Homological Algebra — Chapter 4 Spectral Sequences

Chapter 4
Spectral Sequences

4.1. Spectral sequences

Definition 4.1.1. §

A (cohomological) spectral sequence ${(E_{r}^{p, q}, d_{r}^{p, q})}_{r \geq r_{0}}$ starting at $r = r_{0}$ in an abelian category $𝒞$ consists of a nonnegative integer $r_{0}$ , and for each $r \geq r_{0}$ and $p, q \in ℤ$ ,

i.: an object $E_{r}^{p, q}$ of $𝒞$
ii.: a morphism $d_{r}^{p, q} : E_{r}^{p, q} \to E_{r}^{p + r, q - r + 1}$ , and
iii.: an isomorphism
$\frac{\ker (d_{r}^{p, q})}{im (d_{r}^{p - r, q + r - 1})} \to_{}^{\sim} E_{r + 1}^{p, q}$

such that we have $d_{r}^{p + r, q - r + 1} \circ d_{r}^{p, q} = 0$ for all $r$ , $p$ , and $q$ .

Definition 4.1.2. §

Let $𝔼 = {(E_{r}^{p, q}, d_{r}^{p, q})}_{r \geq r_{0}}$ be a spectral sequence starting at $r = r_{0}$ . Fix $(p, q) \in ℤ^{2}$ and $r \geq r_{0}$ .

a.: The object $E_{r}^{p, q}$ is called the $(p, q)$ -term in the $r$ th layer of $𝔼$ .
b.: The morphism $d_{r}^{p, q}$ is called the $r$ th differential at $(p, q)$ .
c.: The degree of the $(p, q)$ -term $E_{r}^{p, q}$ is $p + q$ .

As one might expect, there exists a category of spectral sequences in $𝒞$ . We define a morphism between spectral sequences as follows.

Definition 4.1.3. §

A morphism of spectral sequences $ψ : 𝔼 \to 𝔻$ with $𝔼 = {(E_{r}^{p, q}, d_{r}^{p, q})}_{r \geq r_{0}}$ and $𝔻 = {(D_{r}^{p, q}, δ_{r}^{p, q})}_{r \geq s_{0}}$ spectral sequences in $𝒞$ is a family of morphisms

ψ_{r}^{p, q} : E_{r}^{p, q} \to D_{r}^{p, q}

in $𝒞$ for sufficiently large $r$ such that

ψ_{r}^{p + r, q - r + 1} \circ d_{r}^{p, q} = δ_{r}^{p, q} \circ ψ_{r}^{p, q}

and $ψ_{r + 1}^{p, q}$ is the map induced by $ψ_{r}^{p, q}$ on subquotients for all $r$ , $p$ , and $q$ .

The reader will check the following.

Lemma 4.1.4. §

Let $ψ : 𝔼 \to 𝔻$ be a morphism of spectral sequences such that there exists an integer $r_{1}$ for which the morphisms $ψ_{r_{1}}^{p, q}$ are all isomorphisms. Then the morphisms $ψ_{r}^{p, q}$ are isomorphisms for all $r \geq r_{1}$ .

Throughout the remainder of this section $𝔼 = {(E_{r}^{p, q}, d_{r}^{p, q})}_{r \geq r_{0}}$ will denote a spectral sequence starting at $r = r_{0}$ in an abelian category $𝒞$ .

Remark 4.1.5. §

Given $p, q \in ℤ$ , the object $E_{r}^{p, q}$ is for any $r \geq r_{0}$ a subquotient of $E_{r_{0}}^{p, q}$ , equal to $Z_{r}^{p, q} ∕ B_{r}^{p, q}$ for subobjects $B_{r}^{p, q} \subseteq Z_{r}^{p, q}$ of $E_{r_{0}}^{p, q}$ that satisfy

\begin{array}{rcl} B_{r}^{p, q} \subseteq B_{r + 1}^{p, q} & and & Z_{r + 1}^{p, q} \subseteq Z_{r}^{p, q} \end{array}

for each $r \geq r_{0}$ .

Notation 4.1.6. §

We refer to $B_{r}^{p, q}$ and $Z_{r}^{p, q}$ as the $r$ -coboundaries and $r$ -cocycles in the $(p, q)$ -term of $𝔼$ .

Definition 4.1.7. §

For a spectral sequence $𝔼$ , we set

\begin{array}{rcl} B_{\infty}^{p, q} = ⋃_{r \geq r_{0}} B_{r}^{p, q} & and & Z_{\infty}^{p, q} = ⋂_{r \geq r_{0}} Z_{r}^{p, q} \end{array}

if these colimits and limits exist, in which case we set $E_{\infty}^{p, q} = Z_{\infty}^{p, q} ∕ B_{\infty}^{p, q}$ .

Terminology 4.1.8. §

When they exist, the objects $B_{\infty}^{p, q}$ , $Z_{\infty}^{p, q}$ , and $E_{\infty}^{p, q}$ are respectively called the limit coboundaries, cocycles, and term at $(p, q)$ .

