3) to A subscript (i) (row 2, column 3), labelled f subscript (i).
An arrow from A subscript (i minus 1) (row 1, column 4) to ellipsis (row 1, column 5), without a label.
An arrow from A subscript (i minus 1) (row 1, column 4) to A subscript (i minus 1) (row 2, column 4), labelled f subscript (i minus 1).
An arrow from ellipsis (row 2, column 1) to A subscript (i plus 1) (row 2, column 2), without a label.
An arrow from A subscript (i plus 1) (row 2, column 2) to A subscript (i) (row 2, column 3), labelled d subscript (i plus 1) superscript (B).
An arrow from A subscript (i) (row 2, column 3) to A subscript (i minus 1) (row 2, column 4), labelled d subscript (i) superscript (B).
An arrow from A subscript (i minus 1) (row 2, column 4) to ellipsis (row 2, column 5), without a label.

commutes.

12.2. The snake and five lemmas

The following result on maps between short exact sequences is the key to much of homological algebra.

Theorem 12.2.1 (Snake lemma). §

Let $𝒞$ be an abelian category, and let

The snake lemma's input diagram. A full diagram description follows.

Diagram description: The snake lemma's input diagram

The two displayed rows are exact, and the squares commute.

Objects, listed by row and column:

Row 1, from left to right: column 1: 0; column 2: A; column 3: B; column 4: C; column 5: 0.
Row 2, from left to right: column 1: 0; column 2: A prime; column 3: B prime; column 4: C prime; column 5: 0.

Arrows and lines:

An arrow from 0 (row 1, column 1) to A, without a label.
An arrow from A to B, labelled f.
An arrow from A to A prime, labelled alpha.
An arrow from B to C, labelled g.
An arrow from B to B prime, labelled beta.
An arrow from C to 0 (row 1, column 5), without a label.
An arrow from C to C prime, labelled gamma.
An arrow from 0 (row 2, column 1) to A prime, without a label.
An arrow from A prime to B prime, labelled f prime.
An arrow from B prime to C prime, labelled g prime.
An arrow from C prime to 0 (row 2, column 5), without a label.

be a commutative diagram in $𝒞$ with exact rows. Then there is an exact sequence

0 \to \ker f \to \ker α \to \ker β \to \ker γ \to_{}^{δ} coker α \to coker β \to coker γ \to coker g^{'} \to 0

such that the resulting diagram

The snake lemma's connecting map. A full diagram description follows.

Diagram description: The snake lemma's connecting map

The middle two rows are the original exact sequences. The blue path from zero through kernel f, kernel alpha, kernel beta, kernel gamma, cokernel alpha, cokernel beta, cokernel gamma, cokernel g prime, and zero is exact. Its long curved arrow is the connecting map delta from kernel gamma to cokernel alpha. The structural squares commute.

Objects, listed by row and column:

Row 1, from left to right: column 3: kernel alpha; column 4: kernel beta; column 5: kernel gamma.
Row 3, from left to right: column 1: 0; column 2: kernel f; column 3: A; column 4: B; column 5: C; column 6: 0.
Row 5, from left to right: column 2: 0; column 3: A prime; column 4: B prime; column 5: C prime; column 6: cokernel g prime; column 7: 0.
Row 7, from left to right: column 3: cokernel alpha; column 4: cokernel beta; column 5: cokernel gamma.

Arrows and lines:

A blue arrow from kernel alpha to kernel beta, without a label.
An arrow from kernel alpha to A, without a label.
A blue arrow from kernel beta to kernel gamma, without a label.
An arrow from kernel beta to B, without a label.
An arrow from kernel gamma to C, without a label.
A curved blue arrow from kernel gamma to cokernel alpha, labelled delta.
An arrow from 0 (row 3, column 1) to kernel f, without a label.
A curved blue arrow from kernel f to kernel alpha, without a label.
An arrow from kernel f to A, without a label.
An arrow from A to B, labelled f.
An arrow from A to A prime, labelled alpha.
An arrow from B to C, labelled g.
An arrow from B to B prime, labelled beta.
An arrow from C to 0 (row 3, column 6), without a label.
An arrow from C to C prime, labelled gamma.
An arrow from 0 (row 5, column 2) to A prime, without a label.
An arrow from A prime to B prime, labelled f prime.
An arrow from A prime to cokernel alpha, without a label.
An arrow from B prime to C prime, labelled g prime.
An arrow from B prime to cokernel beta, without a label.
An arrow from C prime to cokernel gamma, without a label.
An arrow from C prime to cokernel g prime, without a label.
An arrow from cokernel g prime to 0 (row 5, column 7), without a label.
A blue arrow from cokernel alpha to cokernel beta, without a label.
A blue arrow from cokernel beta to cokernel gamma, without a label.
A curved blue arrow from cokernel gamma to cokernel g prime, without a label.

with the natural inclusion and quotient maps is commutative.

Proof.

We work in the category of modules over a ring $R$ . We define $δ$ as follows. For $c \in \ker γ$ , find $b \in B$ with $g (b) = c$ . Then $g^{'} \circ β (b) = γ (c) = 0$ , so $β (b) = f^{'} (a^{'})$ for some $a^{'} \in A^{'}$ . Let $δ (c)$ denote the image ${\bar{a}}^{'}$ of $a^{'}$ in $coker α$ . To see that this is well-defined, note that if $b_{2} \in B$ also satisfies $g (b_{2}) = c$ , then $g (b - b_{2}) = 0$ , so $b_{2} - b = f (a)$ for some $a \in A$ . We then have

β (b_{2}) = β (b) + β \circ f (a) = β (b) + f^{'} \circ α (a),

so $b_{2} = f^{'} (a^{'} + α (a))$ . But $a^{'} + α (a)$ has image ${\bar{a}}^{'}$ in $coker α$ , so $δ$ is well-defined. That $δ$ is an $R$ -module homomorphism follows easily from the construction.

We now check that the other maps are well-defined. Since

β \circ f (\ker α) = f^{'} \circ α (\ker α) = 0,

we have $f (\ker α) \subseteq \ker β$ . Similarly, $g (\ker β) \subset \ker γ$ . Also, if ${\bar{a}}^{'} \in coker α$ , then we may lift it to $a^{'} \in A^{'}$ , map to $b^{'} \in B^{'}$ , and then project to ${\bar{b}}^{'} \in coker β$ . This is well-defined as any other choice of $a^{'}$ differs by some $a \in A$ , which causes ${\bar{b}}^{'}$ to change by the image of $β (f (a))$ , which is zero. Thus $f^{'}$ induces a well-defined homomorphism

{\bar{f}}^{'} : coker α \to coker β .

Similarly, we have a well-defined surjection

{\bar{g}}^{'} : coker β \to coker γ .

We next check that our sequence is a complex. Note that $g \circ f = 0$ , so the same is true on $\ker α$ , and $g^{'} \circ f^{'} = 0$ , so ${\bar{g}}^{'} \circ {\bar{f}}^{'} = 0$ as well. Let $b \in \ker β$ . Then $δ (g (b))$ is given by considering $β (b) = 0$ , lifting it to some $a^{'} \in A^{'}$ , which we may take to be $0$ , and projecting to $coker α$ . Hence $δ (g (\ker β)) = 0$ . On the other hand, if $c \in \ker γ$ , then ${\bar{f}}^{'} (δ (c))$ is given by definition by projecting $β (b)$ to $coker β$ , where $g (b) = c$ , hence is zero. Hence, the image of one map is contained in the kernel of the next at each term of the six term sequence.

Finally, we check exactness at each term. If $a \in \ker α$ , then $f (a) = 0$ implies $a \in \ker f$ . Inclusion then provides a map $\ker f \to \ker α$ that is by definition injective. If $b \in \ker β \cap \ker g$ , then there exists $a \in A$ with $f (a) = b$ . Since $f^{'} \circ α (a) = β \circ f (a) = 0$ and $f^{'}$ is injective, we have $α (a) = 0$ , or $a \in \ker α$ . Hence

f (\ker α) = \ker (\ker β \to \ker γ),

and we have exactness at $\ker β$ .

If $c \in \ker δ$ , then whenever $g (b) = c$ and $f^{'} (a^{'}) = β (b)$ , we have ${\bar{a}}^{'} = 0$ , letting ${\bar{a}}^{'}$ denote the image of $a^{'} \in coker α$ . We then have $a^{'} = f (a)$ for some $a \in A$ , so $b_{2} = b - f (a)$ still satisfies $f (b_{2}) = c$ , but $β (b_{2}) = 0$ . So $b_{2} \in \ker β$ , and we have exactness at $\ker γ$ .

If ${\bar{a}}^{'} \in coker α$ is the image of $a^{'} \in A^{'}$ and ${\bar{f}}^{'} ({\bar{a}}^{'}) = 0$ , then there exists $b \in B$ with $β (b) = f^{'} (a^{'})$ . Now

γ (g (b)) = g^{'} (β (b)) = g^{'} (f^{'} (a^{'})) = 0,

so $g (b) \in \ker γ$ , and $δ (g (b)) = {\bar{a}}^{'}$ . Hence, we have exactness at $coker α$ .

If ${\bar{b}}^{'} \in coker β$ is the image of $b^{'} \in B^{'}$ and ${\bar{g}}^{'} ({\bar{b}}^{'}) = 0$ , then there exists $c \in C$ with $g (b^{'}) = γ (c)$ . Now $c = g (b)$ for some $b \in B$ . And $b_{2}^{'} = b^{'} - b$ has image ${\bar{b}}^{'}$ in $coker β$ . On the other hand, $f^{'} (b_{2}^{'}) = 0$ , so $b_{2}^{'} = f^{'} (a^{'})$ for some $a^{'} \in A^{'}$ . If ${\bar{a}}^{'} \in coker α$ is the image of $a^{'}$ , then ${\bar{f}}^{'} ({\bar{a}}^{'}) = {\bar{b}}^{'}$ as the image of $b_{2}^{'}$ in $coker β$ . Thus, we have exactness at $coker β$ . Finally, if ${\bar{c}}^{'} \in coker γ$ is the image of $c^{'} \in C^{'}$ with trivial image in $coker g^{'}$ , then $c^{'} = g^{'} (b^{'})$ for some $b^{'} \in B^{'}$ , then the image ${\bar{b}}^{'} \in coker β$ of $b^{'}$ satisfies ${\bar{g}}^{'} ({\bar{b}}^{'}) = {\bar{c}}^{'}$ . □

Next, we state another useful result on maps between exact sequences, known as the five lemma.

Theorem 12.2.2 (Five lemma). §

Let

The five lemma. A full diagram description follows.

Diagram description: The five lemma

The two displayed rows are exact, and the squares commute.

Objects, listed by row and column:

Row 1, from left to right: column 1: A; column 2: B; column 3: C; column 4: D; column 5: E.
Row 2, from left to right: column 1: A prime; column 2: B prime; column 3: C prime; column 4: D prime; column 5: E prime.

Arrows and lines:

An arrow from A to B, labelled e.
An arrow from A to A prime, labelled alpha.
An arrow from B to C, labelled f.
An arrow from B to B prime, labelled beta.
An arrow from C to D, labelled g.
An arrow from C to C prime, labelled gamma.
An arrow from D to E, labelled h.
An arrow from D to D prime, labelled delta.
An arrow from E to E prime, labelled epsilon.
An arrow from A prime to B prime, labelled e prime.
An arrow from B prime to C prime, labelled f prime.
An arrow from C prime to D prime, labelled g prime.
An arrow from D prime to E prime, labelled h prime.

be a commutative diagram with exact rows in an abelian category $𝒞$ .

a.: If $β$ and $δ$ are epimorphisms and $𝜖$ is a monomorphism, then $γ$ is an epimorphism.
b.: If $β$ and $δ$ are monomorphisms and $α$ is an epimorphism, then $γ$ is a monomorphism.
c.: If $β$ and $δ$ are isomorphisms, $α$ is an epimorphism, and $𝜖$ is a monomorphism, then $γ$ is an isomorphism.

Proof.

We work in the category of modules over a ring $R$ . It is immediate that parts a and b imply part c (the actual five lemma). We prove part a and note that it, if proven in an arbitrary abelian category, implies b in the opposite category, which is also abelian. Suppose that $β$ and $δ$ are surjective and $𝜖$ is injective. Let $c^{'} \in C^{'}$ , and note that $g^{'} (c^{'}) = δ (d)$ for some $d \in D$ by surjectivity of $δ$ . Also,

𝜖 (h (d)) = h^{'} (δ (d)) = h^{'} (g^{'} (c^{'})) = 0,

so $h (d) = 0$ by injectivity of $𝜖$ . By exactness of the top row at $D$ , we then have $c \in C$ such that $g (c) = d$ . Now,

g^{'} (γ (c) - c^{'}) = δ (g (c)) - g^{'} (c^{'}) = δ (d) - δ (d) = 0,

so by exactness of the bottom row at $C$ , there exists $b^{'} \in B^{'}$ such that $f^{'} (b^{'}) = c^{'} - γ (c)$ . As $β$ is surjective, there also exists $b \in B$ such that $β (b) = b^{'}$ . Set $x = c + f (b) \in C$ . Then

γ (x) - γ (c) = γ (f (b)) = f^{'} (β (b)) = f^{'} (b^{'}) = c^{'} - γ (c)

so $γ (c) = c^{'}$ . Thus, $γ$ is surjective and part a is proven. □

12.3. Homology and cohomology

Definition 12.3.1. §

Let $𝒞$ be an abelian category.

a.: A chain complex, or more simply complex, in $𝒞$ is a sequence $A_{\cdot} = {(A_{i}, d_{i}^{A})}_{i \in ℤ}$ in $𝒞$ such that $d_{i}^{A} \circ d_{i + 1}^{A} = 0$ for all $i \in ℤ$ .
b.: For a chain complex $A_{\cdot}$ and $i \in ℤ$ , the morphism $d_{i}^{A} : A_{i} \to A_{i - 1}$ is called the $i$ th differential in the complex $A_{\cdot}$ .

Notation 12.3.2. §

Unless otherwise specified, the $i$ th object in a chain complex $A_{\cdot}$ will be denoted $A_{i}$ and the $i$ th differential by $d_{i} : A_{i} \to A_{i - 1}$ . If we have multiple complexes, we will use $d_{i}^{A}$ to specify the differential on $A_{\cdot}$ .

Remark 12.3.3. §

Unlike with sequences in general (or exact sequences in particular), if the terms and morphisms of complex are specified only for some interval of integers, then we complete it to a complex by declaring all remaining objects and morphisms to be zero.

Definition 12.3.4. §

A morphism of complexes in an abelian category is a morphism of sequences between complexes.

Definition 12.3.5. §

The category of chain complexes $𝐂𝐡 (𝒞)$ for an abelian category $𝒞$ is the category with objects the complexes ${(A_{i}, d_{i}^{A})}_{i \in ℤ}$ in $𝒞$ and morphisms the morphisms of complexes in $𝒞$ .

Remark 12.3.6. §

A sequence of complexes is exact in $𝐂𝐡 (𝒞)$ if and only if it the resulting sequence of objects in each fixed degree is exact in $𝒞$ . For instance, a sequence of complexes

0 \to A_{\cdot} \to_{}^{f_{\cdot}} B_{\cdot} \to_{}^{g_{\cdot}} C_{\cdot} \to 0

in $𝐂𝐡 (𝒞)$ is short exact if and only if each

0 \to A_{i} \to_{}^{f_{i}} B_{i} \to_{}^{g_{i}} C_{i} \to 0

is a short exact sequence.

Note that the category of chain complexes in $𝒞$ is a fully faithful subcategory of the category of sequences with defining interval $ℤ$ .

Definition 12.3.7. §

A complex is often said to be acyclic if it is an exact sequence.

The reader can easily check the following.

Proposition 12.3.8. §

Let $𝒞$ be an abelian category. Then the category $𝐂𝐡 (𝒞)$ is an abelian category as well.

Definition 12.3.9. §

The $i$ th homology of a complex $A_{\cdot}$ in an abelian category is the object

H_{i} (A) = \frac{\ker d_{i}^{A}}{im d_{i + 1}^{A}} .

Remark 12.3.10. §

A complex $A_{\cdot}$ is exact if and only if $H_{i} (A) = 0$ for all $i \in ℤ$ .

Example 12.3.11. §

The complex

\dots \to ℤ ∕ 8 ℤ \to_{}^{4} ℤ ∕ 8 ℤ \to_{}^{4} ℤ ∕ 8 ℤ \to \dots

of abelian groups has $i$ th homology group $2 ℤ ∕ 4 ℤ≅ℤ ∕ 2 ℤ$ for every $i$ .

Lemma 12.3.12. §

Any morphism $f_{\cdot} : A_{\cdot} \to B_{\cdot}$ of complexes in $𝒞$ induces natural morphisms

f_{i}^{*} : H_{i} (A) \to H_{i} (B)

for each $i \in ℤ$ . More specifically, these satisfy

{\bar{ι}}_{i}^{B} \circ f_{i}^{*} \circ {\bar{π}}_{i}^{A} = π_{i}^{B} \circ f_{i} \circ ι_{i}^{A},

where $π_{i}^{B} : B_{i} \to coker d_{i + 1}^{B}$ and ${\bar{π}}_{i}^{A} : \ker d_{i}^{A} \to H_{i} (A)$ are the canonical epimorphisms, and where $ι_{i}^{A} : \ker d_{i}^{A} \to A_{i}$ and ${\bar{ι}}_{i}^{B} : H_{i} (B) \to coker d_{i + 1}^{B}$ are the canonical monomorphisms.

Proof.

We prove this in the case that $𝒞$ is a category of $R$ -modules. If $a \in \ker d_{i}^{A}$ , then $d_{i}^{B} (f_{i} (a)) = f_{i - 1} (d_{i}^{A} (a)) = 0$ , so $f_{i} (a) \in \ker d_{i}^{B}$ . If $a = d_{i + 1}^{A} (a^{'})$ , then $f_{i} (a) = d_{i + 1}^{B} (f_{i + 1} (a^{'})) \in im d_{i + 1}^{B}$ . Thus, the composite map

\ker d_{i}^{A} \to_{}^{f_{i}} \ker d_{i}^{B} ↠ H_{i} (B)

factors through $H_{i} (A)$ , inducing the stated map $f_{i}^{*}$ . □

Theorem 12.3.13. §

Let $𝒞$ be an abelian category, and let

0 \to A_{\cdot} \to_{}^{f_{\cdot}} B_{\cdot} \to_{}^{g_{\cdot}} C_{\cdot} \to 0

be a short exact sequence in $𝐂𝐡 (𝒞)$ . There there are morphisms $δ_{i} : H_{i} (C) \to H_{i - 1} (A)$ for all $i \in ℤ$ , natural in the short exact sequence, that fit into a long exact sequence

\dots \to H_{i} (A) \to_{}^{f_{i}^{*}} H_{i} (B) \to_{}^{g_{i}^{*}} H_{i} (C) \to_{}^{δ_{i}} H_{i - 1} (A) \to \dots .

Proof.

First, the snake lemma applied to the diagram

Degreewise short exact sequences of complexes. A full diagram description follows.

Diagram description: Degreewise short exact sequences of complexes

The two displayed rows are exact, and the squares commute.

Objects, listed by row and column:

Row 1, from left to right: column 1: 0; column 2: A subscript (i); column 3: B subscript (i); column 4: C subscript (i); column 5: 0.
Row 2, from left to right: column 1: 0; column 2: A subscript (i minus 1); column 3: B subscript (i minus 1); column 4: C subscript (i minus 1); column 5: 0.

Arrows and lines:

An arrow from 0 (row 1, column 1) to A subscript (i), without a label.
An arrow from A subscript (i) to B subscript (i), labelled f subscript (i).
An arrow from A subscript (i) to A subscript (i minus 1), labelled d subscript (i) superscript (A).
An arrow from B subscript (i) to C subscript (i), labelled g subscript (i).
An arrow from B subscript (i) to B subscript (i minus 1), labelled d subscript (i) superscript (B).
An arrow from C subscript (i) to 0 (row 1, column 5), without a label.
An arrow from C subscript (i) to C subscript (i minus 1), labelled d subscript (i) superscript (C).
An arrow from 0 (row 2, column 1) to A subscript (i minus 1), without a label.
An arrow from A subscript (i minus 1) to B subscript (i minus 1), labelled f subscript (i minus 1).
An arrow from B subscript (i minus 1) to C subscript (i minus 1), labelled g subscript (i minus 1).
An arrow from C subscript (i minus 1) to 0 (row 2, column 5), without a label.

provides exact sequences

\begin{array}{rcl} 0 \to \ker d_{i}^{A} \to \ker d_{i}^{B} \to \ker d_{i}^{C} & and & coker d_{i}^{A} \to coker d_{i}^{B} \to coker d_{i}^{C} \to 0 . \end{array}

Then, we wish to apply the snake lemma to the diagram

Kernels and cokernels in the homology sequence. A full diagram description follows.

