STEENROD OPERATIONS IN ALGEBRAIC GEOMETRY

ALEXANDER MERKURJEV

1. Introduction

Let p be a prime integer. For a pair of topological spaces A ⊂ X we write H^{i}(X, A; ℤ/pℤ) for the i-th singular cohomology group with coefficients in ℤ/pℤ. A cohomological operation of degree r is a collection of group homomorphisms

H^{i}(X, A; ℤ/pℤ) → H^{i+r}(X, A; ℤ/pℤ)

satisfying certain naturality conditions. In particular, they commute with the pull-back homomorphisms.

The operations form the Steenrod algebra A_{p} modulo p. It is generated by the reduced power operations P^{k}, k ≥ 1, of degree 2(p − 1)k and the Bockstein operation of degree 1.

The idea of the definition of P^{k} is as follows. Modulo p, the p-th power operation α ↦ α^{p} is additive. It can be described as follows. Let d : X → X^{p} be the diagonal embedding. The composition

H^{∗}(X; ℤ/pℤ) → H^{∗}(X^{p}; ℤ/pℤ) ⟶^{d^{∗}} H^{∗}(X; ℤ/pℤ),

where the first map is the p-th exterior power, takes a class α to the power α^{p}. Note that the composition is a homomorphism, although the first map is not.

Let G be the symmetric group S_{p}. It acts on X^{p} and we can define the composition

H^{∗}(X; ℤ/pℤ) → H_{G}^{∗}(X^{p}; ℤ/pℤ) ⟶^{d^{∗}} H_{G}^{∗}(X; ℤ/pℤ) = H^{∗}(X; ℤ/pℤ)⊗H_{G}^{∗}(pt; ℤ/pℤ).

The ring H_{G}^{∗}(pt; ℤ/pℤ) is isomorphic to the polynomial ring (ℤ/pℤ)[t]. Thus, the image of α is a polynomial with coefficient in H^{∗}(X; ℤ/pℤ). The coefficients are the reduced powers P^{k}(α) of a.

Let a space X be embedable into ℝ^{n}. The group

H_{i}^{BM}(X; ℤ/pℤ) = H^{n−i}(ℝ^{n}, ℝ^{n} − X; ℤ/pℤ)

is independent on the embedding and is called the Borel-Moore homology group with coefficients in ℤ/pℤ. An operation S of degree r defines an operation

H_{i}^{BM}(X; ℤ/pℤ) = H^{n−i}(ℝ^{n}, X; ℤ/pℤ) ⟶^{S} H^{n−i+r}(ℝ^{n}, X; ℤ/pℤ) = H_{i−r}^{BM}(X; ℤ/pℤ)

on the Borel-Moore homology groups. These operations commute with pushforward homomorphisms with respect to proper morphisms. We can them operations of homological type.

Date: May, 2003.

V. Voevodsky has defined operations on the motivic cohomology groups H^{i,j}(X, U; ℤ/pℤ) for a smooth variety X and an open subset U ⊂ X. The reduced power operation P^{k} has bidegree (2(p−1)k, (p−1)k) and the Bockstein operation has bidegree (1, 0).

If d = dim X we have

H^{2i,i}(X, U; ℤ/pℤ) ≃ CH_{d−i}(X − U)/p.

In particular, the reduced power operations (but not the Bockstein operation) act on the Chow groups modulo p. Our aim is to give an “elementary” definition of these operations. We follows P. Brosnan’s approach.

Notation: A scheme is a quasi-projective scheme over a field. A variety is an integral scheme.

2. Chow groups

2.1. Chow groups of homological type. Let X be a scheme over a field F and let i be an integer. The Chow groups of i-dimensional cycles CH_{i}(X) is the factor group of the free group generated by dimension i closed subvarieties of X modulo rational equivalence. In particular, CH_{i}(X) is trivial if i < 0 or i > dim X.

Properties:

1. For a projective morphism f : X → Y there is the push-forward homomorphism

f_{∗} : CH_{i}(X) → CH_{i}(Y).

2. For a flat morphism f : Y → X of relative dimension d there is the pull-back homomorphism

f^{∗} : CH_{i}(X) → CH_{i+d}(Y).

3. (Homotopy invariance) For a vector bundle f : E → X of rank r the pull-back homomorphism f_{∗} : CH_{i}(X) → CH_{i+r}(E) is an isomorphism.

4. For two schemes X and Y over F there is an exterior product homomorphism

CH_{i}(X) ⊗ CH_{j}(Y) → CH_{i+j}(X × Y), α ⊗ β ↦ α × β.

Remark 2.1. We can view Chow groups CH_{∗}(X) as a Borel-Moore type theory associated to the motivic cohomology groups of bidegree (2i, i).

2.2. Chern classes. Let E → X be a vector bundle over a scheme X. For every i ≥ 0 there is a group homomorphism

c_{i}(E) : CH_{k}(X) → CH_{k−i}(X), α ↦ c_{i}(E) ∩ α

called the i-th Chern class of E. We can view c_{i}(E) as an endomorphism of CH_{∗}(X). The class c_{•} = c_{0} + c_{1} + c_{2} + … is called the total Chern class.