Remark 4.1.9. §

If $𝒞$ is a complete category, then $B_{\infty}^{p, q}$ exists. If $𝒞$ is a cocomplete category, then $Z_{\infty}^{p, q}$ exists.

Often, we have that for each pair $(p, q)$ , there exists $r \geq r_{0}$ such that $B_{r}^{p, q} = B_{\infty}^{p, q}$ and $Z_{r}^{p, q} = Z_{\infty}^{p, q}$ . As an example, we have the first quadrant spectral sequences.

Definition 4.1.10. §

We say that $𝔼$ is a $n$ th quadrant spectral sequence, for $1 \leq n \leq 4$ , if $E_{r_{0}}^{p, q} = 0$ for all $(p, q) \in ℤ^{2}$ lying outside of the closed $n$ th quadrant of the plane $ℝ^{2}$ .

Lemma 4.1.11. §

Let $𝔼$ be a first or third quadrant spectral sequence. Then for each $(p, q) \in ℤ^{2}$ , there exists an $r_{1} \geq r_{0}$ such that $d_{r}^{p, q} = d_{r}^{p - r, q + r - 1} = 0$ for all $r \geq r_{1}$ .

Proof.

One need only take $r_{1} = \max {| p | + 1, | q | + 2}$ . □

Remark 4.1.12. §

That $d_{r}^{p, q} = d_{r}^{p - r, q + r - 1} = 0$ says exactly that

\begin{array}{rcl} \ker (d_{r}^{p, q}) = E_{r}^{p, q} & and & im (d_{r}^{p - r, q + r - 1}) = 0, \end{array}

which is to say that the morphism given by property (iii) of a spectral sequence is an isomorphism $E_{r}^{p, q} \to_{}^{\sim} E_{r + 1}^{p, q}$ . In other words, if this holds for all $r \geq r_{1}$ , then $E_{r_{1}}^{p, q}$ is $E_{\infty}^{p, q}$ .

Often, something even stronger occurs.

Definition 4.1.13. §

We say that a spectral sequence $𝔼$ degenerates at $r_{0}$ if $d_{r}^{p, q} = 0$ for all $p, q \in ℤ$ and $r \geq r_{0}$ .

We now define the notion of convergence of a spectral sequence.

Definition 4.1.14. §

Let ${(E^{n})}_{n \in ℤ}$ be a collection of objects in $𝒞$ such that each $E^{n}$ is endowed with a nonincreasing filtration ${(F^{p} E^{n})}_{p \in ℤ}$ by subobjects that satisfy $F^{p} E^{n} = 0$ for sufficiently small $p$ and $F^{p} E^{n} = E^{n}$ for sufficiently large $p$ . For each pair $(p, q) \in ℤ^{2}$ , let us set

{gr}_{p} E^{p + q} = F^{p} E^{p + q} ∕ F^{p + 1} E^{p + q} .

We say that the spectral sequence $𝔼$ converges to $(E^{n})$ and write

E_{r_{0}}^{p, q} \Rightarrow E^{p + q}

if the $E_{\infty}^{p, q}$ exist and there are isomorphisms

α^{p, q} : E_{\infty}^{p, q} \to_{}^{\sim} {gr}_{p} E^{p + q}

for each pair $(p, q) \in ℤ^{2}$ .

Remark 4.1.15. §

There exist definitions of convergence that weaken the conditions that the graded terms in the filtration vanish in sufficiently low and high degrees. We do not treat those here.

Remark 4.1.16. §

In a first quadrant spectral sequence $𝔼$ , each term $E_{r + 1}^{p, 0}$ for $r \geq r_{0}$ is a quotient of the term $E_{r}^{p, 0}$ and each term $E_{r + 1}^{0, q}$ is a subobject of $E_{r}^{0, q}$ . So, if $E_{r_{0}}^{p, q} \Rightarrow E^{p + q}$ , we have epimorphisms $E_{r}^{p, 0} \to E^{p}$ and monomorphisms $E^{q} \to E_{r}^{0, q}$ induced by $α^{p, 0}$ and ${(α^{q, 0})}^{- 1}$ .

Terminology 4.1.17. §

Let $𝔼$ be a first quadrant spectral sequence.

a.

The terms $E_{r}^{p, 0}$ and $E_{r}^{0, q}$ are known as edge terms.

b.

If $E_{r_{0}}^{p, q} \Rightarrow E^{p + q}$ , then the morphisms

\begin{array}{rcl} E_{r}^{p, 0} \to E^{p} & and & E^{q} \to E_{r}^{0, q} \end{array}

are known as edge maps (or morphisms).

Lemma 4.1.18. §

Let $𝔼$ be a first quadrant spectral sequence starting at $r_{0} \leq 2$ and converging to ${(E^{n})}_{n \in ℤ}$ . Then there is an exact sequence

0 \to E_{2}^{1, 0} \to E^{1} \to E_{2}^{0, 1} \to_{}^{d_{2}^{0, 1}} E_{2}^{2, 0} \to E^{2},

in which the maps not labelled are edge maps.