Diagram description: Kernels and cokernels in the homology sequence

The two displayed rows are exact, and the squares commute.

Objects, listed by row and column:

Row 1, from left to right: column 2: cokernel d subscript (i plus 1) superscript (A); column 3: cokernel d subscript (i plus 1) superscript (B); column 4: cokernel d subscript (i) superscript (C); column 5: 0.
Row 2, from left to right: column 1: 0; column 2: kernel d subscript (i minus 1) superscript (A); column 3: kernel d subscript (i minus 1) superscript (B); column 4: kernel d subscript (i minus 1) superscript (C).

Arrows and lines:

An arrow from cokernel d subscript (i plus 1) superscript (A) to cokernel d subscript (i plus 1) superscript (B), labelled bar of (f) subscript (i).
An arrow from cokernel d subscript (i plus 1) superscript (A) to kernel d subscript (i minus 1) superscript (A), labelled bar of (d) subscript (i) superscript (A).
An arrow from cokernel d subscript (i plus 1) superscript (B) to cokernel d subscript (i) superscript (C), labelled bar of (g) subscript (i).
An arrow from cokernel d subscript (i plus 1) superscript (B) to kernel d subscript (i minus 1) superscript (B), labelled bar of (d) subscript (i) superscript (B).
An arrow from cokernel d subscript (i) superscript (C) to 0 (row 1, column 5), without a label.
An arrow from cokernel d subscript (i) superscript (C) to kernel d subscript (i minus 1) superscript (C), labelled bar of (d) subscript (i) superscript (C).
An arrow from 0 (row 2, column 1) to kernel d subscript (i minus 1) superscript (A), without a label.
An arrow from kernel d subscript (i minus 1) superscript (A) to kernel d subscript (i minus 1) superscript (B), labelled f subscript (i minus 1).
An arrow from kernel d subscript (i minus 1) superscript (B) to kernel d subscript (i minus 1) superscript (C), labelled g subscript (i minus 1).

with exact rows (where the “bars” denote morphisms induced on quotients). By the snake lemma, we have an exact sequence

\ker {\bar{d}}_{i}^{A} \to \ker {\bar{d}}_{i}^{B} \to \ker {\bar{d}}_{i}^{C} \to_{}^{δ_{i}} coker {\bar{d}}_{i}^{A} \to coker {\bar{d}}_{i}^{B} \to coker {\bar{d}}_{i}^{C} .

Note that for $X \in {A, B, C}$ , we have

\begin{array}{rcl} \ker {\bar{d}}_{i}^{X} = \ker (X_{i} ∕ im d_{i + 1}^{X} \to X_{i - 1}) ≅ H_{i} (A) & and & coker {\bar{d}}_{i}^{X} ≅ coker (X_{i} \to \ker d_{i - 1}^{X}) ≅ H_{i - 1} (A), \end{array}

and the morphisms induced by $f_{i}$ and $g_{i}$ (resp., $f_{i - 1}$ and $g_{i - 1}$ ) on the first (resp., second) of these are just the morphisms $f_{i}^{*}$ and $g_{i}^{*}$ (resp., $f_{i - 1}^{*}$ and $g_{i - 1}^{*}$ ). Our exact sequence then becomes

H_{i} (A) \to_{}^{f_{i}^{*}} H_{i} (B) \to_{}^{g_{i}^{*}} H_{i} (C) \to_{}^{δ_{i}} H_{i - 1} (A) \to_{}^{f_{i - 1}^{*}} H_{i - 1} (B) \to_{}^{g_{i - 1}^{*}} H_{i - 1} (C) .

Taken for all $i$ , these yield the long exact sequence. □

Example 12.3.14. §

Consider the complexes $A_{\cdot}$ , $B_{\cdot}$ , and $C_{\cdot}$ of abelian groups with $A_{i} = ℤ ∕ 4 ℤ$ , $B_{i} = ℤ ∕ 8 ℤ$ , and $C_{i} = ℤ ∕ 2 ℤ$ for all $i$ and $d_{i}^{A} = d_{i}^{B} = 0$ for all $i$ and $d_{i}^{C}$ equal to multiplication by $4$ for all $i$ . Then $H_{i} (A) = ℤ ∕ 4 ℤ$ , $H_{i} (B) = 2 ℤ ∕ 4 ℤ≅ℤ ∕ 2 ℤ$ , and $H_{i} (C) = ℤ ∕ 2 ℤ$ for all $i$ . We have a short exact sequence

0 \to A_{\cdot} \to_{}^{f_{\cdot}} B_{\cdot} \to_{}^{g_{\cdot}} C_{\cdot} \to 0

for maps $f_{\cdot} : A_{\cdot} \to B_{\cdot}$ induced by multiplication by $2$ on each term and $g_{\cdot} : B_{\cdot} \to C_{\cdot}$ given by reduction modulo $2$ on each term. The resulting long exact sequence has the form

\dots \to_{}^{δ} ℤ ∕ 4 ℤ ↠ ℤ ∕ 2 ℤ \to_{}^{0} ℤ ∕ 2 ℤ \overset{δ}{↪} ℤ ∕ 4 ℤ ↠ ℤ ∕ 2 ℤ \to_{}^{0} ℤ ∕ 2 ℤ \to_{}^{δ} \dots .

We briefly describe cochain complexes and cohomology, which simply amount to a change of indexing from decreasing to increasing.

Definition 12.3.15. §

A cochain complex is a collection $A^{\cdot} = {(A^{i}, d_{A}^{i})}_{i \in ℤ}$ of objects $A^{i}$ and morphisms $d_{A}^{i} : A^{i} \to A^{i + 1}$ such that $d_{A}^{i} \circ d_{A}^{i - 1} = 0$ for all $i \in ℤ$ . The morphism $d_{A}^{i} : A^{i} \to A^{i + 1}$ is called the $i$ th differential of the cochain complex $A^{\cdot}$ .

We then have the notion of cohomology.

Definition 12.3.16. §

The $i$ th cohomology of a cochain complex $A^{\cdot}$ in an abelian category is the object

H^{i} (A) = \frac{\ker d_{A}^{i}}{im d_{A}^{i - 1}} .

Remark 12.3.17. §

Much as with complexes, we can speak of morphisms of cochain complexes $f^{\cdot} : A^{\cdot} \to B^{\cdot}$ , which are collections of morphisms $f^{i} : A^{i} \to B^{i}$ such that $d_{B}^{i} \circ f^{i} = f^{i + 1} \circ d_{A}^{i}$ for all $i \in ℤ$ . Again, short exact sequences

0 \to A^{\cdot} \to_{}^{f^{\cdot}} B^{\cdot} \to_{}^{g^{\cdot}} C^{\cdot} \to 0

of cochain complexes give rise to long exact sequences in cohomology, but now of the form

\dots \to H^{i} (A) \to_{}^{f_{*}^{i}} H^{i} (B) \to_{}^{g_{*}^{i}} H^{i} (C) \to_{}^{δ^{i}} H^{i + 1} (A) \to \dots

Definition 12.3.18. §

Let $A_{\cdot} = (A_{i}, d_{i}^{A})$ and $B_{\cdot} = (B_{i}, d_{i}^{B})$ be chain complexes. Let $f_{\cdot}, g_{\cdot} : A_{\cdot} \to B_{\cdot}$ be morphisms of chain complexes.

a.

A chain homotopy from $f_{\cdot}$ to $g_{\cdot}$ is a sequence $s_{\cdot} = {(s_{i})}_{i \in ℤ}$ of morphisms $s_{i} : A_{i} \to B_{i + 1}$ satisfying

f_{i} - g_{i} = d_{i + 1}^{B} \circ s_{i} + s_{i - 1} \circ d_{i}^{A}

for all $i \in ℤ$ .

b.

We say that $f_{\cdot}$ and $g_{\cdot}$ are chain homotopic, and write $f_{\cdot} \sim g_{\cdot}$ , if there exists a homotopy from $f_{\cdot}$ to $g_{\cdot}$ .

c.

If $f_{\cdot}$ is (chain) homotopic to $0$ , then $f_{\cdot}$ is said to be null-homotopic.

Remark 12.3.19. §

For cochain complexes $A^{\cdot}$ and $B^{\cdot}$ and morphisms $f^{\cdot}, g^{\cdot} : A^{\cdot} \to B^{\cdot}$ , a chain homotopy from $f^{\cdot}$ to $g^{\cdot}$ is a sequence $s^{\cdot} = {(s^{i})}_{i \in ℤ}$ of morphisms $s^{i} : A^{i} \to B^{i - 1}$ such that $f^{i} - g^{i} = d_{B}^{i - 1} \circ s^{i} + s^{i + 1} \circ d_{A}^{i}$ .

The morphisms $s_{\cdot}$ defining a null-homotopy fit into a diagram

A null-homotopy. A full diagram description follows.

Diagram description: A null-homotopy

The diagonal maps are the components of a null-homotopy. This diagram is not generally commutative; the homotopy identity in the surrounding text is a sum of the two routes through a diagonal map.

Objects, listed by row and column:

Row 1, from left to right: column 1: ellipsis; column 2: A subscript (i plus 1); column 3: A subscript (i); column 4: A subscript (i minus 1); column 5: ellipsis.
Row 2, from left to right: column 1: ellipsis; column 2: B subscript (i plus 1); column 3: B subscript (i); column 4: B subscript (i minus 1); column 5: ellipsis.

Arrows and lines:

An arrow from ellipsis (row 1, column 1) to A subscript (i plus 1), without a label.
An arrow from A subscript (i plus 1) to A subscript (i), labelled d subscript (i plus 1) superscript (A).
An arrow from A subscript (i plus 1) to B subscript (i plus 1), labelled f subscript (i plus 1).
An arrow from A subscript (i) to A subscript (i minus 1), labelled d subscript (i) superscript (A).
An arrow from A subscript (i) to B subscript (i), labelled f superscript (i).
An arrow from A subscript (i) to B subscript (i plus 1), labelled s subscript (i).
An arrow from A subscript (i minus 1) to ellipsis (row 1, column 5), without a label.
An arrow from A subscript (i minus 1) to B subscript (i minus 1), labelled f subscript (i minus 1).
An arrow from A subscript (i minus 1) to B subscript (i), labelled s subscript (i minus 1).
An arrow from ellipsis (row 2, column 1) to B subscript (i plus 1), without a label.
An arrow from B subscript (i plus 1) to B subscript (i), labelled d subscript (i plus 1) superscript (B).
An arrow from B subscript (i) to B subscript (i minus 1), labelled d subscript (i) superscript (B).
An arrow from B subscript (i minus 1) to ellipsis (row 2, column 5), without a label.

Proposition 12.3.20. §

Assume that $f_{\cdot}$ and $g_{\cdot}$ are chain homotopic morphisms $A_{\cdot} \to B_{\cdot}$ . Then the morphisms $f_{i}^{*}$ and $g_{i}^{*}$ on homology are equal for all $i \in ℤ$ .

Definition 12.3.21. §

A morphism of complexes $f_{\cdot} : A_{\cdot} \to B_{\cdot}$ is a homotopy equivalence if there exists a morphism $g_{\cdot} : B_{\cdot} \to A_{\cdot}$ such that $g_{\cdot} \circ f_{\cdot} \sim {id}_{A_{\cdot}}$ and $f_{\cdot} \circ g_{\cdot} \sim {id}_{B_{\cdot}}$ .

12.4. Projective and injective objects

We continue to work in an abelian category $𝒞$ .

Definition 12.4.1. §

a.

We say that an epimorphism $g : B \to C$ is split if there exists a morphism $t : C \to B$ with $g \circ t = {id}_{C}$ . In this case, we say that $t$ is a splitting of $g$ .

b.

We say that a monomorphism $f : A \to B$ is split if there exists a morphism $s : B \to A$ with $s \circ f = {id}_{A}$ . In this case, we say that $s$ is a splitting of $f$ .

c.

We say that a short exact sequence

0 \to A \to_{}^{f} B \to_{}^{g} C \to 0

(12.4.1)

splits if there exists an isomorphism $w : A \oplus C \to_{}^{\sim} B$ with $w (a, 0) = f (a)$ and $g (w (0, c)) = c$ for all $a \in A$ and $c \in C$ .

Example 12.4.2. §

The exact sequence of abelian groups

0 \to ℤ ∕ 3 ℤ \to_{}^{\cdot 2} ℤ ∕ 6 ℤ \to_{}^{mod 2} ℤ ∕ 2 ℤ \to 0

is split, but

0 \to ℤ ∕ 2 ℤ \to_{}^{\cdot 2} ℤ ∕ 4 ℤ \to_{}^{mod 2} ℤ ∕ 2 ℤ \to 0

is not.

Proposition 12.4.3. §

The following conditions on a short exact sequence

0 \to A \to_{}^{f} B \to_{}^{g} C \to 0

are equivalent:

The sequence splits.
The monomorphism $f : A \to B$ splits.
The epimorphism $g : B \to C$ splits.

Proof.

We prove this in the category of $R$ -modules.

Suppose we have a splitting map $t : C \to B$ . Then define $s : B \to A$ by $s (b) = a$ where $f (a) = b - t (g (b))$ . This is well-defined as $f$ is injective, and such an $a$ exists since
$g (b - t (g (b))) = g (b) - g (t (g (b))) = g (b) - g (b) = 0 .$
It splits $f$ as
$s (f (a)) = s (b) - s (t (g (b))) = s (b),$
the latter step using the fact that $s \circ t = 0$ , which follows in turn from
$f (s (t (c))) = t (c) - t (g (t (c))) = t (c) - t (c) = 0 .$
Suppose that we have a splitting map $s : B \to A$ . Then define $t : C \to B$ by $t (c) = b - f (s (b))$ where $g (b) = c$ . To see this is well defined, note that $f (a) - f (s (f (a))) = 0$ for any $a \in A$ .
Define $w (a, c) = f (a) + t (c)$ . Its inverse is $w^{'} (b) = (s (b), g (b))$ . To see this, we check that
$w^{'} \circ w (a, c) = (s (f (a) + t (c)), g (f (a) + t (c))) = (s (f (a)), g (t (c))) = (a, c)$
for $a \in A$ and $c \in C$ , and note that
$w \circ w^{'} (b) = w (s (b), g (b)) = f (s (b)) + t (g (b)) .$
for $b \in B$ . Set $c = g (b)$ , so that $g (b - t (c)) = 0$ , and let $a \in A$ be such that $f (a) = b - t (c)$ . Then
$(f \circ s + t \circ g) (b) = f (s (f (a))) + f (s (t (c))) + t (g (f (a))) + t (g (t (c))) = f (a) + t (c) = b .$
Set $t (c) = w (0, c)$ . Then $g (t (c)) = g (w (0, c)) = c$ .

□

Definition 12.4.4. §

An object $P$ of an abelian category is projective if for epimorphism $π : B \to C$ and every morphism $g : P \to C$ , there exists an morphism $f : P \to B$ such that $g = π \circ f$ .

Remark 12.4.5. §

The property of $P$ being projective is represented by the existence of $g$ in the commutative diagram

The lifting property of a projective object. A full diagram description follows.

Diagram description: The lifting property of a projective object

The structural squares and triangles displayed here commute. The lower row is exact, and the dashed arrow f is the requested lift.

Objects, listed by row and column:

Row 1, from left to right: column 2: P.
Row 2, from left to right: column 1: B; column 2: C; column 3: 0.

Arrows and lines:

An arrow from P to C, labelled g.
A dashed arrow from P to B, labelled f.
An arrow from B to C, labelled pi.
An arrow from C to 0, without a label.

with exact lower row.

Proposition 12.4.6. §

Free $R$ -modules are projective.

Proof.

Let $F$ be a free $R$ -module with basis $X$ . Let $π : B \to C$ be a surjective $R$ -module homomorphism, and suppose that $g : F \to C$ is an $R$ -module homomorphism. For each $x \in X$ , let $b$ be an element of $B$ such that $π (b) = g (x)$ . Since $F$ is free, we may defined $f : F \to B$ by $f (x) = b$ for each $x \in X$ . Then $π (f (x)) = g (x)$ for all $x \in X$ , so $π \circ f = g$ as $X$ generates $F$ . □

Example 12.4.7. §

Not every projective module need be free. For example, consider $R = ℤ ∕ 6 ℤ$ . We claim that $P = ℤ ∕ 3 ℤ$ is a projective $R$ -module. To see this, suppose that $B$ is a $ℤ ∕ 6 ℤ$ -module and $g : B \to ℤ ∕ 3 ℤ$ is surjective. Take any $b \in B$ with $g (b) = 1$ . Then the $ℤ ∕ 6 ℤ$ submodule generated by $b$ is isomorphic to $ℤ ∕ 3 ℤ$ , and hence $1 \mapsto b$ defines a splitting of $g$ .

Note that $P$ is not projective as a $ℤ$ -module (abelian group) since the quotient map $ℤ \to ℤ ∕ 3 ℤ$ does not split. In fact, every projective $ℤ$ -module is free.

We describe some equivalent conditions for projectivity.

Proposition 12.4.8. §

The following conditions on an $R$ -module $P$ are equivalent:

i.: $P$ is projective,
ii.: every surjection $π : M \to P$ of $R$ -modules is split, and
iii.: $P$ is a direct summand of a free $R$ -module.

Proof.

If $P$ is projective and $π : M \to P$ is a surjection, then the identity map ${id}_{P} : P \to P$ lifts to a homomorphism $f : P \to M$ such that ${id}_{P} = π \circ f$ , so (i) implies (ii). If (ii) holds, then choose a set of generators $X$ of $P$ , and let $F$ be the free $R$ -module on $X$ , which comes equipped with a surjection $π : F \to P$ that restricts to ${id}_{X}$ . This surjection is split, so $P$ is a direct summand of $F$ by Proposition 12.4.3, and therefore (iii) holds.

If $P$ is a direct summand of a free module $F$ with complement $Q$ , then $π : B \to C$ is a surjection, and $g : P \to C$ is a homomorphism of $R$ -modules, then we can extend $g$ to $\tilde{g} : F \to C$ by setting $\tilde{g} (q) = 0$ for all $q \in Q$ . We then have $\tilde{f} : F \to B$ such that $π \circ \tilde{f} = \tilde{g}$ by the projectivity of $F$ , and the restriction $f = \tilde{f} |_{P}$ satisfies $π \circ f = g$ . Thus, (iii) implies (i). □

Example 12.4.9. §

For $n \geq 1$ , the left $M_{n} (R)$ -module $L$ of column vectors under left multiplication is a direct summand of $M_{n} (R)$ , which is isomorphic to $L^{n}$ as a left $R$ -module. Hence, $L$ is projective, though it is not free for $n \geq 2$ .

In the case that $R$ is a principal ideal domain, we have the following.

Corollary 12.4.10. §

If $R$ is a principal ideal domain, then every projective $R$ -module is free.

Proof.

By the classification of finitely generated modules over a principal ideal domain, it suffices for finitely generated $R$ -modules to show that any $R$ -module of the form

A = R ∕ (a_{1}) \oplus R ∕ (a_{2}) \oplus \dots R ∕ (a_{n})

for nonzero and nonunit $a_{1}, a_{2}, \dots, a_{n} \in R$ is not projective. Consider the obvious quotient map $R^{n} \to A$ . That it splits means that each $R \to R ∕ (a_{i})$ splits. Then $𝑅≅ (a_{i}) \oplus (x)$ for some $x \in R$ , which means that $R$ is free of rank $2$ over itself, which is impossible (e.g., by the classification theorem).

The general case is left as an exercise. □

Definition 12.4.11. §

An object $I$ in an abelian category is injective if for every monomorphism $ι : A \to B$ and every morphism $f : A \to I$ , there exists a morphism $g : B \to I$ such that $f = g \circ ι$ .

Remark 12.4.12. §

The property of $I$ being injective is represented by the existence of $g$ in the commutative diagram

The extension property of an injective object. A full diagram description follows.

Diagram description: The extension property of an injective object

The structural squares and triangles displayed here commute. The upper row is exact, and the dashed arrow g is the requested extension.

Objects, listed by row and column:

Row 1, from left to right: column 1: 0; column 2: A; column 3: B.
Row 2, from left to right: column 2: I.

Arrows and lines:

An arrow from 0 to A, without a label.
An arrow from A to I, labelled f.
An arrow from A to B, labelled iota.
A dashed arrow from B to I, labelled g.

with exact upper row.