Properties:

1. c_{0}(E) is the identity.

2. c_{i}(E) = 0 for i > rank E.

3. (Cartan formula) For an exact sequence 0 → E′ → E → E′′ → 0,

c_{•}(E) = c_{•}(E′) ∘ c_{•}(E′′).

4. The operation c_{1}(L) for a line bundle L → X is the intersection with the Cartier divisor associated with L.

The Chern classes uniquely determined by the properties 1-4.

If E → X is a vector bundle of rank r, the class e(E) = c_{r}(E) is called the Euler class of E. For an exact sequence as above we have

e(E) = e(E′) ∘ e(E′′).

Let A be a commutative ring. A characteristic class over A is a power series w over A in the Chern classes c_{i}. For a vector bundle E → X the operation w(E) on CH_{∗}(X) ⊗ A is well defined.

Let f(x) = 1 + a_{1}x + a_{2}x^{2} + … be a power series over A. A characteristic class w_{f} over A is called a multiplicative class associated with f if the following holds:

1. For a line bundle L → X,

w_{f}(L) = f(c_{1}(L)) = id + a_{1}c_{1}(L) + a_{2}c_{1}(L)^{2} + …;

2. For an exact sequence 0 → E′ → E → E′′ → 0 of vector bundles,

w_{f}(E) = w_{f}(E′) ∘ w_{f}(E′′).

The class w_{f} exists and unique for every f. If E has a filtration by subbundles with line factors L_{i}. Then

w_{f}(E) = ∏_{i} f(c_{1}(L_{i})).

Example 2.2. If f = 1 + x, the class w_{f} coincides with the total Chern class c_{•}.

Note that the operation w_{f}(E) is invertible. We write w_{f}(−E) for w_{f}(E)^{−1}.

2.3. Chow groups of cohomological type. If X is a smooth variety of dimension d we write

CH^{i}(X) = CH_{d−i}(X).

Properties:

1. For a morphism f : Y → X of smooth varieties there is the pull-back homomorphism

f^{∗} : CH^{i}(X) → CH^{i}(Y).

2. For a smooth variety X, the graded group CH^{∗}(X) has a structure of a graded ring with identity 1_{X} = [X] via the pull-back with respect to the diagonal morphism d : X → X × X: α · β = d^{∗}(α × β).

Remark 2.3. The group CH^{i}(X) for a smooth X coincides with the motivic cohomology group H^{2i,i}(X) = H^{2i,i}(X, ∅) and the latter has obvious contravariant properties. We can view the assignment X ↦ CH^{∗}(X) as a cohomology theory.

Example 2.4. Let h ∈ CH^{1}(ℙ^{n}) be the class of a hyperplane section of the projective space ℙ^{n}. Then

CH^{i}(ℙ^{n}) = \begin{cases}ℤh^{i} & if 0 ≤ i ≤ n, \\ 0 & otherwise. \end{cases}

In other words,

CH^{∗}(ℙ^{n}) = ℤ[h]/(h^{n+1}).

2.4. Refined Gysin homomorphisms. Let i : M ↪ P be a closed embedding of smooth varieties of codimension d. Let X be a closed subscheme of P and set Y = M ∩ X. Thus we have a pull-back diagram

\begin{matrix}Y & ⟶ & X \\ ^{f}↓ & & ↓ \\ M & ⟶^{i} & P. \end{matrix}

The pull-back homomorphism i^{∗} (called also a Gysin homomorphism) has a refinement, called the refined Gysin homomorphism

i^{!} : CH_{k}(X) → CH_{k−d}(Y).

In the case X = P the homomorphism i^{!} coincides with i^{∗}.

Remark 2.5. The refined Gysin homomorphism coincide with the pull-back homomorphism

H^{2i.i}(P, P − X) → H^{2i.i}(M, M − Y).

for i = dim P − k.

Suppose we are given two diagrams as above:

\begin{matrix}Y & ⟶ & X \\ ^{f_{l}}↓ & & ↓ \\ M_{l} & ⟶^{i_{l}} & P_{l} \end{matrix}

(l = 1, 2) with i_{l} : M_{l} → P_{l} closed embeddings of smooth varieties of codimension d_{l}. Let N_{l} be the normal bundle of the closed embedding i_{l} over M_{l}. It is a vector bundle on M_{l}.

The following proposition is a variant of so-called excess formula.

Proposition 2.6. For an element α ∈ CH_{k}(X),

e(f_{2}^{∗}N_{2}) ∩ i_{1}^{!}(α) = e(f_{1}^{∗}N_{1}) ∩ i_{2}^{!}(α)

in CH_{k−d_{1}−d_{2}}(Y).

3. Equivariant Chow groups

In topology: Let G be a group and X be a G-space. For a (co)homology theory H we can define an equivariant (co)homology group

H^{G}(X) = H((X × EG)/G)

where EG is a contractible G-space with free G-action.