Proof.

We must have $F^{0} E^{n} = E^{n}$ and $F^{n + 1} E^{n} = 0$ for each $n \geq 0$ and $E^{n} = 0$ for $n < 0$ since $𝔼$ is a first quadrant spectral sequence. Note that $E_{2}^{1, 0} = E_{\infty}^{1, 0}$ , and $E_{3}^{p, q} = E_{\infty}^{p, q}$ for $(p, q) = (0, 1)$ and $(p, q) = (2, 0)$ . The result is then almost immediate from the definition of the edge maps, since we have an exact sequence

0 \to F^{1} E^{1} \to E^{1} \to {gr}^{0} E^{1} \to 0

with $E_{2}^{1, 0} \to_{}^{\sim} F^{1} E^{1}$ and ${gr}^{0} E^{1} \to_{}^{\sim} \ker d_{2}^{0, 1}$ the isomorphisms induced by edge maps. □

Terminology 4.1.19. §

The exact sequence of Lemma 4.1.18 is known as the exact sequence of edge terms of $𝔼$ .

4.2. Filtrations on complexes

In this section, we let $(A^{\cdot}, d^{\cdot})$ be a cochain complex in an abelian category $𝒞$ equipped with a nonincreasing filtration of subcomplexes

\dots \subseteq F^{p + 1} A^{\cdot} \subseteq F^{p} A^{\cdot} \subseteq F^{p - 1} A^{\cdot} \subseteq \dots

such that the differentials on each $F^{p} A^{\cdot}$ are the restrictions of the differentials on $A^{\cdot}$ . Note that $F^{p}$ induces a filtration on the homology of $A^{\cdot}$ .

Definition 4.2.1. §

a.: A filtered cochain complex is a pair, denoted $(A^{\cdot}, F^{\cdot})$ , consisting of a cochain complex $(A^{\cdot}, d^{\cdot})$ and a nonincreasing filtration $F_{p} A^{\cdot}$ of $A^{\cdot}$ by subcomplexes $(F^{p} A^{\cdot}, F^{p} d^{\cdot})$ for $p \in ℤ$ .
b.: We say that a filtered cochain complex is bounded if for each $i \in ℤ$ , there exist $t < s \in ℤ$ such that $F^{s} A^{i} = 0$ and $F^{t} A^{i} = A^{i}$ .
c.: The $p$ th graded piece in the filtration is the complex $({gr}_{p} A^{\cdot}, {gr}_{p} d^{\cdot})$ , where the differentials are induced by the $d^{\cdot}$ via the projection morphism of complexes $π^{p, \cdot} : F^{p} A^{\cdot} \to {gr}_{p} A^{\cdot}$ .

Notation 4.2.2. §

Let $(A^{\cdot}, F^{\cdot})$ be a filtered cochain complex. Let $(p, q) \in ℤ^{2}$ and $r \geq 0$ .

a.: We set ${\tilde{Z}}_{0}^{p, q} = F^{p} A^{p + q}$ and ${\tilde{B}}_{0}^{p, q} = 0$ , and if $r \geq 1$ , we set
$\begin{array}{rcl} {\tilde{Z}}_{r}^{p, q} = {(d^{p + q})}^{- 1} (F^{p + r} A^{p + q + 1}) \cap F^{p} A^{p + q} & and & {\tilde{B}}_{r}^{p, q} = d^{p + q - 1} ({\tilde{Z}}_{r - 1}^{p - r + 1, q + r - 2}) \subseteq {\tilde{Z}}_{r}^{p, q} . \end{array}$
b.: We set
$\begin{array}{rcl} Z_{r}^{p, q} = π^{p, q} ({\tilde{Z}}_{r}^{p, q}) \subseteq {gr}_{p} A^{p + q} & and & B_{r}^{p, q} = π^{p, q} ({\tilde{B}}_{r}^{p, q}) \subseteq Z_{r}^{p, q} . \end{array}$
c.: We set $E_{r}^{p, q} = Z_{r}^{p, q} ∕ B_{r}^{p, q}$ .

Theorem 4.2.3. §

The collection $𝔼 (A^{\cdot}) = (E_{r}^{p, q}, d_{r}^{p, q})$ attacted to a filtered complex $(A^{\cdot}, F^{\cdot})$ , where

d_{r}^{p, q} : E_{r}^{p, q} \to E_{r}^{p + r, q - r + 1}

is induced by ${gr}_{p} d^{p + q}$ , is a spectral sequence. If the filtration is bounded, then $𝔼 (A^{\cdot})$ converges to $H^{p + q} (A^{\cdot})$ .

Proof.

Note that

d^{p + q} (F^{p} A^{p + q}) \subseteq F^{p} A^{p + q + 1},

so the differential $d_{0}^{p, q} : E_{0}^{p, q} \to E_{0}^{p, q + 1}$ is well defined. Moreover, we have that

d^{p + q} ({\tilde{Z}}_{r}^{p, q}) \subseteq F_{p} A^{p + r, q - r + 1} \cap \ker d^{p + q + 1} \subseteq {\tilde{Z}}_{r}^{p + r, q - r + 1}

and $d^{p + q} ({\tilde{B}}_{r}^{p, q}) = 0$ , so $d_{r}^{p, q}$ is well-defined for all $r \geq 0$ .