Dually to the analogous result projective modules, we have the following.

Lemma 12.4.13. §

As $R$ -module $I$ is injective if and only if every monomorphism $ι : I \to R$ is split.

We also have the following interesting criterion, which employs Zorn’s lemma.

Proposition 12.4.14 (Baer’s criterion). §

A left $R$ -module $I$ is injective if and only if every homomorphism $J \to I$ with $J$ a left ideal of $R$ may be extended to a map $R \to I$ .

Proof.

Let $A$ be an $R$ -submodule of an $R$ -module $B$ and $f : A \to I$ be an $R$ -module homomorphism. It suffices to show that we can extend $f$ to $g : B \to I$ with $g |_{A} = f$ . Let $X$ be the set of pairs $(C, h)$ with $C$ an $R$ -submodule of $B$ containing $A$ and $h : C \to I$ an $R$ -module homomorphism. We have a partial ordering on $X$ given by $(C, h) < (C^{'}, h^{'})$ if $C$ is contained in $C^{'}$ and $h^{'} |_{C} = h$ . Given a chain $𝒞$ in $X$ , we have an upper bound $(C, h)$ with $C = ⋃_{(D, k) \in 𝒞} D$ such that if $d \in D$ for $(D, k) \in 𝒞$ , then $h (d) = k (d)$ . By Zorn’s lemma, $𝒞$ has a maximal element $(M, l)$ .

Suppose first that $M \neq B$ , and let $b \in B - M$ . Consider the left ideal

J = {r \in R ∣ 𝑟𝑏 \in M}

of $R$ . Define $s : J \to I$ by $s (r) = l (𝑟𝑏)$ for $r \in J$ . This $R$ -module homomorphism may be extended to $t : R \to I$ by assumption. Then define $N = M + 𝑅𝑏$ and let $q : N \to I$ be the unique $R$ -module homomorphism such that $q |_{M} = l |_{M}$ and $q (𝑟𝑏) = t (r)$ for all $r \in R$ . This exists as $M \cap 𝑅𝑏 = 𝐽𝑏$ , and $l (𝑟𝑏) = s (r) = t (r)$ for $r \in J$ . (Also, if $𝑟𝑏 = 0$ , then $r \in J$ , so $q (𝑟𝑏) = l (𝑟𝑏) = 0$ in this instance.) The existence of $q$ gives a contradiction of the maximality of $M$ . Thus $M = B$ , and we are done. □

Example 12.4.15. §

a.: $ℚ$ is an injective $ℤ$ -module.
b.: $ℤ ∕ 𝑛ℤ$ is an injective $ℤ ∕ 𝑛ℤ$ -module for any $n \geq 1$ .
c.: $ℤ ∕ 3 ℤ$ is an injective $ℤ ∕ 6 ℤ$ -module, but not an injective $ℤ ∕ 9 ℤ$ -module.

We have a very nice description of injective objects in $𝐀𝐛$ .

Definition 12.4.16. §

An abelian group $D$ is called divisible if multiplication by $n$ is surjective on $D$ for every natural number $n$ .

Proposition 12.4.17. §

An abelian group is injective if and only if it is divisible.

Proof.

Let $D$ be injective, and take $d \in D$ . Then there exists a group homomorphism $ϕ : ℤ \to D$ with $1 \mapsto d$ . We also have the multiplication-by- $n$ map on $ℤ$ , which is injective. By injectivity of $D$ , we have a map $𝜃 : ℤ \to D$ with $ϕ = 𝑛𝜃$ . Then $d = 𝑛𝜃 (1)$ , so $D$ is divisible.

Conversely, let $D$ be divisible. By Baer’s criterion, it suffices to show that every homomorphism $ϕ : 𝑛ℤ \to D$ with $n \geq 1$ extends to a homomorphism $𝜃 : ℤ \to D$ . Such a $ϕ$ is determined by $d = ϕ (n)$ . Let $d^{'} \in D$ be such that $n d^{'} = d$ . Set $𝜃 (1) = d^{'}$ . □

12.5. Exact functors

Despite the fact that additive functors preserve direct sums, they may not preserve exact sequences. We make the following definitions.

Definition 12.5.1. §

Let $F : 𝒞 \to 𝒟$ be an additive functor of abelian categories.

a.: We say that $F$ is left exact if for every left short exact sequence $0 \to A \to B \to C$ in $𝒞$ , the sequence $0 \to F (A) \to F (B) \to F (C)$ is exact in $𝒟$ .
b.: We say that $F$ is right exact if for every right short exact sequence $A \to B \to C \to 0$ in $𝒞$ , the sequence $F (A) \to F (B) \to F (C) \to 0$ is exact in $𝒟$ .
c.: We say that $F$ is an exact functor if for every short exact sequence $0 \to A \to B \to C \to 0$ in $𝒞$ , the sequence $0 \to F (A) \to F (B) \to F (C) \to 0$ is exact in $𝒟$ .

Remark 12.5.2. §

A contravariant additive functor $F : 𝒞 \to 𝒟$ is left exact if the resulting covariant functor $𝒞^{op} \to 𝒟$ is left exact.

Example 12.5.3. §

The functor $F : 𝐀𝐛 \to 𝐀𝐛$ by $F (A) = A \oplus A$ with $F (f) = f \oplus f$ is exact.

Terminology 12.5.4. §

We (somewhat loosely) say a functor a certain structure if its takes structures of one sort in a given category (induced from the source category) to those of the same sort in another (induced from the target category). For instance, exact functors are additive functors that preserve short exact sequences.

Lemma 12.5.5. §

Let $F : 𝒞 \to 𝒟$ be an additive functor of abelian categories. The following are equivalent:

i.: $F$ is exact,
ii.: $F$ is both left and right exact,
iii.: $F$ preserves all three-term exact sequences $A \to B \to C$ , and
iv.: $F$ preserves all exact sequences.

Proof.

It is immediate that (iv) implies the other statements and also that (ii) implies (i).

Suppose that $F$ that preserves all three-term exact sequences. To say that

0 \to A \to B \to C \to 0

is short exact is equivalent to saying that the three three-term sequences $0 \to A \to B$ , $A \to B \to C$ , and $B \to C \to 0$ are all exact. Since exactness of these is preserved by $F$ , so is exactness of the original short exact sequence. So, (iii) implies (i).

If $F$ is an exact functor, take any exact sequence $A_{\cdot}$ . Then $0 \to \ker d_{i} \to A_{i} \to im d_{i} \to 0$ is short exact, so

0 \to F (\ker d_{i}) \to F (A_{i}) \to_{}^{F (d_{i})} F (im d_{i}) \to 0

is exact as well. It $F (d_{i})$ follows that image $F (im d_{i})$ and kernel $F (\ker d_{i})$ for all $i$ , so we have

im F (d_{i}) = F (im d_{i}) = F (\ker d_{i - 1}) = \ker F (d_{i - 1}) .

Thus, $F (A_{\cdot})$ is exact. Thus, (i) implies (iv), which finishes the proof. □

Remark 12.5.6. §

The reader may also check that an additive functor of abelian categories is left (resp., right) exact if and only if it sends short exact sequences to left (resp., right) short exact sequences.

Recall that for an additive category $𝒞$ , the functors $h^{X}$ (and $h_{X}$ ) may be viewed as taking values in $𝐀𝐛$ , and clearly such functors are additive. In fact, they are also left exact.

Lemma 12.5.7. §

Let $𝒞$ be an abelian category, and let $X$ be an object of $𝒞$ .

a.: The functor $h_{X} : 𝒞 \to 𝐀𝐛$ is left exact.
b.: The functor $h^{X} : 𝒞^{op} \to 𝐀𝐛$ is left exact.

Proof.

Let

0 \to A \to_{}^{f} B \to_{}^{g} C \to 0

be an exact sequence in $𝒞$ . Applying $h_{X}$ , we obtain homomorphisms

0 \to {Hom}_{𝒞} (X, A) \to_{}^{h_{X} (f)} {Hom}_{𝒞} (X, B) \to_{}^{h_{X} (g)} {Hom}_{𝒞} (X, C)

of abelian groups, and we claim this sequence is exact. If $h_{X} (f) (α) = 0$ , then $f \circ α = 0$ , but $f$ is a monomorphism, so $α = 0$ . Since $h_{X}$ is a functor, we have $h_{X} (g) \circ h_{X} (f) = 0$ , and if $β \in \ker h_{X} (g)$ , then $g \circ β = 0$ . Naturality of the kernel implies that $β$ factors through a morphism $X \to \ker g$ . But we have canonical isomorphisms

A \to_{}^{\sim} coim f \to_{}^{\sim} im f \to_{}^{\sim} \ker g,

the first as $f$ is a monomorphism, and the composite of the composite of these with the canonical morphism $\ker g \to B$ is $g$ . Therefore, we obtain a morphism $α : X \to A$ satisfying $f \circ α = g$ . This proves part a, and part b is just part a with $𝒞$ replaced by $𝒞^{op}$ . □

Lemma 12.5.8. §

Let $R$ be a ring, and let $N$ be a right $R$ -module. The tensor product functor $t_{N} : R -mod \to 𝐀𝐛$ given on objects by $t_{N} (M) = N \otimes_{R} M$ and on morphisms by $t_{N} (g) = {id}_{N} \otimes g$ is right exact.

Proof.

Since tensor products commute with direct sums, $t_{N}$ is additive. Let

A \to_{}^{f} B \to_{}^{g} C \to 0

be a right short exact sequence of $R$ -modules. The group $N \otimes_{R} C$ is generated by simple tensors $n \otimes c$ with $n \in N$ and $c \in C$ and

n \otimes c = n \otimes g (b) = ({id}_{N} \otimes g) (n \otimes b)

for any $b \in B$ with $g (b) = c$ , we have that $t_{N} (g)$ is surjective. We need then only define an inverse to the surjection $\bar{g} : coker t_{N} (f) \to N \otimes_{R} C$ . For this, we consider the map $𝜃 : N \times C \to coker t_{N} (f)$ given on $(n, c) \in N \times C$ by picking $b \in B$ with $g (b) = c$ and then setting $𝜃 (n, c) = n \otimes b + im t_{N} (f)$ . If $g (b^{'}) = c$ , then $g (b - b^{'}) = 0$ , so $b - b^{'} = f (a)$ for some $a \in A$ , and then $n \otimes (b - b^{'}) = t_{N} (f) (n \otimes a)$ , so $𝜃$ is well-defined and then easily seen to be biadditive and $R$ -balanced. The induced map $Θ : N \otimes_{R} C \to coker t_{N} (f)$ is inverse to $\bar{g}$ by definition. □

Lemma 12.5.9. §

Let $𝒞$ be an abelian category. A sequence

0 \to A \to_{}^{f} B \to_{}^{g} C

is exact if every sequence

0 \to {Hom}_{𝒞} (X, A) \to_{}^{h_{X} (f)} {Hom}_{𝒞} (X, B) \to_{}^{h_{X} (g)} {Hom}_{𝒞} (X, C)

is exact.

Proof.

For $X = A$ , we get

g \circ f = h_{X} (g) \circ h_{X} (f) ({id}_{A}) = 0,

so we have a monomorphism $s : im f \to \ker g$ . For $X = \ker g$ and $β : \ker g \to B$ the natural monomorphism defined by the kernel, we have $h^{X} (g) (β) = g \circ β = 0$ , so there exists $α : \ker g \to A$ with $f \circ α = β$ . We then have that $β$ factors a morphism $t : \ker g \to im f$ inverse to $s$ . □

Proposition 12.5.10. §

Any right (resp., left) adjoint to ia functor between abelian categories is left (resp., right) exact.

Proof.

We treat the case of left exactness, the other case simply being the corresponding statement in opposite categories. Let $G : 𝒞 \to 𝒟$ be an additive functor of abelian categories that admits a left adjoint $F$ . Suppose that

0 \to A \to_{}^{f} B \to_{}^{g} C

is a left exact sequence in $𝒞$ . Then for any $D \in Obj (𝒟)$ , the sequence

0 \to h_{F (D)} (A) \to_{}^{h_{F (D)} (f)} h_{F (D)} (B) \to_{}^{h_{F (D)} (g)} h_{F (D)} (C)

is left exact. Since $F$ is left adjoint to $G$ , this sequence is isomorphic to

0 \to h_{D} (G (A)) \to_{}^{h_{D} (G (f))} h_{D} (G (B)) \to_{}^{h_{D} (G (g))} h_{D} (G (C))

as a sequence of abelian groups. Since this holds for all $D$ , the sequence

0 \to G (A) \to_{}^{G (f)} G (B) \to_{}^{G (g)} G (C)

is exact. □

Proposition 12.5.11. §

Let $R$ be a ring, and fix an $R$ -module $M$ .

a.: The covariant homomorphism functor $h_{M} : R -mod \to 𝐀𝐛$ is exact if and only if $M$ is $R$ -projective.
b.: The contravariant homomorphism functor $h^{M} : R -mod \to 𝐀𝐛$ is exact if and only if $M$ is $R$ -injective.

Proof.

We prove part a. Suppose that the functor is exact. Then for any epimorphism $g : B \to P$ we have an epimorphism

{Hom}_{R} (P, B) \to {Hom}_{R} (P, P),

and any inverse image $t$ of ${id}_{P}$ is the desired splitting map of $g$ .

On the other hand, suppose that $P$ is projective. Consider an exact sequence

0 \to A \to_{}^{f} B \to_{}^{g} C \to 0 .

Then we have a diagram

{Hom}_{R} (P, A) \to_{}^{h_{P} (f)} {Hom}_{R} (P, B) \to_{}^{h_{P} (g)} {Hom}_{R} (P, C) \to 0 .

That this is a complex is immediate. Surjectivity of $h_{P} (g)$ follows immediately from the definition of a projective module. Finally, let $h \in \ker h_{P} (g)$ , so $h : P \to \ker g$ . Then $A \to \ker g$ is an epimorphism, and we have by projectivity of $P$ a map $j : P \to A$ with with $f \circ j = h$ , i.e., $h_{P} (f) (j) = h$ . □

Remark 12.5.12. §

If $R$ is a commutative ring, then ${Hom}_{R} (A, B)$ for $R$ -modules $A$ and $B$ may be viewed as an $R$ -module under $(r \cdot f) (a) = r \cdot f (a)$ . It follows easily that ${Hom}_{R} (A, \cdot)$ is an additive functor from the category of $R$ -modules to itself which is exact if $A$ is projective.

The following embedding theorem, the proof of which is beyond the scope of these notes, allows us to do most of the homological algebra that can be done in the category of $R$ -modules for any $R$ in an arbitrary abelian category.

Theorem 12.5.13 (Freyd-Mitchell). §

If $𝒞$ is a small abelian category, then there exists a ring $R$ and an exact, fully faithful functor $𝒞 \to R$ -mod.

In other words, $𝒞$ is equivalent to a full, abelian subcategory of $R -mod$ for some ring $R$ . We can use this as follows: suppose there is a result we can prove about exact diagrams in $R$ -modules for all $R$ , like the snake lemma. We then have the result in all abelian categories, since we can take a small full, abelian subcategory containing the objects in which we are interested and embed it into some category of left $R$ -modules. If the result holds in that category, then by exactness of the embedding, the result will hold in the original category.

12.6. Projective and injective resolutions

Definition 12.6.1. §

Let $𝒞$ be an abelian category, and let $A$ be an object in $𝒞$ .

a.

A resolution of $A$ is a complex $C_{\cdot}$ of objects in $𝒞$ together with an augmentation morphism $𝜖^{C} : C_{0} \to A$ such that the augmented complex

\dots \to C_{1} \to_{}^{d_{1}^{C}} C_{0} \to_{}^{𝜖^{C}} A \to 0

is exact.

b.

A projective resolution of $A$ is a resolution of $A$ by a complex of projective objects.

Definition 12.6.2. §

An abelian category $𝒞$ is said to have sufficiently many (or enough) projectives if for every $A \in Obj (𝒞)$ , there exists a projective object $P \in Obj (𝒞)$ and an epimorphism $P \to A$ .

Since free modules are projective, $R -mod$ has enough projectives.

Remark 12.6.3. §

If $𝒞$ has enough projectives, then every object in $𝒞$ has a projective resolution. (We leave the proof as an exercise.)

Examples 12.6.4. §

We have the following examples of projective resolutions, all of which are in fact resolutions by free modules:

a.

In $𝐀𝐛$ , the abelian group $ℤ ∕ 𝑛ℤ$ has a projective resolution

0 \to ℤ \to_{}^{\cdot n} ℤ \to ℤ ∕ 𝑛ℤ \to 0 .

b.

Consider $R = ℤ [X]$ , and let $A = ℤ [X] ∕ (n, X^{2} + 1)$ . Then we have a projective resolution

0 \to ℤ [X] \to_{}^{(\cdot (X^{2} + 1), \cdot n)} ℤ [X] \oplus ℤ [X] \to_{}^{(\cdot n) - (\cdot (X^{2} + 1))} ℤ [X] \to ℤ [X] ∕ (n, X^{2} + 1) \to 0 .

c.

Consider the ring $R = ℤ [X] ∕ (X^{n} - 1)$ . (This is isomorphic to the group ring $ℤ [ℤ ∕ 𝑛ℤ]$ .) We have a projective resolution of $ℤ$ :

\dots \to R \to_{}^{N} R \to_{}^{X - 1} R \to_{}^{N} R \to_{}^{X - 1} R \to ℤ \to 0,

where $N = \sum_{i = 0}^{n - 1} X^{i}$ .

Proposition 12.6.5. §

Let $P_{\cdot} \to A$ and $Q_{\cdot} \to B$ be projective resolutions in an abelian category, and suppose that $g : A \to B$ is an $R$ -module homomorphism. Then $g$ extends to a morphism $f_{\cdot} : P_{\cdot} \to Q_{\cdot}$ of chain complexes such that

Lifting a map to projective resolutions. A full diagram description follows.

Diagram description: Lifting a map to projective resolutions

The structural squares and triangles displayed here commute. Both horizontal sequences are the displayed augmented projective resolutions.

Objects, listed by row and column:

Row 1, from left to right: column 1: ellipsis; column 2: P subscript (2); column 3: P subscript (1); column 4: P subscript (0); column 5: A; column 6: 0.
Row 2, from left to right: column 1: ellipsis; column 2: Q subscript (2); column 3: Q subscript (1); column 4: Q subscript (0); column 5: B; column 6: 0.

Arrows and lines:

An arrow from ellipsis (row 1, column 1) to P subscript (2), without a label.
An arrow from P subscript (2) to P subscript (1), without a label.
An arrow from P subscript (2) to Q subscript (2), labelled f subscript (2).
An arrow from P subscript (1) to P subscript (0), without a label.
An arrow from P subscript (1) to Q subscript (1), labelled f subscript (1).
An arrow from P subscript (0) to A, without a label.
An arrow from P subscript (0) to Q subscript (0), labelled f subscript (0).
An arrow from A to B, labelled g.
An arrow from A to 0 (row 1, column 6), without a label.
An arrow from ellipsis (row 2, column 1) to Q subscript (2), without a label.
An arrow from Q subscript (2) to Q subscript (1), without a label.
An arrow from Q subscript (1) to Q subscript (0), without a label.
An arrow from Q subscript (0) to B, without a label.
An arrow from B to 0 (row 2, column 6), without a label.

commutes. Furthermore, any other lift of $g$ is chain homotopic to $f_{\cdot}$ .

Proof.

Let $P = (P_{i}, d_{i})$ and $Q = (Q_{i}, d_{i}^{'})$ , and let $𝜖$ and $𝜖^{'}$ denote the respective augmentation maps. Then $g \circ 𝜖 : P_{0} \to B$ . Since $𝜖^{'}$ is an epimorphism, we have a map $f_{0} : P_{0} \to Q_{0}$ lifting $g \circ 𝜖$ . Now $f_{0}$ induces a map

{\bar{f}}_{0} : \ker 𝜖 \to \ker 𝜖^{'},

and since $im d_{1} = \ker 𝜖$ and $\ker d_{1}^{'} = \ker 𝜖^{'}$ , we have an epimorphism $Q_{1} \to \ker d_{1}^{'}$ , and we again use projectivity, this time of $Q_{1}$ , to lift ${\bar{f}}_{0} \circ d_{1}$ to a map $f_{1}$ as in the diagram. We continue in this manner to obtain $f_{\cdot}$ .

