Reduced power operations in motivic cohomology

by Vladimir Voevodsky^{1}^{,}^{2}

Preliminary version March 2001

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Motivic cohomology and operations . . . . . . . . . . . . . . . 3 3 Operations H^{2d,d} → H^{2d+∗,d+i} for i ≤ 0 . . . . . . . . . . . . . 6 4 Thom isomorphism and Euler classes . . . . . . . . . . . . . . 9 5 Total power operations . . . . . . . . . . . . . . . . . . . . . . 11 6 Motivic cohomology of BS_{l} . . . . . . . . . . . . . . . . . . . 14 7 Symmetry theorem . . . . . . . . . . . . . . . . . . . . . . . . 20 8 Power operations and the Bockstein homomorphism . . . . . . 24 9 Individual power operations: formulas . . . . . . . . . . . . . . 26 10 Adem relations . . . . . . . . . . . . . . . . . . . . . . . . . . 30 11 Motivic Steenrod algebra . . . . . . . . . . . . . . . . . . . . . 35 12 Structure of the dual to the motivic Steenrod algebra . . . . . 38 13 Operations Q_{i} and their properties . . . . . . . . . . . . . . . 44

1 Introduction

In this paper we consider cohomological operations in the motivic cohomology of smooth simplicial schemes over a field k. For most part we work with cohomology with coefficients in Z/l where l is a prime different from the characteristics of k. We construct the reduced power operations

P^{i} : H^{∗,∗}(−, Z/l) → H^{∗+2i(l−1),∗+i(l−1)}(−, Z/l)

1Supported by the NSF grants DMS-97-29992 and DMS-9901219 and The Ambrose Monell Foundation

2School of Mathematics, Institute for Advanced Study, Princeton NJ, USA. e-mail:

vladimir@ias.edu

and prove the motivic analogs of the Cartan formula and the Adem relations. We also describe the subalgebra in the algebra of all (bistable) operations in the motivic cohomology with Z/l-coefficients generated by P^{i}’s, the Bockstein homomorphism and the multiplication by the motivic cohomology classes of Spec(k). For odd l this algebra is isomorphic to the twisted tensor product of the usual topological Steenrod algebra with the motivic cohomology ring of the point H^{∗,∗} with respect to the action of the motivic Steenrod operations on H^{∗,∗}. For l = 2, the situation is more complicated since the motivic Adem relations involve nontrivial coefficients from H^{∗,∗}.

To construct the reduced power operations we follow the approach of [6] when one first defines the total power operation and then uses the computation of the cohomology of the product of a space with the classifying space of the symmetric group S_{l} to obtain the individual power operations. We also use the ideas of [6] for the proofs of the relations between power operations and the Bocksteain homomorphism, the Cartan formula and the Adem relations. Our construction of the total power operations is not directly similar to any of the standard topological constructions^{3}. One would get its direct topological analog if one unfolded, to the space-level, the description of power operations in terms of E_{∞}-structure on the Eilenberg-MacLane spectra. In particular, the Thom isomorphism in the motivic cohomology and the Euler classes of vector bundles figure prominently in most of our computations.

Several important results on operations in motivic cohomology are not discussed in this paper and will be proved in a sequel. We do not show that operations P^{i} are unique. We do not show that operations P^{i} and the Bockstein homomorphism generate all bistable cohomological operations. Finally, we do not prove that bistable cohomological operations coincide with the endomorphisms of the Eilenberg-Maclane spectrum in the stable category.

Modulo this identification of bistable operations with the endomorphisms of H_{Z/l}, the present paper contains proofs of the following results used in [7] for the proof of the Milnor conjecture: [7, Theorem 3.14 p.31] is Propositions 3.7 and 3.8; [7, Theorem 3.16 p.32] is Theorem 9.4, Lemma 9.7, Proposition 9.6 and Lemma 9.5. As Example 13.6 shows, the inductive construction used in [7] to define operations Q_{i} is incorrect unless k contains a square root of −1. Instead we define them in a diffrent way and prove [7, Theorem 3.17 p.32] in Proposition 13.2 and Proposition 13.5.