In algebra we don’t have a scheme representing EG. Instead, we consider certain approximations of EG.

Let G be an algebraic group of dimension d_{G} over F and let X be a Gscheme. Let V be a G-vector space (a representation of G) of dimension d_{V} such that G acts regularly on a nonempty open subscheme U ⊂ V. Then for every i ∈ ℤ, the group

CH_{i}^{G}(X) = CH_{i+d_{V} −d_{G}}((X × U)/G)

is independent of V as long as V − U has sufficiently large codimension in V. This group is the i-th equivariant Chow group of X.

A closed G-equivariant subvariety Z ⊂ X has its class [Z]^{G} in CH_{i}^{G}(X) defined as

[(Z × U)/G] ∈ CH_{i+d_{V} −d_{G}}((X × U)/G) = CH_{i}^{G}(X).

If G is the trivial group we have

CH_{i}^{G}(X) = CH_{i}(X).

If H is a subgroup of G, for a G-scheme X there is a natural restriction homomorphism

res_{G/H} : CH_{i}^{G}(X) → CH_{i}^{H}(X).

For a smooth G-variety X of dimension d we write

CH_{G}^{i}(X) = CH_{d−i}^{G}(X).

The equivariant Chow groups have similar functorial properties as the ordinary ones (projective push-forwards, flat pull-backs, refined Gysin homomorphisms, homotopy invariance). In particular, the graded group CH_{G}^{∗}(X) has a structure of a commutative ring.

Example 3.1. Let G = 𝔾_{m} the multiplicative group. Consider the natural G-action on the affine space 𝔸^{r+1}. Clearly G acts regularly on 𝔸^{r+1} −{0} with the factor variety ℙ^{r}. Hence

CH_{G}^{i}(pt) = CH_{−i}^{G}(pt) = CH_{r−i}(ℙ^{r}) = ℤh^{i}

and therefore, CH_{G}^{∗}(pt) = ℤ[h] is a polynomial ring over ℤ. Note that this ring is a domain contrary to the ordinary Chow rings having only nilpotent elements in positive degrees.

Let p be a prime integer. We introduce a new notation:

Ch_{∗}(X) = CH_{∗}(X)/p, Ch_{∗}^{G}(X) = CH_{∗}^{G}(X)/p.

Example 3.2. Let H = μ_{p}. We consider H as a subgroup of G = 𝔾_{m}. The restriction ring homomorphism Ch_{G}^{∗}(pt) → Ch_{H}^{∗}(pt) is an isomorphism, so that

Ch_{H}^{∗}(pt) = (ℤ/pℤ)[h].

More generally, if H acts trivially on a scheme X, we have a canonical isomorphism

Ch_{∗}^{H}(X) = Ch_{∗}(X) ⊗ ℤ/pℤ[h] = Ch_{∗}(X)[h].

Example 3.3. Let G be the semidirect product of μ_{p} and Aut(μ_{p}) = (ℤ/pℤ)^{×}. Since the order of (ℤ/pℤ)^{×} is prime to p, we have for a G-scheme X:

Ch_{∗}^{G}(X) = [Ch_{∗}^{μ_{p}}(X)]^{(ℤ/pℤ)^{×}} .

A class i+pℤ ∈ (ℤ/pℤ)^{×} acts on h by h ↦ h^{i}. The restriction homomorphism

Ch_{G}^{∗}(pt) → Ch_{μ_{p}}^{∗}(pt) = ℤ/pℤ[h]

is injective and identifies the ring Ch_{G}^{∗}(pt) with ℤ/pℤ[h^{p−1}]. More generally, if G acts trivially on a scheme X, we have a canonical isomorphism

Ch_{∗}^{G}(X) = [Ch_{∗}^{μ_{p}}(X)]^{Aut(μ_{p})} = Ch_{∗}(X)[h^{p−1}].

3.1. Equivariant Chern classes. Let X be G-scheme and let E → X be a G-vector bundle over X. For every i ≥ 0 there is a group homomorphism

c_{i}^{G}(E) : CH_{k}^{G}(X) → CH_{k−i}^{G}(X), α ↦ c_{i}^{G}(E) ∩ α

called the equivariant i-th Chern class of E. The equivariant Chern classes satisfy similar properties as the ordinary ones.

Example 3.4. If X = pt, a G-vector bundle over X is a representation V of G. Let G = 𝔾_{m} or μ_{n} and let V be the canonical 1-dimensional representation of G. Then c_{1}^{G}(V) is the multiplication by h in Ch_{G}^{∗}(pt). We will simply write c_{1}^{G}(V) = h.

Example 3.5. Let p be a prime integer. Let G be the semidirect product of μ_{p} and Aut(μ_{p}) = (ℤ/pℤ)^{×}. Consider the algebra F_{p} = F [x]/(x^{p} − 1). It is an étale F -algebra of degree p. If F contains all p-th roots of unity, F_{p} is a product of p copies of F. The group G acts naturally on F_{p} by algebra automorphisms: the group μ_{p} acts on x by multiplication and an automorphism i+pℤ ∈ (ℤ/pℤ)^{×} by x ↦ x^{i}. In particular, F_{p} is a G-module having a (trivial) submodule F. We claim that the mod p equivariant Euler class e^{G}(F̃), where F̃ = F_{p}/F, is the multiplication by −h^{p−1}.