Now, for uniqueness up to chain homotopy, it suffices to show that if $g = 0$ , then $f_{\cdot}$ is chain homotopic to zero. Well, $d_{0}^{'} \circ f_{0} = g \circ d_{0} = 0$ , so $f_{0} (P_{0}) \subseteq im d_{1}^{'}$ . By projectivity of $P_{0}$ , we have $s_{0} : P_{0} \to Q_{1}$ with

f_{0} = d_{1}^{'} \circ s_{0} + s_{- 1} \circ d_{0} = d_{1}^{'} \circ s_{0},

where we have set $s_{i} = 0$ for $i < 0$ (and $d_{i} = 0$ for $i \leq 0$ ). Now $h_{1} = f_{1} - s_{0} \circ d_{1}$ satisfies

d_{1}^{'} \circ h_{1} = d_{1}^{'} \circ f_{1} - d_{1}^{'} \circ s_{0} \circ d_{1} = f_{0} \circ d_{1} - f_{0} \circ d_{1} = 0,

so $im h_{1} \subseteq im d_{2}^{'}$ . Thus, we have $s_{1} : P_{1} \to Q_{2}$ lifting $h_{1}$ , i.e., so that

d_{2}^{'} \circ s_{1} = h_{1} = f_{1} - s_{0} \circ d_{1},

as desired. We continue in this fashion to obtain all $s_{i}$ . □

Proposition 12.6.6 (Horseshoe lemma). §

Suppose that

0 \to A \to_{}^{f} B \to_{}^{g} C \to 0

is a short exact sequence in an abelian category and that $(P_{\cdot}^{A}, 𝜖^{A})$ and $(P_{\cdot}^{C}, 𝜖^{C})$ are projective resolutions of $A$ and $C$ respectively. Then there exists a projective resolution $(P_{\cdot}^{B}, 𝜖^{B})$ of $B$ with $P_{i}^{B} = P_{i}^{A} \oplus P_{i}^{C}$ for each $i$ and such that the diagram

The horseshoe lemma. A full diagram description follows.

Diagram description: The horseshoe lemma

The structural squares and triangles displayed here commute. The rows are the short exact sequences in the horseshoe construction, with the indicated augmentations downward.

Objects, listed by row and column:

Row 1, from left to right: column 1: 0; column 2: P subscript (dot) superscript (A); column 3: P subscript (dot) superscript (B); column 4: P subscript (dot) superscript (C); column 5: 0.
Row 2, from left to right: column 1: 0; column 2: A; column 3: B; column 4: C; column 5: 0.
Row 3, from left to right: column 2: 0; column 3: 0; column 4: 0.

Arrows and lines:

An arrow from 0 (row 1, column 1) to P subscript (dot) superscript (A), without a label.
An arrow from P subscript (dot) superscript (A) to P subscript (dot) superscript (B), labelled iota subscript (dot).
An arrow from P subscript (dot) superscript (A) to A, labelled epsilon superscript (A).
An arrow from P subscript (dot) superscript (B) to P subscript (dot) superscript (C), labelled p subscript (dot).
An arrow from P subscript (dot) superscript (B) to B, labelled epsilon superscript (B).
An arrow from P subscript (dot) superscript (C) to 0 (row 1, column 5), without a label.
An arrow from P subscript (dot) superscript (C) to C, labelled epsilon superscript (C).
An arrow from 0 (row 2, column 1) to A, without a label.
An arrow from A to B, labelled f.
An arrow from A to 0 (row 3, column 2), without a label.
An arrow from B to C, labelled g.
An arrow from B to 0 (row 3, column 3), without a label.
An arrow from C to 0 (row 2, column 5), without a label.
An arrow from C to 0 (row 3, column 4), without a label.

(12.6.1)

commutes, where $ι_{\cdot}$ and $p_{\cdot}$ are the natural maps on each term.

Proof.

Choose a lift $t_{0}$ of $𝜖^{C}$ to $P_{0}^{C} \to B$ , and let

𝜖^{B} = f \circ 𝜖^{A} + t_{0} \circ p_{0} .

Then $𝜖^{B}$ is clearly surjective, and we have the desired commutativity of the “first two” squares. Next, letting $d_{\cdot}^{X}$ denote the boundary maps with $X = A$ , $C$ , we may define the boundary map $d_{1}^{B}$ for $B$ similarly. That is, consider a lift of $d_{1}^{C} : P_{1}^{C} \to \ker 𝜖^{C}$ to a map $t_{1} : P_{1}^{C} \to \ker 𝜖^{B}$ , and define

d_{1}^{B} = ι_{0} \circ d_{1}^{A} + t_{1} \circ p_{1} .

Then $d_{1}^{B}$ maps onto $\ker d_{1}^{B}$ and makes the next two squares commute. We then continue in this fashion. □

Lemma 12.6.7. §

Let $𝒞$ be an abelian category and $P_{\cdot}$ a split long exact sequence of projectives with $P_{i} = 0$ for $i < 0$ . Then $P_{\cdot}$ is a projective object in $𝐂𝐡 (𝒞)$ .

Proof.

Let $P_{\cdot}$ be a split exact sequence of projectives in $𝒞$ . In other words, we may write each $P_{0} = Q_{0}$ and $P_{i} = Q_{i} \oplus Q_{i - 1}$ for $i \geq 1$ , where $Q_{i}$ is a projective object in $𝒞$ , and the morphism $P_{i} \to P_{i - 1}$ is simply the composition of the projection $P_{i} \to Q_{i - 1}$ with the inclusion $Q_{i - 1} \to P_{i}$ . Suppose that $π_{\cdot} : A_{\cdot} \to P_{\cdot}$ is a epimorphism of complexes. Since $Q_{i}$ is projective, there exists a splitting $s_{i} : Q_{i} \to A_{i}$ of the composition of $π_{i}$ with projection to $Q_{i}$ . Then

t_{i} = s_{i} \oplus s_{i - 1} : P_{i} = Q_{i} \oplus Q_{i - 1} \to A_{i}

is a splitting of $π_{i}$ . Since $t_{\cdot}$ is a morphism of complexes, it is a splitting of $π_{\cdot}$ . □

Remark 12.6.8. §

Every split exact complex is the cone of a complex with zero differentials.

Remark 12.6.9. §

Though we shall not prove it, every projective object in the category of chain complexes over an abelian category is a split exact sequence of projectives. Also, the projective objects in the category of bounded below chain complexes (or those in nonnegative degrees) are the bounded below exact sequences of projectives (which automatically split).

Definition 12.6.10. §

We say that a functor $F : 𝒞 \to 𝒟$ between categories preserves projectives if it takes projective objects in $𝒞$ to projective objects in $𝒟$ .

Proposition 12.6.11. §

Let $𝒞$ and $𝒟$ be an abelian category. Let $F : 𝒞 \to 𝒟$ be a functor that is left adjoint to an exact functor $G : 𝒟 \to 𝒞$ . Then $F$ preserves projectives.

Proof.

Let $P$ be a projective object in $𝒞$ . Let $f : A \to B$ be a epimorphism in $𝒟$ . We must show that $h_{F (P)} (f) : F (A) \to F (B)$ is an epimorphism. Note that we have a commutative diagram

An adjunction preserving projectives. A full diagram description follows.

Diagram description: An adjunction preserving projectives

The structural squares and triangles displayed here commute.

Objects, listed by row and column:

Row 1, from left to right: column 1: Hom subscript (script C)(P,G(A)); column 2: Hom subscript (script C)(P,G(B)).
Row 2, from left to right: column 1: Hom subscript (script D)(F(P),A); column 2: Hom subscript (script D)(F(P),B).

Arrows and lines:

An arrow from Hom subscript (script C)(P,G(A)) to Hom subscript (script C)(P,G(B)), labelled h subscript (P)(G(f)).
An arrow from Hom subscript (script C)(P,G(A)) to Hom subscript (script D)(F(P),A), labelled isomorphism symbol.
An arrow from Hom subscript (script C)(P,G(B)) to Hom subscript (script D)(F(P),B), labelled isomorphism symbol.
An arrow from Hom subscript (script D)(F(P),A) to Hom subscript (script D)(F(P),B), labelled h subscript (F(P))(f).

Exactness of $G$ tells us that $G (f)$ is an epimorphism, and the upper horizontal map is then an epimorphism by the projectivity of $P$ , hence the result. □

12.7. Derived functors

Suppose that $F : 𝒞 \to 𝒟$ is a right exact functor between abelian categories $𝒞$ and $𝒟$ and that $𝒞$ has enough projectives. Then we could try to define the $i$ th left derived functor of $F$ (for $i \geq 0$ ) on an object $A$ of $𝒞$ by $H_{i} (F (P_{\cdot}))$ , where $P_{\cdot} \to A$ is a projective resolution of $A$ . Of course, we must check that this definition is independent of the projective resolution chosen, and that we obtain induced maps on morphisms so that our map becomes a functor.

In the following, one may suppose that $𝒞$ is the category of $R -mod$ , but it is often the case that $𝒟$ is some other category, like $𝐀𝐛$ or $S$ -mod for some ring $S$ .

Proposition 12.7.1. §

Let $F : 𝒞 \to 𝒟$ be an additive functor between abelian categories $𝒞$ and $𝒟$ . For $i \geq 0$ , there are functors $L_{i} F : 𝒞 \to 𝒟$ given on $A \in 𝒞$ by $L_{i} F (A) = H_{i} (F (P_{\cdot}))$ for $P_{\cdot} \to A$ a projective resolution of $A \in 𝒞$ and given on $g : A \to B$ in $𝒞$ by $L_{i} F (g) = F {(f)}_{*} : H_{i} (F (P_{\cdot})) \to H_{i} (F (Q_{\cdot}))$ for $P_{\cdot} \to A$ and $Q_{\cdot} \to B$ projective resolutions and $f_{\cdot} : P_{\cdot} \to Q_{\cdot}$ a morphism of complexes $f : P_{\cdot} \to Q_{\cdot}$ compatible with the augmentations to $A$ and $B$ . These functors are dependent on the choices made up only to unique isomorphism.

Proof.

The key point is that Proposition 12.6.5 tells us that $F {(f)}_{*}$ is independent of the choice of $f$ , since any two choices are chain homotopic. In particular, given any two choices of $P_{\cdot}$ and $Q_{\cdot}$ of projective resolutions of $A$ , and any choices of $f_{\cdot} \to P_{\cdot} \to Q_{\cdot}$ $f_{\cdot}^{'} \to Q_{\cdot} \to P_{\cdot}$ augmenting the identity morphism on $A$ gives rise to morphisms on cohomology, the resulting maps $F {(f)}_{*}$ and $F {(f^{'})}_{*}$ must be mutually inverse, since $f \circ f^{'}$ and $f^{'} \circ f$ augment the identity on $A$ , as to the identity morphisms on $Q_{\cdot}$ and $P_{\cdot}$ . Thus, $H_{i} (F (P_{\cdot}))$ is unique up to unique isomorphism, so $L_{i} F$ is well-defined (up to unique isomorphism), and $L_{i} F ({id}_{A}) = {id}_{L_{i} F (A)}$ . Similarly, it is easy to check that the uniqueness also implies that $L_{i} F$ is compatible with compositions. □

Definition 12.7.2. §

For an additive functor $F : 𝒞 \to 𝒟$ , the $i$ th left derived functor $L_{i} F : 𝒞 \to 𝒟$ of $F$ is the functor defined by Proposition 12.7.1.

We have the following obvious corollaries of Lemma 12.7.1.

Lemma 12.7.3. §

Let $F : 𝒞 \to 𝒟$ be a right exact functor of abelian categories. Then we have a canonical, natural isomorphism $L_{0} F \to_{}^{\sim} F$ of functors.

Proof.

Since $F$ is right exact, the sequence

F (P_{1}) \to F (P_{0}) \to F (A) \to 0

is exact. Hence, we have

L_{0} F (A) = H_{0} (F (P_{\cdot})) ≅𝐹 (A) .

The reader will easily check the independence of the choice of resolution and naturality, as in Lemma 12.7.1. □

Corollary 12.7.4. §

If $P$ is a projective object, then $L_{i} F (P) = 0$ for $i \geq 1$ .

Proof.

Consider the projective resolution that is $P$ in degree zero and $0$ elsewhere, where the augmentation map $P \to P$ is the identity. This has the desired homology. □

Next, we prove that the $L_{i} F$ are functors.

Proposition 12.7.5. §

To each morphism $f : A \to B$ in $𝒞$ , we can associate morphisms

L_{i} F (f) : L_{i} F (A) \to L_{i} F (B)

for all $i \geq 0$ in such a way that $L_{i} F : 𝒞 \to 𝒟$ becomes a functor and $L_{0} F (f) = F (f)$ . Furthermore, each $L_{i} F$ is additive.

Proof.

The unique morphism $L_{i} F (f)$ is induced on homology by the morphism of chain complexes given in Proposition 12.6.5. Functoriality follows by canonicality of the map of homology.

To see additivity, note that $L_{i} F (0_{A})$ is induced by the zero morphism of chain complexes and hence is is zero map on $L_{i} F (A)$ . Similarly $L_{i} F (f + g)$ , for $f, g : A \to B$ , can be given by the sum of the induced maps on chain complexes, hence is given by the sum of the maps on homology. □

Definition 12.7.6. §

For a right exact functor $F : 𝒞 \to 𝒟$ of abelian categories, the functor $L_{i} F$ is called $i$ th left derived functor of $F$ .

We see that $L_{0} F$ and $F$ are canonically naturally isomorphic functors.

Definition 12.7.7. §

A homological $δ$ -functor is a sequence of additive functors $F_{i} : 𝒞 \to 𝒟$ for $i \in ℤ$ , together with, for every exact sequence

0 \to A \to_{}^{f} B \to_{}^{g} C \to 0

in $𝒞$ , morphisms $δ_{i} : F_{i} (C) \to F_{i - 1} (A)$ fitting in a long exact sequence

\dots \to F_{i} (A) \to_{}^{F_{i} (f)} F_{i} (B) \to_{}^{F_{i} (g)} F_{i} (C) \to_{}^{δ_{i}} F_{i - 1} (A) \to \dots

which are natural in the sense that if we have a morphism of short exact sequences in $𝒞$ ,

A morphism of short exact sequences. A full diagram description follows.

Diagram description: A morphism of short exact sequences

The two displayed rows are exact, and the squares commute.

Objects, listed by row and column:

Row 1, from left to right: column 1: 0; column 2: A; column 3: B; column 4: C; column 5: 0.
Row 2, from left to right: column 1: 0; column 2: A prime; column 3: B prime; column 4: C prime; column 5: 0.

Arrows and lines:

An arrow from 0 (row 1, column 1) to A, without a label.
An arrow from A to B, without a label.
An arrow from A to A prime, without a label.
An arrow from B to C, without a label.
An arrow from B to B prime, without a label.
An arrow from C to 0 (row 1, column 5), without a label.
An arrow from C to C prime, without a label.
An arrow from 0 (row 2, column 1) to A prime, without a label.
An arrow from A prime to B prime, without a label.
An arrow from B prime to C prime, without a label.
An arrow from C prime to 0 (row 2, column 5), without a label.

then we obtain a morphism of long exact sequences in $𝒟$ ,

Naturality of a homological delta-functor. A full diagram description follows.

Diagram description: Naturality of a homological delta-functor

The two displayed rows are exact, and the squares commute.

Objects, listed by row and column:

Row 1, from left to right: column 1: ellipsis; column 2: F subscript (i)(A); column 3: F subscript (i)(B); column 4: F subscript (i)(C); column 5: F subscript (i minus 1)(A); column 6: ellipsis.
Row 2, from left to right: column 1: ellipsis; column 2: F subscript (i)(A prime ); column 3: F subscript (i)(B prime ); column 4: F subscript (i)(C prime ); column 5: F subscript (i minus 1)(A prime ); column 6: ellipsis.

Arrows and lines:

An arrow from ellipsis (row 1, column 1) to F subscript (i)(A), without a label.
An arrow from F subscript (i)(A) to F subscript (i)(B), without a label.
An arrow from F subscript (i)(A) to F subscript (i)(A prime ), without a label.
An arrow from F subscript (i)(B) to F subscript (i)(C), without a label.
An arrow from F subscript (i)(B) to F subscript (i)(B prime ), without a label.
An arrow from F subscript (i)(C) to F subscript (i minus 1)(A), without a label.
An arrow from F subscript (i)(C) to F subscript (i)(C prime ), without a label.
An arrow from F subscript (i minus 1)(A) to ellipsis (row 1, column 6), without a label.
An arrow from F subscript (i minus 1)(A) to F subscript (i minus 1)(A prime ), without a label.
An arrow from ellipsis (row 2, column 1) to F subscript (i)(A prime ), without a label.
An arrow from F subscript (i)(A prime ) to F subscript (i)(B prime ), without a label.
An arrow from F subscript (i)(B prime ) to F subscript (i)(C prime ), without a label.
An arrow from F subscript (i)(C prime ) to F subscript (i minus 1)(A prime ), without a label.
An arrow from F subscript (i minus 1)(A prime ) to ellipsis (row 2, column 6), without a label.

Example 12.7.8. §

Define functors $F_{0}$ , $F_{1} : 𝐀𝐛 \to 𝐀𝐛$ by $F_{0} (A) = A ∕ 𝑝𝐴$ and

F_{1} (A) = A [p] = {a \in A ∣ 𝑝𝑎 = 0}

for any abelian group $A$ , and set $F_{i} = 0$ otherwise. Given an exact sequence

0 \to A \to B \to C \to 0

in $𝐀𝐛$ , we obtain a long exact sequence

0 \to A [p] \to B [p] \to C [p] \to_{}^{δ_{1}} A ∕ 𝑝𝐴 \to B ∕ 𝑝𝐵 \to C ∕ 𝑝𝐶 \to 0

from the snake lemma applied to the diagram

Multiplication by p on a short exact sequence. A full diagram description follows.

Diagram description: Multiplication by p on a short exact sequence

The two displayed rows are exact, and the squares commute.

Objects, listed by row and column:

Row 1, from left to right: column 1: 0; column 2: A; column 3: B; column 4: C; column 5: 0.
Row 2, from left to right: column 1: 0; column 2: A; column 3: B; column 4: C; column 5: 0.

Arrows and lines:

An arrow from 0 (row 1, column 1) to A (row 1, column 2), without a label.
An arrow from A (row 1, column 2) to B (row 1, column 3), without a label.
An arrow from A (row 1, column 2) to A (row 2, column 2), labelled dot p.
An arrow from B (row 1, column 3) to C (row 1, column 4), without a label.
An arrow from B (row 1, column 3) to B (row 2, column 3), labelled dot p.
An arrow from C (row 1, column 4) to 0 (row 1, column 5), without a label.
An arrow from C (row 1, column 4) to C (row 2, column 4), labelled dot p.
An arrow from 0 (row 2, column 1) to A (row 2, column 2), without a label.
An arrow from A (row 2, column 2) to B (row 2, column 3), without a label.
An arrow from B (row 2, column 3) to C (row 2, column 4), without a label.
An arrow from C (row 2, column 4) to 0 (row 2, column 5), without a label.

This defines a $δ$ -functor.

Theorem 12.7.9. §

For every short exact sequence

0 \to A \to B \to C \to 0

in $𝒞$ , there exist morphisms $δ_{i} : L_{i} F (C) \to L_{i - 1} F (A)$ such that the functors $L_{\cdot} F$ together with the maps $δ_{\cdot}$ form a homological $δ$ -functor.

Proof.

By the Horseshoe lemma, we have a projective resolution $P_{\cdot}^{X} \to X$ for $X = A$ , $B$ , $C$ fitting in a diagram (12.6.1). Now, applying $F$ to the resolutions, we have split exact sequences

0 \to F (P_{i}^{A}) \to F (P_{i}^{B}) \to F (P_{i}^{C}) \to 0

for each $i$ . The resulting exact sequence of complexes (which need not be split) yields a long exact sequence in homology

\dots L_{1} F (B) \to L_{1} F (C) \to_{}^{δ_{1}} F (A) \to F (B) \to F (C) \to 0,

as desired.

It remains to check naturality. Consider a morphism of short exact sequences

A morphism of short exact sequences for derived functors. A full diagram description follows.

Diagram description: A morphism of short exact sequences for derived functors

The two displayed rows are exact, and the squares commute.

Objects, listed by row and column:

Row 1, from left to right: column 1: 0; column 2: A; column 3: B; column 4: C; column 5: 0.
Row 2, from left to right: column 1: 0; column 2: A prime; column 3: B prime; column 4: C prime; column 5: 0.

Arrows and lines:

An arrow from 0 (row 1, column 1) to A, without a label.
An arrow from A to B, labelled f.
An arrow from A to A prime, labelled q superscript (A).
An arrow from B to C, labelled g.
An arrow from B to B prime, labelled q superscript (B).
An arrow from C to 0 (row 1, column 5), without a label.
An arrow from C to C prime, labelled q superscript (C).
An arrow from 0 (row 2, column 1) to A prime, without a label.
An arrow from A prime to B prime, labelled f prime.
An arrow from B prime to C prime, labelled g prime.
An arrow from C prime to 0 (row 2, column 5), without a label.