Since

{\tilde{Z}}_{r - 1}^{p + 1, q - 1} = {\tilde{Z}}_{r}^{p, q} \cap F^{p + 1} A^{p + q},

we have

E_{r}^{p, q} = \frac{Z_{r}^{p, q}}{B_{r}^{p, q}} ≅ \frac{{\tilde{Z}}_{r}^{p, q}}{{\tilde{B}}_{r}^{p, q} + {\tilde{Z}}_{r - 1}^{p + 1, q - 1}} .

From this, the reader may check that

\ker d_{r}^{p, q} ≅ \frac{{(d^{p + q})}^{- 1} ({\tilde{B}}_{r}^{p + r, q - r + 1} + {\tilde{Z}}_{r - 1}^{p + r + 1, q - r}) \cap {\tilde{Z}}_{r}^{p, q}}{{\tilde{B}}_{r}^{p, q} + {\tilde{Z}}_{r - 1}^{p + 1, q - 1}} ≅ \frac{{\tilde{Z}}_{r + 1}^{p, q} + {\tilde{Z}}_{r - 1}^{p + 1, q - 1}}{{\tilde{B}}_{r}^{p, q} + {\tilde{Z}}_{r - 1}^{p + 1, q - 1}} ≅ \frac{Z_{r + 1}^{p, q}}{B_{r}^{p, q}} .

and

im d_{r}^{p - r, q + r - 1} ≅ \frac{{\tilde{B}}_{r + 1}^{p, q} + {\tilde{Z}}_{r - 1}^{p + 1, q - 1}}{{\tilde{B}}_{r}^{p, q} + {\tilde{Z}}_{r - 1}^{p + 1, q - 1}} ≅ \frac{B_{r + 1}^{p, q}}{B_{r}^{p, q}} .

Therefore, we have

E_{r + 1}^{p, q} = \frac{Z_{r + 1}^{p, q}}{B_{r + 1}^{p, q}} ≅ \frac{\ker d_{r}^{p, q}}{im d_{r}^{p - r, q + r - 1}} .

Therefore, taken together with these isomorphisms, $𝔼 (A^{\cdot})$ forms a spectral sequence.

Now suppose that the filtration is bounded. Then the $E_{r}^{p, q}$ are successive subquotients which eventually stabilize for sufficiently large $r$ , with the ${\tilde{Z}}_{r}^{p, q}$ stabilizing to

{\tilde{Z}}_{\infty}^{p, q} = \ker d^{p + q} \cap F^{p} A^{p + q}

and the

{\tilde{B}}_{r}^{p, q} = d^{p + q - 1} ({(d^{p + q - 1})}^{- 1} (F^{p} A^{p + q}) \cap F^{p - r + 1} A^{p + q - 1})

stabilizing to

{\tilde{B}}_{\infty}^{p, q} = im d^{p + q - 1} \cap F^{p} A^{p + q},

with $Z_{\infty}^{p, q} = π^{p, q} ({\tilde{Z}}_{\infty}^{p, q})$ and $B_{\infty}^{p, q} = π^{p, q} ({\tilde{B}}_{\infty}^{p, q})$ . Thus, we have

{gr}^{p} H^{p + q} (A^{\cdot}) ≅ \frac{F^{p} H^{p + q} (A^{\cdot})}{F^{p + 1} H^{p + q} (A^{\cdot})} ≅ \frac{\ker d^{p + q} \cap F^{p} A^{p + q}}{(\ker d^{p + q} \cap F^{p + 1} A^{p + q}) + (im d^{p + q - 1} \cap F^{p} A^{p + q})} ≅ \frac{Z_{\infty}^{p, q}}{B_{\infty}^{p, q}} = E_{\infty}^{p, q} .

□

We next consider a setting in which filtrations on complexes naturally arise.

Definition 4.2.4. §

Let $C^{⋅⋅}$ be a double (cochain) complex in $𝒞$ , and let $n \in ℤ$ .

a.: Let $^{I} σ^{\geq n} C$ be the double subcomplex of $C$ with
$^{I} σ^{\geq n} C^{i, j} = {\begin{matrix} C^{i, j} & if i \geq n \\ 0 & if i < n . \end{matrix}$
b.: Let $^{𝐼𝐼} σ^{\geq n} C$ be the double subcomplex of $C$ with
$^{𝐼𝐼} σ^{\geq n} C^{i, j} = {\begin{matrix} C^{i, j} & if j \geq n \\ 0 & if j < n . \end{matrix}$

Notation 4.2.5. §

Let $C^{⋅⋅}$ be a double (cochain) complex, let $A^{\cdot} = {Tot}^{\oplus} (C)$ , and let $p \in ℤ$ .

a.: Let $^{I} F^{p} A^{\cdot}$ be the subcomplex of $A^{\cdot}$ that is the total (sum) complex of $^{I} σ^{\geq p} C^{i, j}$ .
b.: Let $^{𝐼𝐼} F^{p} A^{\cdot}$ be the subcomplex of $A^{\cdot}$ that is the total (sum) complex of $^{𝐼𝐼} σ^{\geq p} C^{i, j}$ .