Indeed, since Ch_{G}^{∗}(pt) is contained in Ch_{μ_{p}}^{∗}(pt) it is sufficient to replace G

by μ_{p}. Then F̃ is a direct sum of 1-dimensional modules F x^{i} ≃ V^{⊗i} for i = 1, 2, …, p − 1. Hence

e^{G}(F̃) = ∏_{i=1}^{p−1} e^{G}(V^{⊗i}) = ∏_{i=1}^{p−1}(ih) = (p − 1)! · h^{p−1} = −h^{p−1}.

We set t = −h^{p−1} so that CH_{G}^{∗}(pt)/p = ℤ/pℤ[t] and t = e^{G}(F̃) modulo p.

4. Definition of reduced powers

We fix a prime integer p. We assume that char F ≠ p.

4.1. Scheme R^{p}(X). Consider again the algebra F_{p} = F [x]/(x^{p} − 1). Let X be a scheme over a field F. Write R^{p}(X) for the scheme R_{F_{p}/F} (X ⊗_{F} F_{p}) representing the functor S ↦ X(S ⊗_{F} F_{p}). If F contains all p-th roots of unity, X is a product of p copies of X. In particular, dim R^{p}(X) = p dim X.

Consider the semidirect product G of μ_{p} with Aut(μ_{p}) = (ℤ/pℤ)^{×}. The action of G on F_{p} extends to an action of G on R^{p}(X).

The embedding X(S) ↪ X(S ⊗_{F} F_{p}) gives rise to the closed embedding of X into R^{p}(X). In fact, X is identified with the subscheme of G-invariant elements of R^{p}(X). We have a well defined map (not a homomorphism!)

P_{X} : CH_{k}(X) → CH_{pk}^{G}(R^{p}(X)), [Z] ↦ [R^{p}(Z)]^{G}.

Now assume that X is a smooth variety. We would like to compute the normal cone N of the closed embedding X → R^{p}(X). One checks that

T_{R^{p}(X)}|_{X} = T_{X} ⊗_{F} F_{p},

hence

N = T_{X} ⊗_{F} F̃,

where as above F̃ = F_{p}/F. The group G acts on N via F̃. Let us compute the Euler class e^{G}(N).

Consider the multiplicative characteristic class w = w_{f} corresponding to the power series f(x) = 1 + x^{p−1}. Recall that for every line bundle L, we have w(L) = 1 + c_{1}(L)^{p−1}. We write

w(y) = 1 + w_{p−1}y + w_{2(p−1)}y^{2} + …,

so that for a line bundle L,

w(L, y) = 1 + c_{1}(L)^{p−1} · y.

Proposition 4.1. For a vector bundle E → X of rank r we have

e^{G}(E ⊗_{F} F̃) = t^{r}w(E, 1/t) ∈ Ch_{∗}^{G}(X) = Ch_{∗}(X)[t].

Proof. Since Ch_{∗}^{G}(X) injects into Ch_{∗}^{μ_{p}}(X) we may replace G by μ_{p}. We also may assume that E = L is a line bundle. Then

e^{G}(L ⊗_{F} F̃) = ∏_{i=1}^{p−1} c_{1}^{G}(L ⊗ V^{⊗i})

= ∏_{i=1}^{p−1}(c_{1}^{G}(L) + c_{1}^{G}(V^{⊗i}))

= ∏_{i=1}^{p−1}(c_{1}^{G}(L) + ih)

= c_{1}^{G}(L)^{p−1} − h^{p−1}

= t(1 + c_{1}^{G}(L)^{p−1}/t) = t · w(L, 1/t).

□

Remark 4.2. It follows from the proposition that the Euler class e^{G}(E ⊗_{F} F̃) is invertible if we invert t.

4.2. Reduced power operations. Let X be a scheme over F. We embed X as a closed subscheme into a smooth variety M of dimension d. The commutative diagram

X ⟶ R^{p}(X)
↓  ↓
M ⟶^{i} R^{p}(M)

gives rise to a refined Gysin homomorphism

i^{!} : CH_{r}^{G}(R^{p}(X)) → CH_{r−(p−1)d}^{G}(X),

where G be the semidirect product of μ_{p} and (ℤ/pℤ)^{×}.

Note that the composition

D_{X}^{M} : Ch_{k}(X) ⟶^{P_{X}} Ch_{pk}^{G}(R^{p}(X)) ⟶^{i^{!}} Ch_{pk−(p−1)d}^{G}(X)

is a group homomorphism.

Since G acts trivially on X, we have

Ch_{∗}^{G}(X) = Ch_{∗}(X)[t] = ∐_{i≥0} Ch_{∗}(X)t^{i}.