3The construction of the total power operations given here is also slightly different from the construction given in my unpublished notes on the operations written in 1996.

The first draft of this paper was written in April 1996 i.e. exactly five years ago. During these years I discussed problems related to operations in motivic cohomology with a lot of people and I am greatful to all of them for these conversations. I would like to especially thank Mike Hopkins, Fabien Morel, Charles Rezk and Alexander Vishik.

This paper was written while I was a member of the Institute for Advanced Study in Princeton and, part of the time, an employee of the Clay Mathematics Institute. I am very grateful to both institutions for their support.

2 Motivic cohomology and operations

For any p, q ∈ Z and any abelian group A the motivic cohomology of a smooth scheme X are defined as hypercohomology

H^{p,q}(X, A) := H^{p}(X_{Nis}, A(q))

where A(q) = Z(q) ⊗ A and Z(q) is a certain complex of sheaves of abelian groups on (Sm/k)_{Nis}. Let K(p, q, A) be the simplicial abelian sheaf corresponding to the complex A(q)[p]. Considered as a pointed simplicial sheaf of sets it defines an object of the pointed motivic homotopy category H_{•}(k). Since motivic cohomology are homotopy invariant the simplicial sheaves K(p, q, A) are A^{1}-local and for a smooth scheme X one has

Hom_{H_{•}(k)}(X_{+}, K(p, q, A)) = H^{p,q}(X, A)

i.e. the objects K(p, q, A) represent motivic cohomology on H_{•}. For any pointed simplicial sheaf F_{•} we define

H^{p,q}(F_{•}, A) := Hom_{H_{•}(k)}(F_{•}, K(p, q, A))

Let R be a commutative ring with unit. Then R(q) are complexes of sheaves of free R-modules and one has multiplication maps

R(q) ⊗ R(q′) → R(q + q′)

which define morphisms of pointed sheaves

K(p, q, R) ∧ K(p′, q′, R) → K(p + p′, q + q′, R)  (2.1)

Recall that for a smooth scheme X over k we denote by Z_{tr}(X) the presheaf on Sm/k which takes U to the group of cycles on U × X which are finite and equidimensional over U. For an abelian group A we denote by A_{tr}(X) the presheaf Z_{tr}(X) ⊗ A.

Let K_{n,A} be the pointed sheaf of sets associated with the presheaf

K_{n,A}^{pre} : U ↦ A_{tr}(A^{n})(U)/A_{tr}(A^{n} − {0})(U)

A section of K_{n}^{pre} on U is an equivalence class of cycles on U × A^{n} with coefficients in A which are equidimensional and finite over U. If R is a commuuative ring then we have the multiplication maps

K_{n,R} ∧ K_{m,R} → K_{n+m,R}  (2.2)

which send the pair of cycles Z, Z′ to the external product cycle Z ⊗Z′. The following result is proved in [2].

Theorem 2.1 There are A^{1}-weak equivalences K_{n,R} → K(2n, n, R) which are compatible with the multiplication maps.

Theorem 2.2 The morphisms (2.1) define, for any F_{•}, a structure of an associative R-algebra on H^{∗,∗}(F_{•}, R) which is graded commutative with respect to the first grading.

Let S_{s}^{1} and S_{t}^{1} be the circles in H_{•}. We have canonical classes

σ_{s} ∈ H^{1,0}(S_{s}^{1}, R)
σ_{t} ∈ H^{1,1}(S_{t}^{1}, R)

Multiplication with these classes gives us suspension morphisms

H^{p,q}(F_{•}, R) → H^{p+1,q}(F_{•} ∧ S_{s}^{1}, R)
H^{p,q}(F_{•}, R) → H^{p+1,q+1}(F_{•} ∧ S_{t}^{1}, R)

Theorem 2.3 The suspension morphisms are isomorphisms.