We may make the analogous definitions with the total product complex $A^{\cdot} = {Tot}^{Π} (C)$ .

Remark 4.2.6. §

If $C^{⋅⋅}$ is a first quadrant cochain complex, then the filtrations $^{I} F^{p}$ and $^{𝐼𝐼} F^{p}$ on $A = Tot (C)$ are bounded. That is, for each $n$ , we have that $F^{n + 1} A^{n} = 0$ , while we always have that $F^{0} A^{n} = A^{n}$ .

Notation 4.2.7. §

Let $C^{⋅⋅}$ be a first quadrant double (cochain) complex. We let $^{I} 𝔼 (C)$ (resp., $^{𝐼𝐼} 𝔼 (C)$ , with terms $^{I} E_{r}^{p, q}$ and $^{𝐼𝐼} E_{r}^{p, q}$ , respectively) denote the spectral sequence attached to the filtration $^{I} F^{p}$ (resp., $^{𝐼𝐼} F^{p}$ ) on ${Tot}^{\oplus} (C)$ by Theorem 4.2.3.

Remark 4.2.8. §

The spectral sequences $^{I} 𝔼 (C)$ and $^{𝐼𝐼} 𝔼 (C)$ both converge to $H^{p + q} (Tot (C))$ . We have $^{I} E_{0}^{p, q} = C^{p, q}$ and $^{I} d_{0}^{p, q} = d_{v}^{p, q}$ , so

^{I} E_{1}^{p, q} = H^{q} (C^{p, \cdot}),

and $^{I} d_{1}^{p, q}$ is induced by $d_{h}^{p, q}$ , so we have

^{I} E_{2}^{p, q} ≅ H_{h}^{p} (H_{v}^{q} (C)) \Rightarrow H^{p + q} (Tot (C)) .

On the other hand, we have $^{𝐼𝐼} E_{0}^{p, q} = C^{q, p}$ and $^{𝐼𝐼} d_{0}^{p, q} = d_{h}^{q, p}$ , and the maps $^{𝐼𝐼} d_{1}^{p, q}$ are induced by the $d_{v}^{q, p}$ . We therefore have

^{𝐼𝐼} E_{2}^{p, q} ≅ H_{v}^{p} (H_{h}^{q} (C)) \Rightarrow H^{p + q} (Tot (C)) .

We can often play these spectral seqeunces off of each other to obtain interesting results. For instance, we may shed new light on our proof of Proposition 3.5.9.

Remark 4.2.9. §

Let $Λ$ be a ring, $A$ be a right $Λ$ -module, and $B$ be a left $Λ$ -module. Let $P_{\cdot} \to A$ be a projective resolution of $A$ by right $Λ$ -modules, and let $Q_{\cdot} \to B$ be a projective resolution of $B$ by left $Λ$ -modules. We form the double complex $P_{\cdot} \otimes_{Λ} Q_{\cdot}$ and consider the resulting (homological) first quadrant spectral sequence with $E^{1}$ -terms

^{I} E_{p, q}^{1} ≅ H_{q} (P_{p} \otimes_{Λ} Q_{\cdot}) ≅ P_{p} \otimes_{Λ} H_{q} (Q_{\cdot}) ≅ {\begin{matrix} P_{p} \otimes_{Λ} B & if q = 0 \\ 0 & if q \geq 1 . \end{matrix}

We then have that the spectral sequences degenerates at $r = 2$ , with

^{I} E_{p, q}^{\infty} =^{I} E_{p, q}^{2} ≅ {\begin{matrix} H_{p} (P_{\cdot} \otimes_{Λ} B) & if q = 0 \\ 0 & if q \geq 1 . \end{matrix}

On the other hand, if we consider the spectral sequence $^{𝐼𝐼} E_{p, q}^{r}$ , we similarly obtain

^{𝐼𝐼} E_{p, q}^{\infty} =^{𝐼𝐼} E_{p, q}^{2} ≅ {\begin{matrix} H_{p} (A \otimes_{Λ} Q_{\cdot}) & if q = 0 \\ 0 & if q \geq 1, \end{matrix}

and both $^{I} E_{p, 0}^{\infty}$ and $^{𝐼𝐼} E_{p, 0}^{\infty}$ are then isomorphic $H_{p} (Tot (P_{\cdot} \otimes_{Λ} Q_{\cdot}))$ . Now, note that

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

by definition. So, we have a reinterpretation of the proof of the assertion

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

of Proposition 3.5.9.

We can generalize this as follows.