The map D_{X}^{M} changes the degree by a multiple of p − 1. Therefore, for an element α ∈ Ch_{k}(X) we can then write

D_{X}^{M}(α)(t) = ∑_{i} S_{i}^{M}(α)t^{d−k−i}

for some elements

S_{i}^{M}(α) ∈ Ch_{k−(p−1)i}(X).

In other words,

S_{•}^{M}(α)(t) =^{def} ∑_{i} S_{i}^{M}(α)t^{i} = t^{d−k}D_{X}^{M}(α)(1/t).

Thus, for any scheme X over F and a smooth variety M containing X as a closed subscheme we get the reduced power operations for the pair (M, X):

S_{i}^{M} : Ch_{k}(X) → Ch_{k−(p−1)i}(X).

Note that S_{i}^{M} lowers the dimension by (p − 1)i.

We also have the total reduced power operation

S_{•}^{M} = S_{•}^{M}(t)|_{t=1} = ∑ S_{i}^{M} : Ch_{∗}(X) → Ch_{∗}(X).

Remark 4.3. On the motivic cohomology groups the operation S_{i}^{M} coincides with the reduced power

S^{i} : H^{2k,k}(M, M − X) → H^{2k−2(p−1)i,k−(p−1)i}(M, M − X).

Properties:

1. Let Y ⟶^{i} X → M be closed embeddings with M smooth. Then

S_{•}^{M}(i_{∗}α) = i_{∗}(S_{•}^{M}α)

for every α ∈ Ch_{∗}(Y).

2. (Cartan formula) Let X → M and Y → N be closed embeddings with M and N smooth. Then

S_{•}^{M×N}(α × β) = S_{•}^{M}α × S_{•}^{N}β

for α ∈ Ch_{∗}(X) and β ∈ Ch_{∗}(Y).

4.3. Steenrod operations of cohomological type. Let X be a smooth variety over F. Set S^{•} = S_{•}^{X}. Thus, S^{i} is an operation

S^{i} : Ch^{k}(X) → Ch^{k+(p−1)i}(X),

called the reduced power of cohomological type.

Properties:

1. Let f : Y → X be a morphism of smooth varieties over F. Then

f^{∗}S^{•}(α) = S^{•}f^{∗}(α)

for every α ∈ Ch^{∗}(X).

2. (Cartan formula) S^{•}(αβ) = S^{•}(α)·S^{•}(β) for α, β ∈ Ch^{∗}(X), S^{•}(1_{X}) = 1_{X}, i.e., S^{•} is a ring endomorphism of Ch^{∗}(X).

3. For every α ∈ Ch^{k}(X),

S^{i}α = \begin{cases}α^{p}, & if i = k, \\ 0, & if i > k or i < 0. \end{cases}

4.4. Steenrod operations of homological type.

Proposition 4.4. Let X be a scheme and let M_{1} and M_{2} be two smooth varieties containing X. Then

w(T_{M_{2}}|_{X}) ∩ S_{•}^{M_{1}} = w(T_{M_{1}}|_{X}) ∩ S_{•}^{M_{2}}.

Proof. The normal bundle N_{l} of the closed embedding M_{l} → R^{p}(M_{l}) is equal to T_{M_{l}}|_{X} ⊗_{F} F̃, (l = 1, 2). Let d_{l} = dim M_{l}. For an element α ∈ Ch_{k}(X), we have

w(T_{M_{2}}|_{X}, 1/t) ∩ S_{•}^{M_{1}}(1/t) = t^{−d_{2}} · e^{G}(N_{2}) ∩ t^{k−d_{1}}D_{X}^{M_{1}}(α)(t) (Prop. 4.1)
= t^{k−d_{1}−d_{2}} · e^{G}(N_{2}) ∩ i_{1}^{!}(P_{X}(α)).

It follows from Proposition 2.6 that the latter is equal to

t^{k−d_{1}−d_{2}} · e^{G}(N_{1}) ∩ i_{2}^{!}(P_{X}(α)).

The result follows. □

Let X be a subscheme of a smooth variety M. For every α ∈ Ch_{k}(X) set

S_{•}(α) = w(T_{M}|_{X})^{−1} ∩ S_{•}^{M}(α).

By Proposition 4.4, S_{•}(α) is independent of M. The operations

S_{i} : Ch_{k}(X) → Ch_{k−(p−1)i}(X)

are called the reduced powers of homological type.

Properties:

1. If i : Y → X is a projective morphism of schemes over F, then

S_{•}(i_{∗}α) = i_{∗}(S_{•}α)

for every α ∈ Ch_{∗}(Y). In fact, this is a variant of a Riemann-Roch theorem by I. Panin and A. Smirnov.

2. S_{0} is the identity.

3. Let a scheme X be embedable in a smooth scheme of dimension d and let α ∈ Ch_{k}(X). Then S_{i}(α) = 0 for i ∉ [0, d − k].