By Proposition 12.6.5, can extend $q^{A}$ and $q^{C}$ to maps of complexes $q_{\cdot}^{A} : P_{\cdot}^{A} \to P_{\cdot}^{A^{'}}$ and $q_{\cdot}^{C} : P_{\cdot}^{C} \to P_{\cdot}^{C^{'}}$ . Suppose we have constructed $P_{\cdot}^{B}$ and $P_{\cdot}^{B^{'}}$ via the Horseshoe lemma. We fit this all into a commutative diagram

Maps between horseshoe resolutions. A full diagram description follows.

Diagram description: Maps between horseshoe resolutions

The structural squares and triangles displayed here commute.

Objects, listed by row and column:

Row 1, from left to right: column 1: 0; column 3: P subscript (dot) superscript (A); column 5: P subscript (dot) superscript (B); column 7: P subscript (dot) superscript (C); column 9: 0.
Row 2, from left to right: column 2: 0; column 4: A; column 6: B; column 8: C; column 10: 0.
Row 3, from left to right: column 1: 0; column 3: P subscript (dot) superscript (A prime); column 5: P subscript (dot) superscript (B prime); column 7: P subscript (dot) superscript (C prime); column 9: 0.
Row 4, from left to right: column 2: 0; column 4: A prime; column 6: B prime; column 8: C prime; column 10: 0.

Arrows and lines:

An arrow from 0 (row 1, column 1) to P subscript (dot) superscript (A), without a label.
An arrow from P subscript (dot) superscript (A) to P subscript (dot) superscript (B), labelled iota subscript (dot).
An arrow from P subscript (dot) superscript (A) to A, labelled epsilon superscript (A).
An arrow from P subscript (dot) superscript (A) to P subscript (dot) superscript (A prime), labelled q subscript (dot) superscript (A).
An arrow from P subscript (dot) superscript (B) to P subscript (dot) superscript (C), labelled p subscript (dot).
An arrow from P subscript (dot) superscript (B) to B, without a label.
An arrow from P subscript (dot) superscript (C) to C, without a label.
An arrow from P subscript (dot) superscript (C) to P subscript (dot) superscript (C prime), labelled q subscript (dot) superscript (C).
An arrow from P subscript (dot) superscript (C) to 0 (row 1, column 9), without a label.
An arrow from 0 (row 2, column 2) to A, without a label.
An arrow from A to B, labelled f.
An arrow from A to A prime, without a label.
An arrow from B to C, labelled g.
An arrow from B to B prime, without a label.
An arrow from C to C prime, labelled q superscript (C).
An arrow from C to 0 (row 2, column 10), without a label.
An arrow from 0 (row 3, column 1) to P subscript (dot) superscript (A prime), without a label.
An arrow from P subscript (dot) superscript (A prime) to P subscript (dot) superscript (B prime), labelled iota prime subscript (dot).
An arrow from P subscript (dot) superscript (A prime) to A prime, without a label.
An arrow from P subscript (dot) superscript (B prime) to B prime, without a label.
An arrow from P subscript (dot) superscript (B prime) to P subscript (dot) superscript (C prime), labelled p prime subscript (dot).
An arrow from P subscript (dot) superscript (C prime) to C prime, without a label.
An arrow from P subscript (dot) superscript (C prime) to 0 (row 3, column 9), without a label.
An arrow from 0 (row 4, column 2) to A prime, without a label.
An arrow from A prime to B prime, labelled f prime.
An arrow from B prime to C prime, labelled g prime.
An arrow from C prime to 0 (row 4, column 10), without a label.

(12.7.1)

We also have splitting maps $j_{i} : P_{i}^{C} \to P_{i}^{B}$ and $k_{i} : P_{i}^{B} \to P_{i}^{A}$ for each $i$ (and, similarly, maps $j_{i}^{'}$ and $k_{i}^{'}$ ). For each $X$ , let us denote the augmentation map by $𝜖^{X}$ .

We must define a map $q_{\cdot}^{B} : P_{\cdot}^{B} \to P_{\cdot}^{B^{'}}$ making the entire diagram (12.7.1) commute. We first note that

\begin{matrix} g^{'} \circ (q^{B} \circ 𝜖^{B} - 𝜖^{B^{'}} \circ j_{0}^{'} \circ q_{0}^{C} \circ p_{0}) = q^{C} \circ g \circ 𝜖^{B} - 𝜖^{C^{'}} \circ p_{0}^{'} \circ j_{0} \circ q_{0}^{C} \circ p_{0} \\ = q^{C} \circ 𝜖^{C} \circ p_{0} - 𝜖^{C^{'}} \circ q_{0}^{C} \circ p_{0} = (q^{C} \circ 𝜖^{C} - 𝜖^{C^{'}} \circ q_{0}^{C}) \circ p_{0} = 0 . \end{matrix}

Hence, there exists a map $β_{0} : P_{0}^{B} \to A^{'}$ with

f^{'} \circ β_{0} = q^{B} \circ 𝜖^{B} - 𝜖^{B^{'}} \circ j_{0}^{'} \circ q_{0}^{C} \circ p_{0} .

Since $𝜖^{A^{'}}$ is an epimorphism, we may choose $α_{0} : P_{0}^{B} \to P_{0}^{A^{'}}$ with $𝜖^{A^{'}} \circ α_{0} = β_{0}$ . Now set

q_{0}^{B} = ι_{0}^{'} \circ q_{0}^{A} \circ k_{0} + ι_{0}^{'} \circ α_{0} \circ p_{0} + j_{0}^{'} \circ q_{0}^{C} \circ p_{0} .

The trickiest check of commutativity is that $𝜖^{B^{'}} \circ q_{0}^{B} = q^{B} \circ 𝜖^{B}$ . We write this mess out:

\begin{array}{l} 𝜖^{B^{'}} \circ q_{0}^{B} & = 𝜖^{B^{'}} \circ ι_{0}^{'} \circ q_{0}^{A} \circ k_{0} + 𝜖^{B^{'}} \circ ι_{0}^{'} \circ α_{0} \circ p_{0} + 𝜖^{B^{'}} \circ j_{0}^{'} \circ q_{0}^{C} \circ p_{0} \\ = f^{'} \circ 𝜖^{A^{'}} \circ q_{0}^{A} \circ k_{0} + f^{'} \circ 𝜖^{A^{'}} \circ α_{0} \circ p_{0} + 𝜖^{B^{'}} \circ j_{0}^{'} \circ q_{0}^{C} \circ p_{0} \\ = f^{'} \circ q^{A} \circ 𝜖^{A} \circ k_{0} + f^{'} \circ β_{0} \circ p_{0} + 𝜖^{B^{'}} \circ j_{0}^{'} \circ q_{0}^{C} \circ p_{0} \\ = f^{'} \circ q^{A} \circ 𝜖^{A} \circ k_{0} + (j_{0}^{'} \circ q^{C} \circ 𝜖^{C} - 𝜖^{B^{'}} \circ j_{0}^{'} \circ q_{0}^{C}) \circ p_{0} + 𝜖^{B^{'}} \circ j_{0}^{'} \circ q_{0}^{C} \circ p_{0} \\ = f^{'} \circ q^{A} \circ 𝜖^{A} \circ k_{0} + j_{0}^{'} \circ q^{C} \circ 𝜖^{C} \circ p_{0} \\ = q^{B} \circ 𝜖^{B} . \end{array}

The other $q_{i}^{B}$ are defined similarly. For instance, one can see there exists a map $β_{1} : P_{1}^{C} \to P_{0}^{A^{'}}$ such that

ι_{0} \circ β_{1} = ι_{0} \circ β_{0} \circ d_{0}^{C} + j_{1}^{'} \circ q_{0}^{C} \circ d_{0}^{C} - d_{0}^{B^{'}} \circ j_{1}^{'} \circ q_{1}^{C} \circ p_{1},

and we set

q_{1}^{B} = ι_{1}^{'} \circ q_{1}^{A} \circ k_{1} + ι_{1}^{'} \circ α_{1} \circ p_{1} + j_{1}^{'} \circ q_{1}^{C} .

□

Definition 12.7.10. §

A (homological) universal $δ$ -functor is a $δ$ -functor $F_{\cdot} = (F_{i}, δ_{i})$ with $F_{i} : 𝒞 \to 𝒟$ such that if $G_{\cdot} = (G_{i}, δ_{i}^{'})$ is any other $δ$ -functor with $G_{i} : 𝒞 \to 𝒟$ for which there exists a natural transformation $η_{0} : G_{0} ⇝ F_{0}$ , then $η_{0}$ extends uniquely to a morphism of $δ$ -functors, i.e., a sequence of natural transformations $η_{i} : G_{i} ⇝ F_{i}$ such that

A morphism of homological delta-functors. A full diagram description follows.

Diagram description: A morphism of homological delta-functors

The structural squares and triangles displayed here commute.

Objects, listed by row and column:

Row 1, from left to right: column 1: G subscript (i)(C); column 2: G subscript (i minus 1)(A).
Row 2, from left to right: column 1: F subscript (i)(C); column 2: F subscript (i minus 1)(A).

Arrows and lines:

An arrow from G subscript (i)(C) to F subscript (i)(C), labelled ( eta subscript (i)) subscript (C).
An arrow from G subscript (i)(C) to G subscript (i minus 1)(A), labelled delta prime subscript (i).
An arrow from G subscript (i minus 1)(A) to F subscript (i minus 1)(A), labelled ( eta subscript (i minus 1)) subscript (A).
An arrow from F subscript (i)(C) to F subscript (i minus 1)(A), labelled delta subscript (i).

commutes for any short exact sequence in $𝒞$ :

0 \to A \to B \to C \to 0 .

(That is, we get a morphism of the associated long exact sequences.)

The $δ$ -functor of left derived functors of $F$ is universal, which will follow as a corollary of Theorem 12.7.14 below.

Theorem 12.7.11. §

The $δ$ -functor $(L_{i} F, δ_{i})$ is universal.

Definition 12.7.12. §

Let $F : 𝒞 \to 𝒟$ be a left exact functor between abelian categories. We say that an object $Q$ in $𝒞$ is $F$ -acyclic if $L_{i} F (Q) = 0$ for all $i \geq 1$ .

Note that the $L_{i} F (A)$ for any $A \in Obj (𝒞)$ may be computed using resolutions by $F$ -acyclic objects, as opposed to just projectives.

Proposition 12.7.13. §

Let $F : 𝒞 \to 𝒟$ be a left exact functor between abelian categories, and let $A$ be an object of $𝒞$ . Suppose that $C_{\cdot} \to A$ is a resolution of $A$ by $F$ -acyclic objects. Then $L_{i} F (A) ≅ H_{i} (F (C_{\cdot}))$ for each $i \geq 0$ .

Proof.

Note that we have an exact sequence

F (C_{1}) \to_{}^{F (d_{1}^{C})} F (C_{0}) \to_{}^{F (𝜖^{C})} F (A) \to 0,

so $F (A) ≅ H_{0} (F (C_{\cdot}))$ . Set $K_{0} = \ker 𝜖^{C}$ . We then have an exact sequence

0 \to L_{1} F (A) \to F (K_{0}) \to F (C_{0}) \to F (A) \to 0,

which yields

L_{1} F (A) ≅ \ker (coker (F (C_{2}) \to F (C_{1})) \to F (C_{0})) ≅ \frac{\ker F (d_{1}^{C})}{im F (d_{2}^{C})} ≅ H_{1} (F (C_{\cdot})) .

We also have isomorphisms $L_{i} F (A) ≅ L_{i - 1} F (K_{0})$ for each $i \geq 2$ .

For $i \geq 1$ , set $K_{i} = \ker d_{i}^{C} ≅ im d_{i + 1}^{C}$ . The exact sequences

0 \to K_{i} \to C_{i} \to K_{i - 1} \to 0,

then yield isomorphisms used in the following for $i \geq 2$ :

L_{i} F (A) ≅ L_{i - 1} F (K_{0}) ≅ \dots ≅ L_{1} F (K_{i - 2}) ≅ \ker (F (K_{i - 1}) \to F (C_{i - 1})) ≅ \frac{\ker F (d_{i}^{C})}{im F (d_{i + 1}^{C})} ≅ H_{i} (F (C_{\cdot})) .

□

More generally, we have the following characterization of universal $δ$ -functors.

Theorem 12.7.14. §

Let $𝒞$ and $𝒟$ be abelian categories such that $𝒞$ has enough projectives. Suppose that $(F_{i}, δ_{i})$ form a $δ$ -functor $F_{i} : 𝒞 \to 𝒟$ and $F_{i} (P) = 0$ for every projective $P \in Obj (𝒞)$ and $i \geq 1$ . Then $(F_{i}, δ_{i})$ is universal.

Proof.

Suppose that $(G_{i}, δ_{i}^{'})$ is another $δ$ -functor and that we have a natural transformation $G_{0} ⇝ F_{0}$ . Let $A \in Obj (𝒞)$ and let $π : P \to A$ be an epimorphism with $P$ projective. Let $K = \ker π$ . Let $i \geq 1$ , and suppose that we have constructed a natural transformation $G_{i - 1} ⇝ F_{i - 1}$ . Since $F_{i} (P) = 0$ is projective, we have a commutative diagram

Extending a natural transformation by dimension shifting. A full diagram description follows.

Diagram description: Extending a natural transformation by dimension shifting

The structural squares and triangles displayed here commute. The dashed map is the unique map making the square commute.

Objects, listed by row and column:

Row 1, from left to right: column 2: G subscript (i)(A); column 3: G subscript (i minus 1)(K); column 4: G subscript (i minus 1)(P).
Row 2, from left to right: column 1: 0; column 2: F subscript (i)(A); column 3: F subscript (i minus 1)(K); column 4: F subscript (i minus 1)(P).

Arrows and lines:

An arrow from G subscript (i)(A) to G subscript (i minus 1)(K), without a label.
A dashed arrow from G subscript (i)(A) to F subscript (i)(A), without a label.
An arrow from G subscript (i minus 1)(K) to G subscript (i minus 1)(P), without a label.
An arrow from G subscript (i minus 1)(K) to F subscript (i minus 1)(K), without a label.
An arrow from G subscript (i minus 1)(P) to F subscript (i minus 1)(P), without a label.
An arrow from 0 to F subscript (i)(A), without a label.
An arrow from F subscript (i)(A) to F subscript (i minus 1)(K), without a label.
An arrow from F subscript (i minus 1)(K) to F subscript (i minus 1)(P), without a label.

The morphism $G_{i} (A) \to F_{i} (A)$ is the unique map which makes the diagram commute.

Now let $f : A \to B$ be a morphism in $𝒞$ . We create a diagram as follows:

Lifting a map between short exact sequences. A full diagram description follows.

Diagram description: Lifting a map between short exact sequences

The structural squares and triangles displayed here commute.

Objects, listed by row and column:

Row 1, from left to right: column 1: 0; column 2: K; column 3: P; column 4: A; column 5: 0.
Row 2, from left to right: column 1: 0; column 2: K prime; column 3: P prime; column 4: B; column 5: 0.

Arrows and lines:

An arrow from 0 (row 1, column 1) to K, without a label.
An arrow from K to P, without a label.
An arrow from K to K prime, without a label.
An arrow from P to A, without a label.
An arrow from P to P prime, without a label.
An arrow from A to 0 (row 1, column 5), without a label.
An arrow from A to B, labelled f.
An arrow from 0 (row 2, column 1) to K prime, without a label.
An arrow from K prime to P prime, without a label.
An arrow from P prime to B, without a label.
An arrow from B to 0 (row 2, column 5), without a label.

by taking $P^{'}$ to be projective, $P$ to be any projective with an epimorphism to the pullback of the diagram $P^{'} \to B \leftarrow A$ , and $K$ and $K^{'}$ to be the relevant kernels. We then have a diagram

Naturality of the dimension-shifting construction. A full diagram description follows.

Diagram description: Naturality of the dimension-shifting construction

The structural squares and triangles displayed here commute.

Objects, listed by row and column:

Row 1, from left to right: column 1: G subscript (i)(A); column 3: G subscript (i minus 1)(K).
Row 2, from left to right: column 2: F subscript (i)(A); column 4: F subscript (i minus 1)(K).
Row 3, from left to right: column 1: G subscript (i)(B); column 3: G subscript (i minus 1)(K prime ).
Row 4, from left to right: column 2: F subscript (i)(B); column 4: F subscript (i minus 1)(K prime ).

Arrows and lines:

An arrow from G subscript (i)(A) to G subscript (i minus 1)(K), without a label.
An arrow from G subscript (i)(A) to F subscript (i)(A), without a label.
An arrow from G subscript (i)(A) to G subscript (i)(B), without a label.
An arrow from G subscript (i minus 1)(K) to F subscript (i minus 1)(K), without a label.
An arrow from G subscript (i minus 1)(K) to G subscript (i minus 1)(K prime ), without a label.
An arrow from F subscript (i)(A) to F subscript (i minus 1)(K), without a label.
An arrow from F subscript (i)(A) to F subscript (i)(B), without a label.
An arrow from F subscript (i minus 1)(K) to F subscript (i minus 1)(K prime ), without a label.
An arrow from G subscript (i)(B) to G subscript (i minus 1)(K prime ), without a label.
An arrow from G subscript (i)(B) to F subscript (i)(B), without a label.
An arrow from G subscript (i minus 1)(K prime ) to F subscript (i minus 1)(K prime ), without a label.
An arrow from F subscript (i)(B) to F subscript (i minus 1)(K prime ), without a label.

We need only see that the leftmost square commutes, but this follows easily from a diagram chase and the fact that the two horizontal maps on the frontmost square are monomorphisms.

Hence, we have constructed a sequence of natural transformations $G_{i} ⇝ F_{i}$ . It remains only to see that these form a morphism of $δ$ -functors. This being an inductive argument of the above sort, we leave it to the reader. □

As a corollary, we have a natural isomorphism of $δ$ -functors between the left derived functors $L_{i} F_{0}$ of a right exact functor $F_{0}$ and any $δ$ -functor $(F_{i}, δ_{i})$ with $F_{i} (P) = 0$ for $P$ projective and $i \geq 1$ .

We next wish to study right derived functors of left exact functors.

Definition 12.7.15. §

An injective resolution of an object $A$ of an abelian category is a cochain complex $I^{\cdot}$ of injective objects with $I^{i} = 0$ for $i < 0$ and a morphism $A \to I^{0}$ such that the resulting diagram

0 \to A \to I^{0} \to I^{1} \to I^{2} \to \dots

is exact.

Definition 12.7.16. §

We say that an abelian category $𝒞$ has enough (or sufficiently many) injectives if for every $A \in Obj (𝒞)$ , there exists an injective object $I \in Obj (𝒞)$ and a monomorphism $A \to I$ .

Remark 12.7.17. §

An abelian category has enough injectives if and only if every object of it has an injective resolution.

Proposition 12.7.18. §

The category $R -mod$ has enough injectives.

Proof.

First take the case that $R = ℤ$ . Let $A$ be an abelian group, and write it as a quotient of a free abelian group

A = (⨁_{j \in J} ℤ) ∕ T

for some indexing set $J$ and submodule $T$ of $\oplus_{j \in J} ℤ$ . Then we may embed $A$ in

I = (⨁_{j \in J} ℚ) ∕ T,

which is divisible as a quotient of a divisible group.

Next, let $A$ be a left $R$ -module. We have an injection of left $R$ -modules,

ϕ : A \to {Hom}_{ℤ} (R, A),

by $ϕ (a) (r) = 𝑟𝑎$ . Now, embed $A$ in a divisible group $D$ , so that the resulting map

{Hom}_{ℤ} (R, A) \to {Hom}_{ℤ} (R, D)

is an injection. The proof that ${Hom}_{ℤ} (R, D)$ is an injective $R$ -module is left to the reader. □

We also have the analogues of Propositions 12.6.5 and 12.6.6 for injective resolutions.

Suppose now that $F : 𝒞 \to 𝒟$ is a left exact functor between abelian categories and that $𝒞$ has enough injectives. For each $i \geq 0$ , we define additive functors $R^{i} F : 𝒞 \to 𝒟$ by

R^{i} F (A) = H^{i} (F (I^{\cdot})),

where $A \to I^{\cdot}$ is any injective resolution of $A \in Obj (𝒞)$ and, for $f : A \to B$ in $𝒞$ , by

R^{i} F (f) : R^{i} F (A) \to R^{i} F (B)

to be the map on homology induced by any morphism of chain complexes $I^{\cdot} \to J^{\cdot}$ extending $f$ , where $A \to I^{\cdot}$ and $B \to J^{\cdot}$ are injective resolutions. We have $R^{0} F = F$ . The functors $R^{\cdot} F$ are called the right-derived functors of $F$ .