Theorem 4.2.10 (Künneth spectral sequence). §

Let $M_{\cdot}$ be a bounded below chain complex of flat right $Λ$ -modules, and let $B$ be a $Λ$ -module. Then there is a convergent spectral sequence

E_{p, q}^{2} = {Tor}_{p}^{Λ} (H_{q} (M_{\cdot}), B) \Rightarrow H_{p + q} (M_{\cdot} \otimes_{Λ} B) .

Proof.

We take a projective resolution $Q_{\cdot} \to B$ of $B$ by left $Λ$ -modules and form the double complex $M_{\cdot} \otimes_{Λ} Q_{\cdot}$ . Since the terms of $M_{\cdot}$ are flat and $H_{q} (Q_{\cdot}) = 0$ for $q \geq 1$ , we obtain

^{I} E_{p, q}^{\infty} =^{I} E_{p, q}^{2} ≅ {\begin{matrix} H_{p} (M_{\cdot} \otimes_{Λ} B) & if q = 0 \\ 0 & if q \geq 1, \end{matrix}

which then implies that

H_{p} (M_{\cdot} \otimes_{Λ} B) ≅ H_{p + q} (Tot (M_{\cdot} \otimes_{Λ} Q_{\cdot})) .

Since the terms of $Q_{\cdot}$ are flat as well, we have

^{𝐼𝐼} E_{p, q}^{2} = H_{p} (H_{q} (M_{\cdot}) \otimes_{Λ} B) ≅ {Tor}_{p}^{Λ} (H_{q} (M_{\cdot}), B),

which finishes the proof. □

Remark 4.2.11. §

If $M_{\cdot}$ is a projective resolution of a right $Λ$ -module $A$ , the spectral sequence degenerates at $r = 2$ , and the resulting isomorphisms are again simply those of Proposition 3.5.9.

4.3. Grothendieck spectral sequences

Let $𝒞$ be an abelian category. Recall that an injective object in $𝐂𝐡 (𝒞)$ is a split exact sequence of injective objects in $𝒞$ . However, unless a complex $A^{\cdot}$ of objects in $𝒞$ is bounded below, it is not clear that there will exists an injective object $I^{\cdot}$ of $𝐂𝐡 (𝒞)$ and a monomorphism $A^{\cdot} \to I^{\cdot}$ . Therefore, we not be able to find injective resolutions of unbounded complexes. However, we do always have the following substitute.

Definition 4.3.1. §

Let $𝒞$ be an abelian category. A Cartan-Eilenberg resolution of a cochain complex $A^{\cdot}$ is a resolution $(I^{⋅⋅}, 𝜖^{\cdot})$ of $A^{\cdot}$ in $𝐂𝐡 (𝒞)$ , where $𝜖^{i} : A^{i} \to I^{i, 0}$ is a morphism in $𝒞$ , such that

i.

each $I^{i, j}$ for $i, j \in ℤ$ is injective,

ii.

$I^{i, \cdot} = 0$ for each $i \in ℤ$ with $A^{i} = 0$ ,

iii.

each of the objects

\begin{array}{rcl} B_{h}^{i, j} (I) = im d_{h}^{i - 1, j} & and & H_{h}^{i, j} (I) = \ker d_{h}^{i, j} ∕ im d_{h}^{i - 1, j} \end{array}

is injective, and

iv.

each of the augmented complexes

im d_{A}^{i - 1} \to B_{h}^{i, \cdot} (I)

and

H^{i} (A) \to H_{h}^{i, \cdot} (I),

with the morphisms induced by the $d_{v}^{i, j}$ and the augmentation maps by $𝜖^{i}$ , is exact.

Remark 4.3.2. §

If $(I^{⋅⋅}, 𝜖^{\cdot})$ is a Cartan-Eilenberg resolution of a complex $A^{\cdot}$ , then the augmented complex $\ker d^{i} \to Z_{h}^{i, \cdot} (I)$ , with $Z_{h}^{i, j} (I) = \ker d_{h}^{i, j}$ , is an injective resolution.

Remark 4.3.3. §

If $A^{\cdot}$ is concentrated in degree $0$ , then a Cartan-Eilenberg resolution of $A^{\cdot}$ consists of a double complex concentrated in the $0$ th column, an injective resolution of $A^{0}$ .

The following is a clever application of the Horseshoe lemma.

Proposition 4.3.4. §

Let $𝒞$ be an abelian categories, and suppose that $𝒞$ has enough injectives. Then every cochain complex in $𝒞$ has a Cartan-Eilenberg resolution.

Proof.