If X is smooth, we have the operations S^{•} and S_{•} on the Chow groups of X modulo p. For every α ∈ Ch_{k}(X) we have

S_{•}(α) = w(T_{X})^{−1} ∩ S^{•}(α) = w(−T_{X}) ∩ S^{•}(α).

In particular,

S_{•}(1_{X}) = w(−T_{X}) ∩ 1_{X}, i.e., S_{i}(1_{X}) = w_{(p−1)i}(−T_{X}) ∩ 1_{X}.

Remark 4.5. The homological operation S_{i} is a special case of the cohomological operation S_{i}^{M} if X is affine. Let i : X ↪ 𝔸^{n} be a closed embedding. Since the tangent bundle of 𝔸^{n} is trivial, we have

S_{i} = S_{i}^{𝔸^{n}}.

5. Reduced Steenrod algebra

We fix a prime integer p in this section. Consider the polynomial ring

H = (ℤ/pℤ)[b] = (ℤ/pℤ)[b_{1}, b_{2}, …]

in infinitely many variables b_{1}, b_{2}, … as a graded ring with deg b_{i} = p^{i} − 1. The monomials

b^{R} = b_{1}^{r_{1}}b_{2}^{r_{2}} …,

where R ranges over all sequences (r_{1}, r_{2}, …) of non-negative integers such that almost all of the r_{i}’s are zero, form a basis of H over ℤ/pℤ. We set

|R| = ∑_{i≥1} r_{i}(p^{i} − 1).

Clearly, deg b^{R} = |R|.

Denote the scheme Spec H by G. For a commutative ℤ/pℤ-algebra A the set of A-points

G(A) = Hom_{rings}(H, A)

can be identified with the set of sequences (a_{1}, a_{2}, …) of the elements of A and, therefore, with the set of power series of the form

x + a_{1}x^{p} + a_{2}x^{p^{2}} + a_{3}x^{p^{3}} + ⋯ ∈ A[[x]].

The composition law (f_{1} ∗ f_{2})(x) = f_{2}(f_{1}(x)) makes G a group scheme over ℤ/pℤ and H a graded Hopf ℤ/pℤ-algebra. The co-multiplication morphism c : H → H ⊗ H is given by the rule

c(b_{k}) = ∑_{i+j=k} b_{i}^{p^{j}} ⊗ b_{j}.

We write \overline{A}_{p} for the graded Hopf ℤ/pℤ-algebra dual to H. Thus, \overline{A}_{p} is the graded Hopf algebra such that the d-component of \overline{A}_{p} is the ℤ/pℤ-space dual to the d-component of H. The algebra \overline{A}_{p} is the reduced Steenrod ℤ/pℤalgebra, i.e., the Steenrod ℤ/pℤ-algebra modulo the ideal generated by the Bockstein element. It can be also considered as the subalgebra of the Steenrod algebra A_{p} generated by the reduced power operations.

Let \{P^{R}\} be the basis of \overline{A}_{p} dual to the basis \{b^{R}\} of H.

We write P^{i} for the basis element P^{R} with R = (i, 0, …). The element P^{0} is the identity of \overline{A}_{p} and the algebra \overline{A}_{p} is generated by the P^{i}, i ≥ 1. These generators are subject to the set of defining Adem relations:

P^{b} · P^{c} = ∑_{i=0}^{[b/p]}(−1)^{b+i}\binom{(p − 1)(c − i) − 1}{b − pi} · P^{b+c−i} · P^{i}.

P. Brosnan proved that the power operations S^{i} and S_{i} satisfy the Adem relations respectively. Hence, for any smooth variety X, the group Ch^{∗}(X) has a structure of a left \overline{A}_{p}-module (the generator P^{i} acts as S^{i})), and for every scheme X, the group Ch_{∗}(X) has a structure of a left \overline{A}_{p}-module (the generator P^{i} acts as S_{i})). Denote by S^{R} (resp. S_{R}) the operation induced by P^{R} on CH_{∗}(X) if X is smooth (resp. arbitrary scheme). The operations S^{R} commute with the pull-back homomorphisms and the operations S_{R} commute with the push-forward homomorphisms.

Consider the “generic” power series f_{gen}(x) = 1+b_{1}x^{p−1} +b_{2}x^{p^{2}−1} +… over the ring ℤ[b]. Write the corresponding multiplicative class w_{gen} = w_{f_{gen}} in the form

w_{gen} = ∑_{R} w_{R}b^{R}

for unique characteristic classes w_{R} over ℤ. In particular, if R = (i, 0, …), we have w_{R} = w_{(p−1)i} in the old notation.

If X is a smooth variety, the equality S_{i}(1_{X}) = w_{(p−1)i}(−T_{X})∩1_{X} generalizes as follows: S_{R}(1_{X}) = w_{R}(−T_{X}) ∩ 1_{X} for every sequence R.

6. Degree formulas

Let X be a scheme over F. For a closed point x ∈ X we define its degree as the integer deg(x) = [F (x) : F] and set

n_{X} = gcd deg(x),

where the gcd is taken over all closed points of X.