Definition 12.7.19. §

A cohomological $δ$ -functor is a sequence of additive functors $F^{i} : 𝒞 \to 𝒟$ for $i \in ℤ$ , together with, for every exact sequence

0 \to A \to_{}^{f} B \to_{}^{g} C \to 0

in $𝒞$ , morphisms $δ_{i} : F^{i} (C) \to F^{i + 1} (A)$ fitting in a long exact sequence

\dots \to F^{i} (A) \to_{}^{F^{i} (f)} F^{i} (B) \to_{}^{F^{i} (g)} F^{i} (C) \to_{}^{δ_{i}} F^{i + 1} (A) \to \dots

which are natural in the sense that if we have a morphism of short exact sequences in $𝒞$ ,

Diagram description: A morphism of short exact sequences

The two displayed rows are exact, and the squares commute.

Objects, listed by row and column:

Row 1, from left to right: column 1: 0; column 2: A; column 3: B; column 4: C; column 5: 0.
Row 2, from left to right: column 1: 0; column 2: A prime; column 3: B prime; column 4: C prime; column 5: 0.

Arrows and lines:

An arrow from 0 (row 1, column 1) to A, without a label.
An arrow from A to B, without a label.
An arrow from A to A prime, without a label.
An arrow from B to C, without a label.
An arrow from B to B prime, without a label.
An arrow from C to 0 (row 1, column 5), without a label.
An arrow from C to C prime, without a label.
An arrow from 0 (row 2, column 1) to A prime, without a label.
An arrow from A prime to B prime, without a label.
An arrow from B prime to C prime, without a label.
An arrow from C prime to 0 (row 2, column 5), without a label.

then we obtain a morphism of long exact sequences in $𝒟$ ,

Naturality of a cohomological delta-functor. A full diagram description follows.

Diagram description: Naturality of a cohomological delta-functor

The two displayed rows are exact, and the squares commute.

Objects, listed by row and column:

Row 1, from left to right: column 1: ellipsis; column 2: F superscript (i)(A); column 3: F superscript (i)(B); column 4: F superscript (i)(C); column 5: F superscript (i plus 1)(A); column 6: ellipsis.
Row 2, from left to right: column 1: ellipsis; column 2: F superscript (i)(A prime ); column 3: F superscript (i)(B prime ); column 4: F superscript (i)(C prime ); column 5: F superscript (i plus 1)(A prime ); column 6: ellipsis.

Arrows and lines:

An arrow from ellipsis (row 1, column 1) to F superscript (i)(A), without a label.
An arrow from F superscript (i)(A) to F superscript (i)(B), without a label.
An arrow from F superscript (i)(A) to F superscript (i)(A prime ), without a label.
An arrow from F superscript (i)(B) to F superscript (i)(C), without a label.
An arrow from F superscript (i)(B) to F superscript (i)(B prime ), without a label.
An arrow from F superscript (i)(C) to F superscript (i plus 1)(A), without a label.
An arrow from F superscript (i)(C) to F superscript (i)(C prime ), without a label.
An arrow from F superscript (i plus 1)(A) to ellipsis (row 1, column 6), without a label.
An arrow from F superscript (i plus 1)(A) to F superscript (i plus 1)(A prime ), without a label.
An arrow from ellipsis (row 2, column 1) to F superscript (i)(A prime ), without a label.
An arrow from F superscript (i)(A prime ) to F superscript (i)(B prime ), without a label.
An arrow from F superscript (i)(B prime ) to F superscript (i)(C prime ), without a label.
An arrow from F superscript (i)(C prime ) to F superscript (i plus 1)(A prime ), without a label.
An arrow from F superscript (i plus 1)(A prime ) to ellipsis (row 2, column 6), without a label.

Remark 12.7.20. §

A cohomological $δ$ -functor $(F^{i}, δ^{i})$ is universal if there exists a unique extension of any natural transformation $F^{0} ⇝ G^{0}$ , where $(G^{i}, {(δ^{'})}^{i})$ is another $δ$ -functor, to a morphism of $δ$ -functors.

Theorem 12.7.21. §

The functors $R^{\cdot} F$ form a cohomological universal $δ$ -functor.

The proof is dual to that of Theorems 12.7.9 and 12.7.11. We also have the following.

Theorem 12.7.22. §

Let $𝒞$ be an abelian category that has enough injectives. Then the cohomology functors $H^{i} : {𝐂𝐡}^{\geq 0} (𝒞) \to 𝒞$ for $i \geq 0$ on complexes in nonnegative degrees together with the connecting homomorphisms $δ^{i}$ attached to a short exact sequence of complexes form a universal $δ$ -functor.

12.8. Tor and Ext

Example 12.8.1. §

Take the abelian group $M = ℤ ∕ 𝑛ℤ$ . If we apply the functor $t_{M} : 𝐀𝐛 \to 𝐀𝐛$ to the exact sequence of abelian groups $0 \to ℤ \to_{}^{n} ℤ \to ℤ ∕ 𝑛ℤ \to 0$ for some $n \geq 2$ , we obtain the right, but not left, exact sequence

ℤ ∕ 𝑛ℤ \to_{}^{0} ℤ ∕ 𝑛ℤ \to_{}^{\sim} ℤ ∕ 𝑛ℤ \to 0 .

If we apply $h_{M} : 𝐀𝐛 \to 𝐀𝐛$ to the same exact sequence, we obtain the left, but not right, exact sequence

0 \to 0 \to 0 \to ℤ ∕ 𝑛ℤ,

noting that ${Hom}_{ℤ} (ℤ ∕ 𝑛ℤ, ℤ) = 0$ .

Definition 12.8.2. §

A right $R$ -module $N$ is $R$ -flat, or just flat, if the tensor product functor $t_{N} : R -mod \to 𝐀𝐛$ is exact.

Remarks 12.8.3. §

a.: An $S$ - $R$ -bimodule $N$ is flat as a right $R$ -module if and only if the functor $t_{N} : R -mod \to S -mod$ is exact.
b.: A left $R$ -module $M$ is defined to be flat if it is flat as a right $R^{op}$ -module (which is equivalent to the right tensor product functor with $M$ being exact on $R^{op} -mod$ ).

Proposition 12.8.4. §

Projective right $R$ -modules are $R$ -flat.

Proof.

Let $P$ be a projective $R$ -module, and let $Q$ be a complement in a free $R$ -module $F$ on a basis $X$ . Let $f : A \to B$ be an injection of $R$ -modules. We have a commutative diagram

Tensor products of a projective summand. A full diagram description follows.

Diagram description: Tensor products of a projective summand

The structural squares and triangles displayed here commute.

Objects, listed by row and column:

Row 1, from left to right: column 1: P tensor subscript (R) A; column 2: P tensor subscript (R) B.
Row 2, from left to right: column 1: F tensor subscript (R) A; column 2: F tensor subscript (R) B.
Row 3, from left to right: column 1: direct sum subscript (x in X) A; column 2: direct sum subscript (x in X) B.

Arrows and lines:

A hooked arrow from P tensor subscript (R) A to F tensor subscript (R) A, without a label.
An arrow from P tensor subscript (R) A to P tensor subscript (R) B, labelled id subscript (P) tensor f.
A hooked arrow from P tensor subscript (R) B to F tensor subscript (R) B, without a label.
An arrow from F tensor subscript (R) A to F tensor subscript (R) B, labelled id subscript (F) tensor f.
An arrow from F tensor subscript (R) A to direct sum subscript (x in X) A, labelled isomorphism symbol.
An arrow from F tensor subscript (R) B to direct sum subscript (x in X) B, labelled isomorphism symbol.
A hooked arrow from direct sum subscript (x in X) A to direct sum subscript (x in X) B, labelled (f) subscript (x in X).

the vertical isomorphisms following from the commutativity of direct sums and tensor products. Since $f$ is injective, so is the lowermost vertical map. Since $P$ is a direct sum of $F$ , the map

P \otimes_{R} A \to (P \otimes_{R} A) \oplus (Q \otimes_{R} A) \to_{}^{\sim} F \otimes_{R} A

is injective, and similarly with $A$ replaced by $B$ . Thus, commutativity of the diagram yields the injectivity ${id}_{P} \otimes f$ . Since left tensor product with $P$ preserves injective homomorphisms and right exact sequences, it preserves short exact sequences and is therefore exact. □

For modules over a principal ideal domain, we can characterize flat modules as follows.

Proposition 12.8.5. §

Let $R$ be a PID. An $R$ -module $M$ is flat if and only if $M$ is $R$ -torsion-free.

Proof.

Let $M$ be a flat $R$ -module. Let $r \in R$ be a nonzero element, and let $ϕ_{r} : R \to R$ be the injective map $ϕ_{r} (x) = 𝑟𝑥$ for $x \in R$ . The tensor product map ${id}_{M} \otimes_{R} ϕ_{r}$ is injective as $A$ is $R$ -flat. Under the identification $M \otimes_{R} 𝑅≅𝑀$ of Corollary 8.4.24, determined by $m \otimes r \mapsto 𝑟𝑚$ , the map ${id}_{M} \otimes_{R} ϕ_{r}$ becomes identified with the map $ψ_{r} : M \to M$ that is left multiplication by $r$ . Since $ψ_{r}$ is then injective for every nonzero $r$ , we see that $M$ has no nonzero $R$ -torsion.

Next, let $M$ be $R$ -torsion free. It is the union (which is also the direct limit) of its finitely generated, necessarily torsion-free $R$ -submodules. We omit here a check of the fact that direct limits and tensor products commute. Given this, we may assume that $M$ is finitely generated, in which case it follows from Proposition 12.8.5 that $M$ is free, hence projective, and hence flat. □

Remark 12.8.6. §

It follows from Corollary 8.9.3 and Proposition 12.8.5 that finitely generated flat modules over a PID $R$ are $R$ -free.

Definition 12.8.7. §

Let $R$ and $S$ be rings, and let $A$ be an $S$ - $R$ -bimodule. For $i \geq 0$ , the $i$ th $Tor$ -functor

{Tor}_{i}^{R} (A, \cdot) : R -mod \to S -mod

is the $i$ th left derived functor of $t_{A}$ .

Remark 12.8.8. §

If $R$ is a commutative ring, then an $R$ -module $A$ provides functors

{Tor}_{i}^{R} (A, \cdot) : R -mod \to R -mod

since $R$ -modules are automatically $R$ - $R$ -bimodules.

Remark 12.8.9. §

As ${Tor}_{i}^{R} (A, B) = H_{i} (A \otimes_{R} Q_{\cdot})$ for any projective resolution $Q_{\cdot}$ of $B$ by $R$ -modules, the composition of the functor

{Tor}_{i}^{R} (A, \cdot) : R -mod \to S -mod

with the forgetful functor $F : R -mod \to 𝐀𝐛$ agrees with the functor

{Tor}_{i}^{R} (F (A), \cdot) : R -mod \to 𝐀𝐛,

hence the omission of the notation for $S$ in the definition of ${Tor}_{i}^{R} (A, \cdot)$ .

Example 12.8.10. §

In $𝐀𝐛$ , consider the projective resolution

0 \to ℤ \to_{}^{n} ℤ \to ℤ ∕ 𝑛ℤ \to 0

of $B$ . Computing the homology of $0 \to A \to_{}^{n} A \to 0$ , we obtain

{Tor}_{i}^{ℤ} (A, ℤ ∕ 𝑛ℤ) ≅ {\begin{matrix} A ∕ 𝑛𝐴 & if i = 0 \\ A [n] = {a \in A ∣ 𝑛𝑎 = 0} & if i = 1 \\ 0 & if i \geq 2 . \end{matrix}

Lemma 12.8.11. §

Let $R$ be a ring. The following conditions on a right $R$ -module $A$ are equivalent:

i.: $A$ is flat,
ii.: ${Tor}_{1}^{R} (A, \cdot) = 0$ ,
iii.: ${Tor}_{i}^{R} (A, \cdot) = 0$ for all $i \geq 1$ .

Proof.

Clearly, (iii) implies (ii). If

0 \to B_{1} \to B_{2} \to B_{3} \to 0

is an exact sequence of right $R$ -modules, then we have a long exact sequence for any $R$ -module that ends with

{Tor}_{1}^{R} (A, B_{3}) \to A \otimes_{R} B_{1} \to A \otimes_{R} B_{2} \to A \otimes_{R} B_{3} \to 0,

from which it is clear that (ii) implies (i).

Finally, if (i) holds and $Q_{\cdot}$ is a projective resolution of $B$ in $R -mod$ , then the complex

\dots \to A \otimes_{R} Q_{1} \to A \otimes_{R} Q_{0} \to A \otimes_{R} B \to 0

is exact by the flatness of $A$ . It follows that

{Tor}_{i}^{R} (A, B) = H_{i} (A \otimes_{R} Q_{\cdot}) = 0

for all $i \geq 1$ . □

Proposition 12.8.12. §

Let $A$ be a right $R$ -module and $B$ a left $R$ -module. Let $P_{\cdot} \to A$ be a resolution of $A$ by projective right $R$ -modules. Then

{Tor}_{i}^{R} (A, B) ≅ H_{i} (P_{\cdot} \otimes_{R} B)

for all $i \geq 0$ . In particular, the functors ${Tor}_{i}^{R} (\cdot, B)$ are the left derived functors of $R$ -tensor product with $B$ .

Proof.

We sketch a proof. Form projective resolutions $P_{\cdot} \to A$ and $Q_{\cdot} \to B$ . We then have a double complex $P_{\cdot} \otimes_{R} Q_{\cdot}$ , and we can consider homology of the total complex

Tot {(P_{\cdot} \otimes_{R} Q_{\cdot})}_{k} = ⨁_{i + j = k} P_{i} \otimes_{R} Q_{j},

where the boundary maps from each term $P_{i} \otimes_{R} Q_{j}$ are given by the sums

d_{i}^{A} \otimes {id}_{Q_{j}} + {(- 1)}^{i} {id}_{P_{i}} \otimes d_{j}^{B} .

We claim that the homology of this chain complex is isomorphic to the homology of the complexes $P_{\cdot} \otimes_{R} B$ and $A \otimes_{R} Q_{\cdot}$ , from which the lemma follows.

We have maps of complexes

Tot (P_{\cdot} \otimes_{R} Q_{\cdot}) \to P_{\cdot} \otimes_{R} B

(12.8.1)

and

Tot (P_{\cdot} \otimes_{R} Q_{\cdot}) \to A \otimes_{R} Q_{\cdot}

(12.8.2)

induced by augmentation morphisms (up to sign, and zero maps otherwise). The double complex $P_{\cdot} \otimes_{R} Q_{\cdot} \to P_{\cdot} \otimes_{R} B$ (i.e., with $P_{i} \otimes_{R} B$ in the $(i, - 1)$ -position) has exact columns, since each projective module is flat. One can show that this implies that the total complex of this cpomplex is exact. This says precisely that the map in (12.8.1) induces an isomorphism on homology. Similarly, so does the map in (12.8.2). □

We have the following almost immediate corollary, since left and right tensor product with a module over a commutative ring are naturally isomorphic functors.

Corollary 12.8.13. §

Let $R$ be commutative. We have ${Tor}_{i}^{R} (A, B) ≅ {Tor}_{i}^{R} (B, A)$ for all $R$ -modules $A$ , $B$ and $i \geq 0$ .

We now give an alternate proof of Proposition 12.8.12.

Proof.

Let $Q_{\cdot} \to B$ be a projective resolution of $A$ by right $R$ -modules. Suppose that

0 \to A_{1} \to A_{2} \to A_{3} \to 0

is an exact sequence. Then

0 \to A_{1} \otimes_{R} Q_{\cdot} \to A_{2} \otimes_{R} Q_{\cdot} \to A_{3} \otimes_{R} Q_{\cdot} \to 0

is exact. This yields a long exact sequence in homology of the form

\dots \to {Tor}_{i}^{R} (A_{1}, B) \to {Tor}_{i}^{R} (A_{2}, B) \to {Tor}_{i}^{R} (A_{3}, B) \to {Tor}_{i - 1}^{R} (A_{1}, B) \to \dots,

so the functors ${Tor}_{i}^{R} (\cdot, B)$ do in fact form a $δ$ -functor. Futhermore, since any projective right $R$ -module $P$ is flat, we have that ${Tor}_{i}^{R} (P, B) = 0$ for all $i \geq 1$ . By Theorem 12.7.14, it follows that the ${Tor}_{i}^{R} (\cdot, B)$ are a universal $δ$ -functor extending $t_{B}$ . The proposition therefore follows by Theorem 12.7.11. □

Remark 12.8.14. §

It follows from Proposition 12.8.12 and Proposition 12.7.13 that the ${Tor}_{i}^{R} (A, B)$ can be computed via a flat resolution of either $A$ or $B$ .

The following explains something more of the name “Tor”.

Lemma 12.8.15. §

The functor ${Tor}_{1}^{ℤ} (A, \cdot) = 0$ if and only if $A$ is torsion-free.

Proof.

We prove this for finitely generated abelian groups. (The general result then follows from the fact that left derived functors commute with colimits.) By Proposition 12.8.13, we may compute ${Tor}_{1}^{ℤ} (A, B)$ by finding a projective resolution of $A$ . Say

𝐴≅ ℤ^{m} \oplus ℤ ∕ n_{1} ℤ \oplus \dots ℤ ∕ n_{r} ℤ

with $r \geq 0$ and the $n_{i} \geq 2$ . Then we have a projective resolution of the form

0 \to ℤ^{m + r} \to_{}^{(1, \dots, 1, n_{1}, \dots, n_{r})} ℤ^{m + r} \to A \to 0 .

Tensoring with $B$ and computing $H_{1}$ , we obtain $B [n_{1}] \oplus \dots \oplus B [n_{r}]$ . This will always be trivial if and only if $r = 0$ . □

By Lemma 12.8.15, a $ℤ$ -module is flat if and only if it is torsion-free. This is seen to hold in the same manner with $ℤ$ replaced by any PID. Note that this does not hold for all commutative rings.

Example 12.8.16. §

Consider $R = ℚ [x, y]$ . Then the exact sequence

0 \to R \to_{}^{(y, - x)} R^{2} \to_{}^{(a, b) \mapsto 𝑎𝑥 + 𝑏𝑦} R \to ℚ \to 0

is a free resolution of $ℚ$ . Let $J$ be the ideal $(x, y)$ of $R$ , so $ℚ≅𝑅 ∕ J$ . Then we have isomorphisms

{Tor}_{1}^{R} (J, ℚ) ≅ {Tor}_{2}^{R} (ℚ, ℚ) ≅ \ker (ℚ \to_{}^{0} ℚ^{2}) = ℚ .

Thus $J$ is not flat as an $R$ -module, even though it is torsion-free.

Here is another class of examples.

Lemma 12.8.17. §

Let $S$ be a subset of $R$ that is multiplicatively closed. Then the localization $S^{- 1} R$ is a flat $R$ -module.

Proof.

Recall that we have natural isomorphisms $S^{- 1} 𝐴≅ S^{- 1} R \otimes_{R} A$ for $R$ -modules $A$ . Suppose that $f : A \to B$ is an injection of $R$ -modules. Then we obtain an induced $R$ -module homomorphism $\tilde{f} : S^{- 1} A \to S^{- 1} B$ , which we must show is an injection. Suppose $\tilde{f} (s^{- 1} a) = 0$ . Then

0 = s \tilde{f} (s^{- 1} a) = \tilde{f} (a) = f (a),

so $a = 0$ . □

Definition 12.8.18. §

Let $R$ and $S$ be rings, and let $A$ be an $R$ - $S$ -bimodule. For $i \geq 0$ , the $i$ th $Ext$ -functor

{Ext}_{R}^{i} (A, \cdot) : R -mod \to S -mod

is the $i$ th right derived functors of $h_{A}$ .

Example 12.8.19. §

For $R = ℤ$ , we may consider the injective resolution

0 \to ℤ ∕ 𝑛ℤ \to ℚ ∕ ℤ \to_{}^{n} ℚ ∕ ℤ \to 0

of $ℤ ∕ 𝑛ℤ$ . For any abelian group $B$ , we write $B^{\lor} = Hom (B, ℚ ∕ ℤ)$ . We must compute the cohomology of $A^{\lor} \to_{}^{n} A^{\lor}$ . This yields

{Ext}_{ℤ}^{i} (A, ℤ ∕ 𝑛ℤ) ≅ {\begin{matrix} A^{\lor} [n] & if i = 0 \\ A^{\lor} ∕ n A^{\lor} & if i = 1 \\ 0 & if i = 2 . \end{matrix}

One has that ${Ext}_{R}^{i} (P, B) = 0$ for all $B$ and all $i \geq 1$ if $P$ is a projective module, as follows from the exactness of ${Hom}_{R} (P, \cdot)$ . We have the analogous result to Proposition 12.8.13 for Ext-groups, which says that such groups may be computed using projective resolutions.