Set $Z^{i} = \ker d_{A}^{i}$ , $B^{i} = im d_{A}^{i + 1}$ , and $H^{i} = H^{i} (A)$ . We consider $Z^{\cdot}$ , $B^{\cdot}$ , and $H^{\cdot}$ as complexes with zero differentials, since the $d_{A}^{i}$ induce trivial maps on their $i$ th terms. Choose injective resolutions $ι_{B}^{i} : B^{i} \to I_{B}^{i, \cdot}$ and $ι_{H}^{i} : H^{i} \to I_{H}^{i, \cdot}$ for each $i$ , taking $I_{B}^{i} = I_{H}^{i} = 0$ if $B^{i} = 0$ or $I^{i} = 0$ . These form double complexes $I_{B}^{⋅⋅}$ and $I_{H}^{⋅⋅}$ with zero horizontal differentials. Set $I_{Z}^{\cdot, j} = I_{B}^{\cdot, j} \oplus I_{H}^{\cdot, j}$ for each $j \geq 0$ , and then apply the Horseshoe lemma to obtain the vertical differentials that give the augmented complex $Z^{\cdot} \to I_{Z}^{⋅,⋅}$ (again with zero horizontal differentials). Finally, apply the Horseshoe lemma to the exact sequence

0 \to Z^{\cdot} \to A^{\cdot} \to B^{\cdot} \to 0

and the augmented complexes $Z^{\cdot} \to I_{Z}^{⋅⋅}$ and $B^{\cdot} \to I_{B}^{⋅⋅}$ to obtain the Cartan-Eilenberg resolution. □

The use of Cartan-Eilenberg resolutions is seen in the following lemmas.

Lemma 4.3.5. §

Let $A^{\cdot}$ be a complex and $(I^{⋅⋅}, 𝜖^{\cdot})$ a Cartan-Eilenberg resolution of $A^{\cdot}$ . Suppose that either $A^{\cdot}$ is bounded below or $𝒞$ admits direct products. Then the induced augmentation morphism $A^{\cdot} \to {Tot}^{Π} (I^{⋅⋅})$ is a quasi-isomorphism. In particular, every bounded below complex admits a quasi-isomorphism to a bounded below complex of injective objects.

Proof.

Consider the augmented complex $B^{⋅⋅}$ attached to $A^{\cdot} \to I^{⋅⋅}$ (with $A^{\cdot}$ in the row of degree $- 1$ ). Then ${Tot}^{Π} (B^{⋅⋅})$ is exact by Proposition 2.8.10. Since the latter total complex is isomorphic to the cone of the morphism $A^{\cdot} \to {Tot}^{Π} (I^{⋅⋅})$ , we have the result. □

Lemma 4.3.6. §

Let $f^{\cdot} : A^{\cdot} \to B^{\cdot}$ be a morphism of cochain complexes in $𝒞$ , and let $A^{\cdot} \to I^{⋅⋅}$ and $B^{\cdot} \to J^{⋅⋅}$ be Cartan-Eilenberg resolutions.

a.: There is a morphism $α^{⋅⋅} : I^{⋅⋅} \to J^{⋅⋅}$ such that the pair $(f^{\cdot}, α)$ is a morphism of the augmented complexes, and any two such morphisms are chain homotopic (as morphisms of double complexes).
b.: If $f^{\cdot} = {id}_{A^{\cdot}}$ , then the morphism $α^{⋅⋅}$ is a homotopy equivalence, as is the induced morphism ${Tot}^{Π} (I^{⋅⋅}) \to {Tot}^{Π} (J^{⋅⋅})$ .

Let $𝒞$ be an abelian category. We will discuss cochain complexes, resolutions by injectives, and cohomological spectral sequences, though everything can be done for chain complexes, resolutions by projectives and homological spectral sequences.

Definition 4.3.7. §

Let $F : 𝒞 \to 𝒟$ be a left exact functor of abelian categories, and suppose that $𝒞$ has enough injectives. For any cochain complex $A^{\cdot}$ in $𝒞$ nonnegative degrees, we define the $i$ th right hyper-derived functor

ℝ^{i} F : 𝐂𝐡 (𝒞) \to 𝒟

ℝ^{i} F (A^{\cdot}) ≅ H^{i} (F (C^{\cdot})),

where $C^{\cdot}$ is the total complex of a Cartan-Eilenberg resolution of $A^{\cdot}$ in $𝐂𝐡 (𝒞)$ .

Proposition 4.3.8. §

Let $F : 𝒞 \to 𝒟$ be a left exact functor of abelian categories, and suppose that $𝒞$ has enough injectives. Then there is a natural isomorphism of $δ$ -functors between the restrictions $ℝ^{i} F : {𝐂𝐡}^{\geq 0} (𝒞) \to 𝒟$ of the right derived hyper-functors of $F$ and the right derived functors of the left exact functors $H^{0} \circ F$ , where $H^{0} : {𝐂𝐡}^{\geq 0} (𝒟) \to 𝒟$ is the $0$ th cohomology functor.

Proposition 4.3.9. §

Let $F : 𝒞 \to 𝒟$ be a left exact functor of abelian categories, and suppose that $𝒞$ has enough injectives. Let $A^{\cdot}$ be an object of ${𝐂𝐡}^{\geq 0} (𝒞)$ . Then we have first quadrant convergent spectral sequences

^{I} E_{2}^{p, q} (A^{\cdot}) = H^{p} (R^{q} F (A^{\cdot})) \Rightarrow ℝ^{p + q} F (A^{\cdot}) .

and

^{𝐼𝐼} E_{2}^{p, q} (A^{\cdot}) = (R^{p} F) (H^{q} (A^{\cdot})) \Rightarrow ℝ^{p + q} F (A^{\cdot}) .