Fix a prime integer p ≠ char F. Let X be a projective variety of dimension d > 0 and let q : X → Spec F be the structure morphism. For every sequence R with |R| = d, the group Ch_{d}(Spec F) in the commutative diagram

Ch_{d}(X) ⟶^{S_{R}} Ch_{0}(X)
q_{∗}↓  ↓^{q_{∗}=deg}
Ch_{d}(Spec F) ⟶^{S_{R}} Ch_{0}(Spec F)  ℤ/p

is trivial. Hence the degree of a cycle α_{R}^{X} ∈ CH_{0}(X) representing the element S_{R}([X]) ∈ Ch_{0}(X) is divisible by p. The class of the integer deg(α_{R}^{X})/p modulo n_{X} is independent on the choice of α_{R}^{X}; we denote it by

u_{R}(X) ∈ ℤ/n_{X}ℤ.

Clearly, p · u_{R}(X) = 0.

If X is a smooth variety, S_{R}(1_{X}) = w_{R}(−T_{X}) ∩ 1_{X}, hence we can take α_{R}^{X} = w_{R}(−T_{X}) ∩ 1_{X}. Thus, deg(w_{R}(−T_{X}) ∩ 1_{X}) is divisible by p and

u_{R}(X) = \frac{deg(w_{R}(−T_{X}) ∩ 1_{X})}{p} + n_{X}ℤ.

Remark 6.1. The degree of the 0-cycle w_{R}(−T_{X}) ∩ 1_{X} does not change under field extensions. In particular, it can be computed over an algebraic closure of F.

Let f : X → Y be a morphism of varieties over F of the same dimension. We set

deg(f) = \{ \begin{matrix}[F (X) : F (Y)], & if f is dominant; \\ 0, & otherwise. \end{matrix}

Thus, if f is projective, then f_{∗}(1_{X}) = deg(f) · 1_{Y}.

Theorem 6.2. (Degree formula) Let f : X → Y be a morphism of projective varieties over F of dimension d > 0. Then n_{Y} divides n_{X} and for every sequence R with |R| = d, and for any prime integer p ≠ char F, we have

u_{R}(X) = deg(f) · u_{R}(Y) ∈ ℤ/n_{Y} ℤ.

In particular, if X and Y are smooth, then

\frac{deg(w_{R}(−T_{X}) ∩ 1_{X})}{p} ≡ deg(f) · \frac{deg(w_{R}(−T_{Y}) ∩ 1_{X})}{p} (mod n_{Y} ).

Proof. It follows from the commutativity of the diagram

Ch_{d}(X) ⟶^{S_{R}} Ch_{0}(X)
f_{∗}↓  ↓^{f_{∗}}
Ch_{d}(Y) ⟶^{S_{R}} Ch_{0}(Y)

and the equality f_{∗}(1_{X}) = deg(f) · 1_{Y} that

f_{∗}S_{R}(1_{X}) = deg(f) · S_{R}(1_{Y}) ∈ Ch_{0}(Y)

and therefore,

f_{∗}α_{R}^{X} ≡ deg(f) · α_{R}^{Y} (mod p CH_{0}(Y)).

Applying the degree homomorphism, we get

deg(α_{R}^{X}) = deg(f_{∗}α_{R}^{X}) ≡ deg(f) · deg(α_{R}^{Y}) (mod pn_{Y}),

whence the result. □

Remark 6.3. The case of sequences R = (i, 0, 0, …) was considered by M. Rost.

7. Applications

Let X and Y be varieties over a field F, d = dim(X). A correspondence from X to Y, denoted α: X ⇝ Y, is a dimension d algebraic cycle on X × Y. A correspondence α is called prime if α is given by a prime (irreducible) cycle. Any correspondence is a linear combination with integer coefficients of prime correspondences.

Let α : X ⇝ Y be a prime correspondence. Suppose that α is given by a closed subvariety Z ⊂ X × Y. We define multiplicity of α as the degree of the projection Z → X. We extend the notion of multiplicity to arbitrary correspondences by linearity.

A rational morphism X → Y defines a multiplicity 1 prime correspondence X ⇝ Y given by the closure of its graph. One can think of a correspondence of multiplicity m as a “generically m-valued morphism”.

We fix a prime integer p and a field F such that char F ≠ p. Let R be a nonzero sequence and let X be a projective variety over F of dimension |R|. The variety X is called R^{p}-rigid if u_{R}(X) ≠ 0 ∈ ℤ/n_{X}ℤ.

We write v_{p} for the p-adic valuation of ℚ. Theorem 7.1. Let X and Y be projective varieties over F and let R be a sequence such that dim(X) = |R| > 0. Suppose that

(1) There is a correspondence α : X ⇝ Y of multiplicity not divisible by p;

(2) X is R^{p}-rigid;

(3) v_{p}(n_{X}) ≤ v_{p}(n_{Y}). Then

(1) dim(X) ≤ dim(Y);

(2) If dim(X) = dim(Y), (2a) There is a correspondence β : Y ⇝ X of multiplicity not divisible by p; (2b) Y is R^{p}-rigid; (2c) v_{p}(n_{X}) = v_{p}(n_{Y}).