Proposition 12.8.20. §

We have ${Ext}_{R}^{i} (A, B) ≅ H^{i} ({Hom}_{R} (P_{\cdot}, B))$ , where $P_{\cdot} \to A$ is any projective resolution of $A$ .

We end with a characterization of ${Ext}_{R}^{1}$ in terms of extensions.

Definition 12.8.21. §

An extension of an $R$ -module $A$ by an $R$ -module $B$ is an exact sequence $0 \to B \to E \to A \to 0$ , where $E$ is an $R$ -module. Two extensions of $A$ by $B$ are called equivalent if there is an isomorphism of exact sequences between them that is the identity on $A$ and $B$ .

Note that all split extensions (i.e., those with split exact sequences) are split.

Example 12.8.22. §

There are $p$ equivalence classes of extensions of $ℤ ∕ 𝑝ℤ$ by $ℤ ∕ 𝑝ℤ$ as $ℤ$ -modules:

0 \to ℤ ∕ 𝑝ℤ \to_{}^{𝑝𝑖} ℤ ∕ p^{2} ℤ \to_{}^{mod p} ℤ ∕ 𝑝ℤ \to 0

with $1 \leq i \leq p - 1$ , and

0 \to ℤ ∕ 𝑝ℤ \to ℤ ∕ 𝑝ℤ \oplus ℤ ∕ 𝑝ℤ \to ℤ ∕ 𝑝ℤ \to 0 .

Theorem 12.8.23. §

There is a one-to-one correspondence between equivalence classes of extensions of $A$ by $B$ and ${Ext}_{R}^{1} (A, B)$ .

Proof.

Suppose that $E$ is an equivalence class of extensions of $A$ by $B$ , represented by an exact sequence

0 \to B \to E \to A \to 0 .

(12.8.3)

We then have an exact sequence

{Hom}_{R} (E, B) \to {Hom}_{R} (B, B) \to_{}^{\partial_{E}} {Ext}_{R}^{1} (A, B),

and we set $Φ (E) = \partial_{E} ({id}_{B}) \in {Ext}_{R}^{1} (A, B)$ . This is clearly independent of the choice of representative.

Conversely, suppose $u \in {Ext}_{R}^{1} (A, B)$ . Fix an exact sequence

0 \to K \to_{}^{ι} P \to A \to 0

with $P$ projective. We then have an exact sequence

{Hom}_{R} (P, B) \to {Hom}_{R} (K, B) \to_{}^{\partial} {Ext}_{R}^{1} (A, B) \to 0 .

Let $t \in {Hom}_{R} (K, B)$ with $\partial (t) = u$ . Let $E$ be the pushout

E = P ∐_{K} B = P \oplus B ∕ {(ι (k), t (k)) ∣ k \in K} .

We have a commutative diagram

Constructing an extension by a pushout. A full diagram description follows.

Diagram description: Constructing an extension by a pushout

The two displayed rows are exact, and the squares commute.

Objects, listed by row and column:

Row 1, from left to right: column 1: 0; column 2: K; column 3: P; column 4: A; column 5: 0.
Row 2, from left to right: column 1: 0; column 2: B; column 3: E; column 4: A; column 5: 0.

Arrows and lines:

An arrow from 0 (row 1, column 1) to K, without a label.
An arrow from K to P, without a label.
An arrow from K to B, labelled t.
An arrow from P to A (row 1, column 4), without a label.
An arrow from P to E, without a label.
An arrow from A (row 1, column 4) to 0 (row 1, column 5), without a label.
Equality joins A (row 1, column 4) and A (row 2, column 4), without a label.
An arrow from 0 (row 2, column 1) to B, without a label.
An arrow from B to E, without a label.
An arrow from E to A (row 2, column 4), without a label.
An arrow from A (row 2, column 4) to 0 (row 2, column 5), without a label.

(12.8.4)

Here, the map $E \to A$ is defined by universality of the pushout (via the map $P \to A$ and the zero map $B \to A$ ). We define $Ψ (u)$ to be the equivalence class $E^{'}$ of the extension given by the lower row. Though it is not immediately clear that this is independent of the choice of $t$ with $\partial (t) = u$ , this follows if we can show that $Ψ$ and $Φ$ as constructed are mutually inverse.

To see that $Φ (Ψ (u)) = u$ , set $E = Ψ (u)$ , again choosing any $t$ with $\partial (t) = u$ . The diagram

Naturality of the extension connecting homomorphism. A full diagram description follows.

Diagram description: Naturality of the extension connecting homomorphism

The structural squares and triangles displayed here commute.

Objects, listed by row and column:

Row 1, from left to right: column 1: Hom subscript (R)(B,B); column 2: Ext subscript (R) superscript (1)(A,B).
Row 2, from left to right: column 1: Hom subscript (R)(K,B); column 2: Ext subscript (R) superscript (1)(A,B).

Arrows and lines:

An arrow from Hom subscript (R)(B,B) to Ext subscript (R) superscript (1)(A,B) (row 1, column 2), labelled partial subscript (script E).
An arrow from Hom subscript (R)(B,B) to Hom subscript (R)(K,B), labelled h superscript (B)(t).
Equality joins Ext subscript (R) superscript (1)(A,B) (row 1, column 2) and Ext subscript (R) superscript (1)(A,B) (row 2, column 2), without a label.
An arrow from Hom subscript (R)(K,B) to Ext subscript (R) superscript (1)(A,B) (row 2, column 2), labelled partial.

(12.8.5)

commutes. Hence, we have

Φ (Ψ (u)) = Φ (E) = \partial_{E} ({id}_{B}) = \partial (t) = u,

as desired.

On the other hand, suppose given $E$ with exact sequence (12.8.3). By projectivity of $P$ , the map $P \to A$ lifts to a map $P \to E$ . Hence, we have a diagram as in (12.8.4). Furthermore, the map $t$ in the diagram (12.8.4) satisfies $\partial (t) = \partial_{E} ({id}_{B})$ by the commutativity of (12.8.5). Now, there exists a map $P ∐_{K} B \to E$ by universality of the pushout, and it is the identity on $A$ and $B$ , hence an isomorphism by the $5$ -lemma. It follows by construction that

Ψ (Φ (E)) = Ψ (\partial_{E} ({id}_{B})) = E .

□

12.9. Group cohomology

In this section, we let $G$ denote a group.

Definition 12.9.1. §

The augmentation map $𝜀 : ℤ [G] \to ℤ$ is the unique ring homomorphism with $𝜀 (g) = 1$ for all $g \in G$ .

Definition 12.9.2. §

The augmentation ideal $I_{G}$ of $ℤ [G]$ is the kernel of the augmentation map.

Lemma 12.9.3. §

The augmentation ideal $I_{G}$ is generated by ${g - 1 ∣ g \in G}$ .

Proof.

We have

I_{G} = {\sum_{g \in G} a_{g} g \in ℤ [G] ∣ \sum_{g \in G} a_{g} = 0} .

For $α = \sum_{g \in G} a_{g} g \in I_{G}$ , we have

α = α - \sum_{g \in G} a_{g} = \sum_{g \in G} a_{g} (g - 1) .

□

Definition 12.9.4. §

Let $A$ be an $ℤ [G]$ -module.

a.

The $G$ -invariant group of $A$ is the $ℤ$ -module

A^{G} = {a \in A ∣ 𝑔𝑎 = a for all g \in G},

the maximal $ℤ [G]$ -submodule of $A$ on which all elements of $G$ act trivially.

b.

The $G$ -coinvariant group of $A$ is the $ℤ$ -module $A_{G} = A ∕ I_{G} A$ , the maximal $ℤ [G]$ -quotient of $A$ on which all elements of $G$ act trivially.

Examples 12.9.5. §

a.

If we view $ℤ$ as a $ℤ [G]$ -trivial module, we have $ℤ^{G} = ℤ$ and $ℤ_{G} ≅ℤ$ .

b.

We have $ℤ {[G]}_{G} ≅ℤ$ via the augmentation map, and

ℤ {[G]}^{G} = {\begin{matrix} ℤ \cdot N_{G} & if G is finite \\ 0 & otherwise, \end{matrix}

where $N_{G} = \sum_{g \in G} g$ is the norm element in a finite group $G$ . The computation of the invariant group follows from the fact that the action of $G$ on itself by left multiplication is transitive, so for an element of $ℤ [G]$ to be $G$ -fixed, its coefficients must all be equal.

c.

Let $K ∕ F$ be a finite Galois extension of fields, and let $G = Gal (K ∕ F)$ . Then $K^{G} = F$ and ${(K^{\times})}^{G} = F^{\times}$ .

Example 12.9.6. §

For $n \geq 2$ , let $S_{n}$ act on $𝒜 = ℤ [x_{1}, x_{2}, \dots, x_{n}]$ by

σ \cdot p (x_{1}, x_{2}, \dots, x_{n}) = p (x_{σ (1)}, x_{σ (2)}, \dots, x_{σ (n)})

for $σ \in S_{n}$ and $p \in 𝒜$ . This action is $ℤ$ -bilinear so it gives $𝒜$ the structure of a left $𝒜 [S_{n}]$ -module. Then $𝒜^{S_{n}}$ is the $ℤ$ -module of symmetric polynomials in $𝒜$ , which is the $ℤ$ -module generated by the elementary symmetric polynomials (see Definition 6.13.4). On the other hand, $𝒜_{S_{n}} ≅ℤ [x]$ , with the isomorphism induced by the $ℤ$ -linear map $𝒜 \to ℤ [x]$ taking each $x_{i}$ to $x$ .

Remark 12.9.7. §

We have a left exact invariant functor $A \mapsto A^{G}$ as a functor $ℤ [G] -mod \to 𝐀𝐛$ , with the map on homomorphisms being the restriction to invariant subgroups. This functor is naturally isomorphic to the functor $h_{ℤ}$ , where $ℤ$ is viewed as the trivial $ℤ [G]$ -module. In particular

η_{A} : {Hom}_{ℤ [G]} (ℤ, A) \to A^{G}, η_{A} (ϕ) = ϕ (1)

for $ϕ \in {Hom}_{ℤ [G]} (ℤ, A)$ is a natural isomorphism. Thus, the invariant factor is left exact.

Similarly, $A \mapsto A_{G}$ defines a right exact coinvariant functor which is isomorphic to the functor $t_{ℤ}$ , in that we have natural isomorphisms

ℤ \otimes_{ℤ [G]} A \to_{}^{\sim} A_{G}, 1 \otimes a \mapsto a + I_{G} A .

In particular, the coinvariant functor is right exact.

Definition 12.9.8. §

a.: The cohomology $H^{*} (G, \cdot)$ of $G$ is the $δ$ -functor given by the right derived functors of the $G$ -invariant functor. The $i$ th cohomology group of $G$ with coefficients in a $ℤ [G]$ -module $A$ is $H^{i} (G, A)$ .
b.: The homology $H_{*} (G, \cdot)$ of $G$ is the $δ$ -functor given by the left derived functor of the $G$ -coinvariant functor. The $i$ th homology group of $G$ with coefficients in a $ℤ [G]$ -module $A$ is $H_{i} (G, A)$ .

Remark 12.9.9. §

By definition, we have natural isomorphisms

\begin{array}{rcl} H^{i} (G, A) ≅ {Ext}_{ℤ [G]}^{i} (ℤ, A) & and & H_{i} (G, A) ≅ {Tor}_{i}^{ℤ [G]} (ℤ, A) \end{array}

for $i \geq 0$ and $ℤ [G]$ -modules $A$ .

Let us give a more explicit description of group cohomology.

Definition 12.9.10. §

The bar resolution of $ℤ$ as a $ℤ [G]$ -module is the complex $C_{\cdot}$ with $C_{i} = ℤ [G^{i + 1}]$ for $i \geq 0$ , differentials $d_{i} : C_{i} \to C_{i - 1}$ given on $(g_{0}, \dots, g_{i}) \in G^{i + 1}$ by

d_{i} ((g_{0}, \dots, g_{i})) = \sum_{j = 0}^{i} {(- 1)}^{j} (g_{0}, \dots, g_{j - 1}, g_{j + 1}, \dots, g_{i})

and augmentation $𝜀 : C_{0} \to ℤ$ the augmentation map.

Remark 12.9.11. §

As follows from Remark 12.9.9, the group $H^{i} (G, A)$ is the $i$ th cohomology group of the complex

\begin{matrix} 0 \to {Hom}_{ℤ [G]} (ℤ [G], A) \to_{}^{D^{0}} {Hom}_{ℤ [G]} (ℤ [G^{2}], A) \to \dots \\ \to {Hom}_{ℤ [G]} (ℤ [G^{i}], A) \to_{}^{D^{i - 1}} {Hom}_{ℤ [G]} (ℤ [G^{i + 1}], A) \to \dots \end{matrix}

with ${Hom}_{ℤ [G]} (ℤ [G], A) ≅ A$ in degree $0$ . Similarly, $H_{i} (G, A)$ is the $i$ th homology group of the complex

\dots \to ℤ [G^{i + 1}] \otimes_{ℤ [G]} A \to \dots \to ℤ [G^{2}] \otimes_{ℤ [G]} A \to ℤ [G] \otimes_{ℤ [G]} A \to 0,

with $ℤ [G] \otimes_{ℤ [G]} 𝐴≅𝐴$ in degree $0$ .

There is another complex which computes the cohomology of $G$ , that of the inhomogeneous $G$ -cocycles, which has a more complicated differential but is more amenable to computation.

Definition 12.9.12. §

Let $A$ be a $G$ -module, and let $i \geq 0$ .

a.: The group of $i$ -cochains of $G$ with coefficients in $A$ is the set of functions from $G^{i}$ to $A$ :
$C^{i} (G, A) = {f : G^{i} \to A} .$
b.: The $i$ th differential $d^{i} = d_{A}^{i} : C^{i} (G, A) \to C^{i + 1} (G, A)$ is the map
$\begin{matrix} d^{i} (f) (g_{0}, g_{1}, \dots, g_{i}) = g_{0} \cdot f (g_{1}, \dots g_{i}) \\ + \sum_{j = 1}^{i} {(- 1)}^{j} f (g_{0}, \dots, g_{j - 2}, g_{j - 1} g_{j}, g_{j + 1}, \dots, g_{i}) + {(- 1)}^{i + 1} f (g_{0}, \dots, g_{i - 1}) . \end{matrix}$

We remark that $C^{0} (G, A)$ is taken simply to be $A$ , as $G^{0}$ is a singleton set. The proof of the following, which tells us that $C^{\cdot} (G, A)$ is a cochain complex, is left to the reader.

Lemma 12.9.13. §

For any $i \geq 0$ , one has $d^{i + 1} \circ d^{i} = 0$ .

We consider the cohomology groups of $C^{\cdot} (G, A)$ .

Definition 12.9.14. §

Let $i \geq 0$ .

a.: We set $Z^{i} (G, A) = \ker d^{i}$ , the group of $i$ -cocycles of $G$ with coefficients in $A$ .
b.: We set $B^{0} (G, A) = 0$ and $B^{i} (G, A) = im d^{i - 1}$ for $i \geq 1$ . We refer to $B^{i} (G, A)$ as the group of $i$ -coboundaries of $G$ with coefficients in $A$ .

Theorem 12.9.15. §

The maps

ψ^{i} : {Hom}_{ℤ [G]} (ℤ [G^{i + 1}], A) \to C^{i} (G, A)

defined by

ψ^{i} (φ) (g_{1}, \dots, g_{i}) = φ (1, g_{1}, g_{1} g_{2}, \dots, g_{1} g_{2} \dots g_{i})

are isomorphisms for all $i \geq 0$ . This provides isomorphisms of complexes in the sense that $ψ^{i + 1} \circ D^{i} = d^{i} \circ ψ^{i}$ for all $i \geq 0$ . Moreover, these isomorphisms are natural in the $G$ -module $A$ .

Proof.

If $ψ^{i} (φ) = 0$ , then

φ (1, g_{1}, g_{1} g_{2}, \dots, g_{1} g_{2} \dots g_{i}) = 0

for all $g_{1}, \dots, g_{i} \in G$ . Let $h_{0}, \dots, h_{i} \in G$ , and define $g_{j} = h_{j - 1}^{- 1} h_{j}$ for all $1 \leq j \leq i$ . We then have

φ (h_{0}, h_{1}, \dots, h_{i}) = h_{0} φ (1, h_{0}^{- 1} h_{1}, \dots, h_{0}^{- 1} h_{i}) = h_{0} φ (1, g_{1}, \dots, g_{1} \dots g_{i}) = 0 .

Therefore, $ψ^{i}$ is injective. On the other hand, if $f \in C^{i} (G, A)$ , then defining

φ (h_{0}, h_{1}, \dots, h_{i}) = h_{0} f (h_{0}^{- 1} h_{1}, \dots, h_{i - 1}^{- 1} h_{i}),

we have

φ (g h_{0}, g h_{1}, \dots, g h_{i}) = g h_{0} f ({(g h_{0})}^{- 1} g h_{1}, \dots, {(g h_{i - 1})}^{- 1} g h_{i}) = 𝑔𝜑 (h_{0}, h_{1}, \dots, h_{i})

and $ψ^{i} (φ) = f$ . Therefore, $ψ^{i}$ is an isomorphism of groups.

That $ψ^{\cdot}$ forms a map of complexes is shown in the following computation:

\begin{array}{l} ψ^{i + 1} (D^{i} (φ)) (g_{1}, \dots, g_{i + 1}) & = D^{i} (φ) (1, g_{1}, \dots, g_{1} \dots g_{i + 1}) \\ = φ \circ d_{i + 1} (1, g_{1}, \dots, g_{1} \dots g_{i + 1}) \\ = \sum_{j = 0}^{i + 1} {(- 1)}^{j} φ (1, g_{1}, \dots, g_{1} \dots g_{j - 2}, g_{1} \dots g_{j}, \dots, g_{1} \dots g_{i + 1}) . \end{array}

The latter term equals

\begin{matrix} g_{1} ψ^{i} (φ) (g_{2}, \dots, g_{i + 1}) + \sum_{j = 1}^{i} {(- 1)}^{j} ψ^{i} (φ) (g_{1}, \dots, g_{j - 2}, g_{j - 1} g_{j}, g_{j + 1}, \dots, g_{i + 1}) \\ + {(- 1)}^{i + 1} ψ^{i} (φ) (g_{1}, \dots, g_{i}), \end{matrix}

which is $d^{i} (ψ^{i} (φ))$ .

Finally, suppose that $α : A \to B$ is a $G$ -module homomorphism. We then have

α \circ ψ^{i} (φ) (g_{1}, \dots, g_{i}) = α \circ φ (1, g_{1}, \dots, g_{1} \dots g_{i}) = ψ^{i} (α \circ φ) (g_{1}, \dots, g_{i}),

hence the desired naturality. □

Corollary 12.9.16. §

The $i$ th cohomology group of the complex $({Hom}_{ℤ [G]} (ℤ [G^{i + 1}], A), D_{A}^{i})$ is naturally isomorphic to $H^{i} (G, A)$ .

Corollary 12.9.17. §

The $i$ th cohomology group of $G$ with coefficients in $A$ is

H^{i} (G, A) = Z^{i} (G, A) ∕ B^{i} (G, A) .

The cohomology groups measure how far the cochain complex $C^{\cdot} (G, A)$ is from being exact. We give some examples of cohomology groups in low degree.

Lemma 12.9.18. §

We have

Z^{1} (G, A) = {f : G \to A ∣ f (𝑔h) = 𝑔𝑓 (h) + f (g) for all g, h \in G}

and $B^{1} (G, A)$ is the subgroup of $f : G \to A$ for which there exists $a \in A$ such that $f (g) = 𝑔𝑎 - a$ for all $g \in G$ . In particular, if $A$ is a $ℤ [G]$ -module with trivial $G$ -action, then $H^{1} (G, A) = Hom (G, A)$ .

Proof.