Proof.

These are simply the spectral sequences $^{I} 𝔼 (F (I^{⋅⋅}))$ and $^{𝐼𝐼} 𝔼 (F (I^{⋅⋅}))$ , where $A^{\cdot} \to I^{⋅⋅}$ is a Cartan-Eilenberg resolution. As explained in Remark 4.2.8, these spectral sequences both converge to $H^{n} (Tot (F (I^{⋅⋅})))$ , and we have that they satisfy

^{I} E_{2}^{p, q} (A^{\cdot}) ≅ H_{h}^{p} (H_{v}^{q} (F (I^{⋅⋅}))) ≅ H^{p} (R^{q} F (A^{\cdot}))

and

^{𝐼𝐼} E_{2}^{p, q} (A^{\cdot}) ≅ H_{v}^{p} (H_{h}^{q} (F (I^{⋅⋅}))) ≅ H_{v}^{p} (F (H_{h}^{q} (I^{⋅⋅}))) ≅ R^{p} F (H^{q} (A^{\cdot})) .

□

We are now able to construct Grothendieck spectral sequences.

Theorem 4.3.10 (Grothendieck). §

Let $B$ , $𝒞$ , and $𝒟$ be abelian categories such that both $B$ and $𝒞$ have enough injectives. Let $F : 𝒞 \to 𝒟$ and $G : B \to 𝒞$ be left exact functors of abelian categories, and suppose that $G$ sends injective objects in $B$ to $F$ -acyclic objects in $𝒞$ . Let $B$ be an object of $B$ . Then there exists a first quadrant convergent cohomological spectral sequence

E_{2}^{p, q} = (R^{p} F) (R^{q} G) (B) \Rightarrow R^{p + q} (F \circ G) (B),

and this construction is natural in $B$ .

Proof.

Let $B \to I^{\cdot}$ be an injective resolution. Then $G (I^{\cdot})$ is an object of ${𝐂𝐡}^{\geq 0} (𝒞)$ . Consider the hyper-derived functors $ℝ^{i} F$ applied to $G (I^{\cdot})$ . We have convergent spectral sequences as in Proposition 4.3.9 with $E_{2}$ -terms:

^{I} E_{2}^{p, q} = H^{p} ((R^{q} F) (G (I^{\cdot})))

and

^{𝐼𝐼} E_{2}^{p, q} = R^{p} F (H^{q} (G (I^{\cdot}))) ≅ (R^{p} F) (R^{q} G) (B) .

Since $G (I^{\cdot})$ is $F$ -acyclic, the spectral sequence has $^{I} E_{2}^{p, q} = 0$ for $q > 0$ and

^{I} E_{2}^{p, 0} = H^{p} (F (G (I^{\cdot}))) ≅ R^{p} (F \circ G) (B)

so degenerates at $r = 2$ and therefore converges to the sequence of $R^{n} (F \circ G) (B)$ . It follows immediately that the second spectral sequence converges to the sequence of $R^{n} (F \circ G) (B)$ as well, which is what we required. □

Remark 4.3.11. §

In view of Proposition 4.3.8, the spectral sequence $^{𝐼𝐼} E_{2}^{p, q} (A^{\cdot})$ of Proposition 4.3.9 is the Grothendieck spectral sequence for the functors $H^{0}$ and $F$ . Note that $F$ takes injective objects in ${𝐂𝐡}^{\geq 0} (𝒞)$ , which are exact complexes of injectives in nonnegative degrees, to complexes in ${𝐂𝐡}^{\geq 0} (𝒟)$ that are exact in degree $0$ by left exactness of $F$ . Of course, the complexes that are exact in degree $0$ are exactly the $H^{0}$ -acyclic complexes.

We give an extremely useful example.

Theorem 4.3.12 (Hochschild-Serre). §

Let $G$ be a group, $N$ a normal subgroup, and $A$ a $ℤ [G]$ -module. Then there is a first quadrant convergent cohomological spectral sequence

E_{2}^{p, q} = H^{p} (G ∕ N, H^{q} (N, A)) \Rightarrow H^{p + q} (G, A) .

Proof.

We consider the functors $ℤ [N] -mod \to 𝐀𝐛$ and $ℤ [G] -mod \to ℤ [G ∕ N] -mod$ given by taking $(G ∕ N)$ -invariants and $N$ -invariants, respectively. We claim that the $N$ -invariant functor preserves injectives. The Hochschild-Serre spectral sequence is then simply the Grothendieck spectral sequence for these two functors.

So, let $B$ be a $ℤ [G]$ -module and $A$ a $ℤ [G ∕ N]$ -module, and note that we have isomorphisms

{Hom}_{ℤ [G ∕ N]} (A, B^{N}) ≅ {Hom}_{ℤ [G]} (A, B)

natural in $A$ and $B$ . In other words, the (additive) $N$ -invariant functor is right adjoint to the forgetful functor, which is exact. Therefore, Proposition 3.4.2 yields the claim. □