Proof. Suppose that m = dim(X)−dim(Y) ≥ 0 and set Y′ = Y ×ℙ_{F}^{m}. Clearly, n_{Y′} = n_{Y}. We embed Y into Y′ as Y × z where z is a rational point of ℙ_{F}^{m}.

We may assume that α is a prime correspondence, replacing if necessary, α by one of its prime components. Let Z ⊂ X × Y be the closed subvariety representing α. We have two natural morphisms f : Z → X and g : Z → Y ↪ Y′. By assumption, deg(f) is not divisible by p.

We write the degree formulas of Theorem 6.2 for the morphisms f and g:

(1)  u_{R}(Z) = deg(f) · u_{R}(X) ∈ ℤ/n_{X}ℤ,
(2)  u_{R}(Z) = deg(g) · u_{R}(Y′) ∈ ℤ/n_{Y} ℤ.

The variety X is R^{p}-rigid and the degree deg(f) is not divisible by p, hence it follows from (1) that u_{R}(Z) ≠ 0 in ℤ/n_{X}ℤ. Since v_{p}(n_{X}) ≤ v_{p}(n_{Y}) and p · u_{R}(Z) = 0, we have u_{R}(Z) ≠ 0 in ℤ/n_{Y} ℤ and it follows from the degree formula (2) that deg(g) is not divisible by p, so that g is surjective, and u_{R}(Y′) ≠ 0 in ℤ/n_{Y} ℤ. The image of g is contained in Y, therefore Y = Y′, i.e., m = 0 and dim(X) = dim(Y).

The variety Z defines a correspondence β : Y ⇝ X of multiplicity deg(g) not divisible by p. Since u_{R}(Y) = u_{R}(Y′) ≠ 0 in ℤ/n_{Y} ℤ, it follows that Y is R^{p}-rigid. Finally, p · u_{R}(Z) = 0 and u_{R}(Z) is nonzero in both ℤ/n_{X}ℤ and ℤ/n_{Y} ℤ, therefore, we must have v_{p}(n_{X}) = v_{p}(n_{Y}). □ Corollary 7.2. The classes u_{R}(X) ∈ ℤ/n_{X}ℤ are birational invariants of a smooth projective variety X.

8. Examples

Let X be a complete intersection in ℙ^{n} of m smooth hypersurfaces of degrees d_{1}, d_{2}, …, d_{m}. Let L = L_{can} be the restriction on X of the canonical line bundle on ℙ^{n}. Then

[T_{X}] = [T_{ℙ^{n}}|_{X}] − ∑_{i=1}^{m}[L^{⊗d_{i}}] ∈ K_{0}(X).

We have

[T_{ℙ^{n}}|_{X}] = (n + 1)[L] − 1_{X} ∈ K_{0}(X).

and

w_{gen}(L) = 1 + h^{p−1}b_{1} + h^{p^{2}−1}b_{2} + …,

where h = c_{1}(L) ∈ Ch^{1}(X) is the class of a hyperplane section. Therefore,

w_{gen}(−T_{X}) = \frac{∏ w_{gen}(L^{⊗d_{i}})}{w_{gen}(L)^{n+1}} = \frac{∏(1 + (d_{i}h)^{p−1}b_{1} + (d_{i}h)^{p^{2}−1}b_{2} + …)}{(1 + h^{p−1}b_{1} + h^{p^{2}−1}b_{2} + …)^{n+1}}.

Let R be a sequence such that |R| = dim X = n − m. Since

deg h^{n−m} = ∏ d_{i},

we have

deg w_{R}(−T_{X}) = ∏ d_{i} · [\frac{∏(1 + d_{i}^{p−1}b_{1} + d_{i}^{p^{2}−1}b_{2} + …)}{(1 + b_{1} + b_{2} + …)^{n+1}}]_{R} .

Example 8.1. Let X be the Severi-Brauer variety of a central simple algebra A of degree p^{n}. Then X is a twisted form of the projective space ℙ^{p^{n}−1}. For

R = (0, …, 0, 1^{n}, 0, …) we have

deg w_{R}(−T_{X}) = −p^{n}.

Thus, X is R^{p}-rigid if n_{X} = p^{n}, i.e., A is a division algebra.

Example 8.2. Let X be a smooth hypersurface of degree p in ℙ^{p^{n}}. For a sequence R as above we have

deg w_{R}(−T_{X}) = p(p^{p^{n}−1} − p^{n} − 1).

Thus, X is R^{p}-rigid if n_{X} = p.

Example 8.3. Let X be a smooth intersection of two quadrics in ℙ^{4}. Then for R = (2, 0, 0, …) and p = 2 we have

deg w_{R}(−T_{X}) = −4.

Thus, X is R^{2}-rigid if n_{X} = 4.

Alexander Merkurjev, Department of Mathematics, University of California, Los Angeles, CA 90095-1555, USA

  E-mail address: merkurev@math.ucla.edu