Let $a \in A$ . Then $d^{0} (a) (g) = 𝑔𝑎 - a$ for $g \in G$ , so $\ker d^{0} = A^{G}$ . That proves part a, and part b is simply a rewriting of the definitions. Part c follows immediately, as the definition of $Z^{1} (G, A)$ reduces to $Hom (G, A)$ , and $B^{1} (G, A)$ is clearly $(0)$ , in this case. □

We remark that, as $A$ is abelian, we have $Hom (G, A) = Hom (G^{ab}, A)$ , where $G^{ab}$ is the maximal abelian quotient of $G$ (i.e., its abelianization).

We turn briefly to an interesting use for second cohomology groups.

Definition 12.9.19. §

A group extension of $G$ by a $G$ -module $A$ is a short exact sequence of groups

0 \to A \to_{}^{ι} E \to_{}^{π} G \to 1

such that, choosing any section $s : G \to E$ of $π$ , one has

s (g) 𝑎𝑠 {(g)}^{- 1} = g \cdot a

for all $g \in G$ , $a \in A$ . Two such extensions $E \to E^{'}$ are said to be equivalent if there is an isomorphism $𝜃 : E \to_{}^{\sim} E^{'}$ fitting into a commutative diagram

Equivalence of group extensions. A full diagram description follows.

Diagram description: Equivalence of group extensions

The two displayed rows are exact, and the squares commute.

Objects, listed by row and column:

Row 1, from left to right: column 1: 0; column 2: A; column 3: script E; column 4: G; column 5: 0.
Row 2, from left to right: column 1: 0; column 2: A; column 3: script E prime; column 4: G; column 5: 0.

Arrows and lines:

An arrow from 0 (row 1, column 1) to A (row 1, column 2), without a label.
An arrow from A (row 1, column 2) to script E, without a label.
Equality joins A (row 1, column 2) and A (row 2, column 2), without a label.
An arrow from script E to G (row 1, column 4), without a label.
An arrow from script E to script E prime, labelled theta.
An arrow from G (row 1, column 4) to 0 (row 1, column 5), without a label.
Equality joins G (row 1, column 4) and G (row 2, column 4), without a label.
An arrow from 0 (row 2, column 1) to A (row 2, column 2), without a label.
An arrow from A (row 2, column 2) to script E prime, without a label.
An arrow from script E prime to G (row 2, column 4), without a label.
An arrow from G (row 2, column 4) to 0 (row 2, column 5), without a label.

We denote the set of equivalence classes of such extensions by $E (G, A)$ .

Definition 12.9.20. §

A factor set of a group $G$ valued in a $ℤ [G]$ -module $A$ is a $2$ -cocycle $f : G^{2} \to A$ satisfying $f (1, g) = f (g, 1) = 0$ for all $g \in G$ .

Lemma 12.9.21. §

Every $2$ -cocycle of a group is cohomologous to, i.e., has the same cohomology class as, a factor set.

Proof.

The condition that $F : G^{2} \to A$ is a $2$ -cocycle is that

𝑔𝐹 (h, k) + F (g, h𝑘) = F (𝑔h, k) + F (g, h)

for all $g, h, k \in G$ . In particular, taking $g = h = e$ , we have $F (e, k) = F (e, e)$ and taking $h = k = e$ , we have $𝑔𝐹 (e, e) = 𝑓𝐹𝑔, e)$ . Note that for a $1$ -cochain $c$ , we have

𝑑𝑐 (g, h) = 𝑔𝑐 (h) - c (𝑔h) + c (g) .

In particular, if we set $c (g) = c$ for all $g \in G$ some fixed $c \in A$ , then $𝑑𝑐 (g, h) = 𝑔𝑐$ for all $g \in G$ , so if we take $c = F (e, e)$ and replace $F$ by $f = F - 𝑑𝑐$ , then $f (e, k) = 0$ and $f (g, e) = 0$ for all $g, k \in G$ . □

Theorem 12.9.22. §

The group $H^{2} (G, A)$ is in canonical bijection with $E (G, A)$ via the map induced by that taking a factor set $f : G^{2} \to A$ to the extension $E_{f} = A \times G$ with multiplication given by

(a, g) \cdot (b, h) = (a + 𝑔𝑏 + f (g, h), 𝑔h)

This identification takes the identity to the semi-direct product $A ⋉ G$ determined by the action of $G$ on $A$ .

Proof.

We check that $E_{f}$ so defined is a group. That it has identity $(0, e)$ is clear from the definition. Associativity is as follows:

\begin{matrix} (a + 𝑔𝑏 + f (g, h), 𝑔h) \cdot (c, k) = (a + 𝑔𝑏 + f (g, h) + 𝑔h𝑐 + f (𝑔h, k), 𝑔h𝑘) \\ = (a + g (b + h𝑐) + 𝑔𝑓 (h, k) + f (g, h𝑘), 𝑔h𝑘) = (a, g) \cdot (b + h𝑐 + f (h, k), h𝑘) . \end{matrix}

The inverse of $(a, g)$ clearly has the form $(b, g^{- 1})$ for some $b \in A$ , and we then must have $a + 𝑔𝑏 + f (g, g^{- 1}) = 0$ , so $b = - g^{- 1} a - g^{- 1} f (g, g^{- 1})$ , and the inverse exists. That $E_{f}$ is a group extension of $G$ by $A$ is now nearly immediate. Note also that $E_{f}$ is split if $f = 0$ , ]since in that case $(0, g) (0, h) = (0, 𝑔h)$ , so $G$ is a subgroup.

Let $c$ be a $1$ -cochain with $𝑑𝑐 (e, g) = 𝑑𝑐 (g, e) = 0$ , the latter property occurring if and only if $c (e) = 0$ . Consider the map $ψ_{f, c} : E_{f} \to E_{f + 𝑑𝑐}$ given by

ψ_{f, c} (a, g) = (a - c (g), g) .

We have

\begin{matrix} ψ_{f, c} (a + 𝑔𝑏 + f (g, h), 𝑔h) = (a + 𝑔𝑏 + f (g, h) - c (𝑔h), 𝑔h) \\ = (a + 𝑔𝑏 + f (g, h) + 𝑑𝑐 (g, h) - 𝑔𝑐 (h) + c (g), 𝑔h) = ψ_{c} (a, g) ψ_{c} (a, h), \end{matrix}

so $ψ_{c}$ is a homomorphism, and it is clearly has inverse $ψ_{f + 𝑑𝑐, - c}$ . Thus, we have a well-defined map from $H^{2} (G, A)$ to $E (G, A)$ .

It remains to construct an inverse, which we sketch as the computations all follow from what we have already done. Given a group extension, we indeed always have a $2$ -cochain $f (g, h) : G^{2} \to A$ defining the multiplication. We claim that $f$ is a factor set. For this, associativity again tells us that $f$ is a $2$ -cocycle, and the fact that $(0, e)$ is a two-sided identity forces $f (e, g) = f (g, e) = 0$ for all $g \in G$ . The resulting association is clearly inverse on the level of extensions and cochains. If $𝜃 : E \to E^{'}$ is an isomorphism of group extensions of $G$ by $A$ , then $𝜃 (0, g) = (- c (g), g)$ for some $c : G \to A$ that has the property that if the factor set is associated to $E$ is $𝑑𝑐$ plus the factor set associated to $E^{'}$ . □

Remark 12.9.23. §

Theorem 12.9.22 tells us that $E (G, A)$ also has a group structure, which may also be given an explicit description. Given $E$ and $E^{'}$ extensions of $G$ by $A$ , their product is

E * E^{'} = (E \times_{G} E^{'}) ∕ ⟨ (a, - a) ∣ a \in A ⟩ .

This product is known as the Baer sum of the two extensions.

Let’s give a group-theoretic application of this description of $H^{2} (G, A)$ .

Proposition 12.9.24 (Schur). §

Let $G$ be a group of order $𝑚𝑛$ , where $m$ and $n$ are relatively prime positive integers. Then every abelian normal subgroup of order $n$ has a complement in $G$ of order $m$ .

Proof.

Let $N$ be an abelian normal subgroup of $G$ , and set $H = G ∕ N$ . Let $f : G^{2} \to N$ be a factor set corresponding to $G$ as an extension of $H$ by $N$ by Theorem 12.9.22. For every $h \in H$ , let

t (h) = \prod_{k \in H} f (h, k),

which makes sense as $H$ is abelian. For $h, h^{'} \in H$ , we have

\prod_{k \in H} f (h, h^{'} k) = \prod_{k \in H} f (h, k) = t (h) .

Now

f {(h, h^{'})}^{m} t (h h^{'}) = \prod_{k \in H} f {(h, h^{'})}^{m} f (h h^{'}, k) = \prod_{k \in H} h𝑓 (h^{'}, k) h^{- 1} f (h, h^{'} k) = h𝑡 (h^{'}) h^{- 1} \cdot t (h) .

Let $a, b \in ℤ$ be such that $𝑎𝑚 + 𝑏𝑛 = 1$ . We then have

f (h, h^{'}) = f {(h, h^{'})}^{𝑎𝑚 + 𝑏𝑛} = 𝑑𝑡 {(h, h^{'})}^{a} .

Thus, $f$ is a coboundary, so $G$ is a split extension by Theorem 12.9.22. In particular, it contains a subgroup of order $n$ , isomorphic to $H$ . □

Remark 12.9.25. §

Though we do not prove it, we have $H^{i} (G, A) = 0$ for all $i \geq 1$ whenever $G$ and $A$ are finite of relatively prime order. In fact, for any finite group $G$ and $G$ -module $A$ , the exponent of $H^{i} (G, A)$ divides the order of $G$ .

Definition 12.9.26. §

A Hall subgroup of a finite group is a subgroup with relatively prime order and index.

We can extend Proposition 12.9.24 from abelian to arbitrary normal subgroups.

Theorem 12.9.27 (Schur-Zassenhaus). §

Every normal Hall subgroup of a finite group has a complement.

Proof.

Let $N$ be a normal Hall subgroup of $G$ of order $n$ and index $m$ . If $N$ is abelian, then the result follows from Proposition 12.9.24. Suppose the result holds true in the case of normal subgroups of order less than $n \geq 2$ . Let $p$ be a prime dividing $n$ . Let $P$ be a Sylow $p$ -subgroup of $G$ . Then $𝑃𝑁 ∕ N$ has $p$ -power order dividing $m$ . Since $m$ and $n$ are relatively prime, this forces $P$ to be contained in $N$ . In other words, the Sylow $p$ -subgroups of $N$ and $G$ are the same. Now

[G : N_{G} (P)] = n_{p} (G) = n_{p} (N) = [N : N_{N} (P)],

so $[N_{G} (P) : N_{N} (P)] = m$ . On the other hand, $N_{N} (P) ∕ P$ has order prime to $m$ and less than $n$ , being properly contained in $N ∕ P$ . Furthermore, $N_{N} (P) = N \cap N_{G} (P)$ is normal in $N_{G} (P)$ . By induction on $n$ , we have that there exists a subgroup $K$ of $N_{G} (P)$ with $K ∕ P$ isomorphic to $N_{G} (P) ∕ N_{N} (P)$ and $| K ∕ P | = m$ .

Since $P$ is a $p$ -group, its center $Z = Z (P)$ is nontrivial. It is also a characteristic subgroup of $P$ , so it is normal in $K$ . By induction, $P ∕ Z$ has a has a complement in $K ∕ Z$ , equal to $H ∕ Z$ for some subgroup $H$ of $K$ , which necessarily has order $m$ . This group $H$ is the desired complement to $N$ . □

12.10. Galois cohomology

We briefly consider the cohomology of finite Galois extensions. We have the following generalization of Hilbert’s Theorem 90, which also has the same name.

Theorem 12.10.1 (Hilbert’s Theorem 90). §

Let $L ∕ K$ be a finite Galois extension with Galois group $G$ . Then $H^{1} (G, K^{\times}) = 0$ .

Proof.

Let $f : G \to L^{\times}$ be a $1$ -cocycle. We view the elements $σ \in G$ as abelian characters $L^{\times} \to L^{\times}$ . As distinct characters of $L^{\times}$ , these characters form a linearly independent set. The sum $\sum_{σ \in G} f (σ) σ$ is therefore a nonzero map $L^{\times} \to L$ . Let $α \in L^{\times}$ be such that $z = \sum_{σ \in G} f (σ) σ (α) \neq 0$ . For any $τ \in G$ , we have

\begin{matrix} τ^{- 1} (z) = \sum_{σ \in G} τ^{- 1} (f (σ)) \cdot τ^{- 1} σ (α) = \sum_{σ \in G} τ^{- 1} (f (𝜏𝜎)) σ (α) \\ = \sum_{σ \in G} τ^{- 1} (f (τ) \cdot 𝜏𝑓 (σ)) σ (α) = τ^{- 1} (f (τ)) \sum_{σ \in G} f (σ) σ (α) = τ^{- 1} (f (τ)) z . \end{matrix}

Thus,

f (τ) = \frac{z}{τ (z)},

so $f$ is the $1$ -coboundary of $z^{- 1}$ . □

To see how this implies Hilbert’s theorem 90 in the case of finite cyclic extensions, we prove the following result on the cohomology of cyclic groups.

Proposition 12.10.2. §

Let $G$ be a finite cyclic group and $A$ be a $ℤ [G]$ -module. Then for $i \geq 1$ , we have

H^{i} (G, A) ≅ {\begin{matrix} A^{G} ∕ N_{G} A & if i is even, \\ A [N_{G}] ∕ I_{G} A & if i is odd, \end{matrix}

where $A [N_{G}]$ is the kernel of multiplication by $N_{G}$ on $A$ .

Proof.

Let $g$ be a generator of $G$ , and consider the augmented resolution of $ℤ$ given by

\dots \to ℤ [G] \to_{}^{N_{G}} ℤ [G] \to_{}^{g - 1} ℤ [G] \to_{}^{N_{G}} ℤ [G] \to_{}^{g - 1} ℤ [G] \to_{}^{𝜖} ℤ \to 0,

where $𝜖$ is the augmentation maps. Note that ${Hom}_{ℤ [G]} (ℤ [G], A) ≅ A$ by evaluation at $1$ , and the map $N_{G}$ (resp., $g - 1$ ) on $ℤ [G]$ induces $N_{G}$ (resp., $g - 1$ ) on $A$ via these isomorphisms. The groups $H^{i} (G, A)$ are then the cohomology groups of the complex

A \to_{}^{g - 1} A \to_{}^{N_{G}} A \to_{}^{g - 1} A \to \dots,

which have the desired form. □

Remark 12.10.3. §

Suppose that $L ∕ K$ is finite cyclic with Galois group $G$ having generator $σ$ . Proposition 12.10.2 implies that

H^{1} (G, K^{\times}) ≅ \frac{\ker N_{L ∕ K}}{{σ (α) ∕ α ∣ α \in L^{\times}}},

which is trivial by Theorem 12.10.1. This is exactly the statement of Hilbert’s Theorem 90 for finite cyclic extensions.

We next see how we can use Galois cohomology to study Kummer theory.

Proposition 12.10.4. §

Suppose that $L ∕ K$ is a finite extension with Galois group $G$ . Let $n$ be a positive integer not divisible by the characteristic of $K$ . Then there is an isomorphism

H^{1} (G, μ_{n}) ≅ \frac{K^{\times} \cap L^{\times n}}{K^{\times n}},

where the class of an element $a \in K^{\times} \cap L^{\times n}$ in the quotient corresponds to the class of the cocycle

χ_{a} (σ) = \frac{σ (α)}{α},

where $α \in L^{\times}$ with $α^{n} = a$

Proof.

The short exact sequence

1 \to μ_{n} (L) \to L^{\times} \to_{}^{n} L^{\times n} \to 1

of $G$ -modules gives rise to a long exact sequence

0 \to μ_{n} (K) \to K^{\times} \to_{}^{n} K^{\times} \cap L^{\times n} \to_{}^{\partial} H^{1} (G, μ_{n}) \to 0,

where Hilbert’s Theorem 90 gives the final equality. That $a \in K^{\times} \cap L^{\times n}$ is sent to the class of $χ_{a}$ follows from the definition of the connecting homomorphism by the Snake lemma. □

This leads to the following definition.

Definition 12.10.5. §

Let $n$ be a positive integer not divisible by the characteristic of $K$ . For $a \in K^{\times}$ , a Kummer cocycle attached to $K$ is a map $χ_{a} : G_{K} \to μ_{n}$ given by

χ_{a} (σ) = \frac{σ (α)}{α},

where $α \in {(K^{sep})}^{\times}$ with $α^{n} = a$ .

Remark 12.10.6. §

The Kummer cocycle $χ_{a}$ in Definition 12.10.5 actually depends on the choice of $n$ th root of $a$ up to a $1$ -coboundary of an element of $μ_{n}$ . If $μ_{n} \subseteq K$ , however, it is unique and is the Kummer character of $a$ . In this case, Proposition 12.10.4 reduces to

Hom (G, μ_{n}) ≅ Δ ∕ K^{\times n},

where $Δ = K^{\times} \cap L^{\times n}$ . This in turn yields the perfect pairing of Kummer duality.

We next turn to the question of the structure of $H^{2} (G, L^{\times})$ for a finite Galois extension $L ∕ K$ with Galois group $G$ . We fix such an extension $L ∕ K$ with Galois group $G$ in what follows.

Definition 12.10.7. §

A central simple algebra over a field $K$ is a simple $K$ -algebra with center equal to $K$ .

Example 12.10.8. §

Any matrix algebra $M_{n} (D)$ over a division algebra $D$ is a central simple algebra over the center $Z (D)$ , which is a field. For instance, if $ℍ$ denotes the ring of quaternions, then $M_{n} (ℍ)$ is a central simple $ℝ$ -algebra.

Proposition 12.10.9. §

Let $f \in Z^{2} (G, L^{\times})$ be a factor set. Let $B_{f}$ be an $L$ -vector space with basis $b_{σ}$ for $g \in G$ . Define a multiplication on $B_{f}$ as the unique binary operation extending the scalar multiplication $L \times B_{f} \to B_{f}$ and satisfying

\begin{array}{rcl} b_{σ} α = σ (α) b_{σ} & and & b_{σ} b_{τ} = f (σ, τ) b_{𝜎𝜏} . \end{array}

for $σ, τ \in G$ and $α \in L$ . Then $B_{f}$ is a central simple $K$ -algebra with identity $b_{1}$ .

Proof.

We have

\begin{matrix} (b_{σ} b_{τ}) b_{ρ} = f (σ, τ) (b_{𝜎𝜏} b_{ρ}) = f (σ, τ) f (𝜎𝜏, ρ) b_{𝜎𝜏𝜌} = σ (f (τ, ρ)) f (σ, 𝜏𝜌) b_{𝜎𝜏𝜌} \\ = σ (f (τ, ρ)) (b_{σ} b_{𝜏𝜌}) = b_{σ} (f (𝜏𝜌) b_{𝜏𝜌}) = b_{σ} (b_{τ} b_{ρ}) . \end{matrix}

It follows that $B_{f}$ is an associative $L$ -algebra with $K$ in its center. Note that $b_{1} = 1$ in the ring $B_{f}$ since $f (σ, 1) = f (1, σ) = 1$ for all $σ \in G$ .

Let $β \in L^{\times}$ generate $L ∕ K$ . Let

z = \sum_{σ \in G} α_{σ} b_{σ} \in Z (G) .

. Then $𝑧𝛽 = 𝛽𝑧$ , so

\sum_{σ \in G} (α_{σ} σ (β) - β α_{σ}) b_{g} = 0,

and therefore $σ (β) = β$ for all $σ \in G$ with $α_{σ} = 0$ . Since $β$ is a generator of $L ∕ K$ , this forces $α_{σ} = 0$ for all $σ \neq 1$ , so $z = α_{1} \in K$ . Thus, $Z (B_{f}) = K$ .

Next, let $I$ be a nonzero ideal of $B_{f}$ , and let $x \in B_{f}$ be an element with a minimal number $k$ of nonzero coefficients in its expression as an $L$ -linear combination of elements of $G$ . If $σ$ and $τ$ are distinct elements of $G$ for which $x$ has nonzero coefficients, then $σ (β) \neq τ (β)$ . Then $x - τ (β) x β^{- 1} \in I$ , but its $b_{τ}$ -coefficient is now zero, while its $b_{σ}$ -coefficient is not, and it has no nonzero coefficients that $x$ does not have. This contradicts the minimality of $k$ , forcing it to be $1$ . Thus, $x = α b_{σ}$ for some $α \in L^{\times}$ and $σ \in G$ . But such an $x$ is a unit in $B_{f}$ , so $I = B_{f}$ . Thus, $B_{f}$ is a simple ring. □

Definition 12.10.10. §

For a factor set $f : G \to L^{\times}$ , the $K$ -algebra $B_{f}$ of Proposition 12.10.9 is the crossed product algebra of $f$ .