We define a bistable cohomological operation of bidegree (i, j) in motivic cohomology with coefficients in R as a collection of natural transformations of functors on H_{•}

ϕ_{p,q} : H^{p,q}(−, R) → H^{p+i,q+j}(−, R)

whcih commutes with the supension morphisms i.e. such that for x ∈ H^{p,q} one has

\begin{matrix}ϕ_{p+1,q}(x ∧ σ_{s}) = ϕ_{p,q}(x) ∧ σ_{s} \\ ϕ_{p+1,q+1}(x ∧ σ_{t}) = ϕ_{p,q}(x) ∧ σ_{t} \end{matrix}  (2.3)

Denote by σ_{T} the canonical element in H^{2,1}(T, R) where T = A^{1}/A^{1} − {0}.

Proposition 2.4 There is a bijection between the set of bistable cohomological operation of bidegree (i, j) and the set of natural transformations

ϕ_{n} : H^{2n,n}(−, R) → H^{2n+i,n+j}(−, R)

given for all n ≥ 0 such that

ϕ_{n+1}(x ∧ σ_{T}) = ϕ_{n}(X) ∧ σ_{T}  (2.4)

Proof: Since T = S_{s}^{1} ∧ S_{t}^{1} and σ_{T} = σ_{s} ∧ σ_{t}, the restriction of a bistable operation to groups of degree (2n, n) satisfies (2.4). On the other hand, for a family ϕ_{n} we can construct ϕ_{p,q} as follows. For F_{•} we have

H^{p,q}(F_{•}) = H^{p+a+b,q+b}(S_{t}^{a} ∧ S_{s}^{b} ∧ F_{•})

taking a = 2q−p+b and taking b to be greater or equal to max{0, −q, p−2q} we get:

H^{p,q}(F_{•}) = H^{2(q+b),q+b}(S_{t}^{a} ∧ S_{s}^{b} ∧ F_{•})

where a, b, q + b ≥ 0. Using these isomorphisms, operation ϕ_{q+b} defines a map

ϕ_{p,q} : H^{p,q}(F_{•}) → H^{p+i,q+j}(F_{•})

The condition (2.4) implies that this map does not depend on the choice of b and that the maps ϕ_{p,q} all (p, q) satisfy (2.3).

Combining Theorem 2.1 and Proposition 2.4 we get.

Proposition 2.5 There is a bijection between the set of bistable cohomological operations of bidegree (i, j) and collections of motivic cohomology classes ϕ_{n} ∈ H^{2n+i,n+j}(K_{n,R}, R) such that the restriction of ϕ_{n+1} to K_{n,R} ∧ T is ϕ_{n} ∧ σ_{T}.

Let c : S_{s}^{1} → S_{s}^{1} ∨S_{s}^{1} be the map in H_{•} corresponding to the usual codiagonal on the (simplicial) circle.

Proposition 2.6 Let F_{•} be a pointed simplicial sheaf. Then for any p, q the map

H^{p,q}(S_{s}^{1} ∧ F_{•}) ⊕ H^{p,q}(S_{s}^{1} ∧ F_{•}) → H^{p,q}(S_{s}^{1} ∧ F_{•})

defined by the codiagonal c is of the form (a, b) ↦ a + b.

Proof: It follows by adjunction argument from the fact that the map of the free abelian groups defined by c is isomorphic, in the derived category of sheaves of abelian groups, to the diagonal map Z[1] → Z[1] ⊕ Z[1].

Corollary 2.7 Let α : H^{p,q} → H^{r,s} be a cohomological operation. Then for any pointed simplicial sheaf F_{•} the map:

H^{p,q}(S_{s}^{1} ∧ F_{•}) → H^{r,s}(S_{s}^{1} ∧ F_{•})

defined by α is a homomorphism of abelian groups.

Proof: Follows from Proposition 2.6 using the naturality of α with respect to the map defined by the codiagonal c.

Corollary 2.8 Let α : H^{∗,∗} → H^{∗+i,∗+j} be a bistable cohomological operation. Then for any F_{•} the map

H^{∗,∗}(F_{•}) → H^{∗+i,∗+j}(F_{•})

defined by α is a homomorphism of abelian groups.

Proof: Follows from Corollary 2.7.

3 Operations H^{2d,d} → H^{2d+∗,d+i} for i ≤ 0

In this section it will be convenient for us to use a diffrent model for the space K_{n,A}. We define K′_{n,A} as the sheaf which sends U to the group of cycles with coefficients in R on U × A^{n} which are equidimensional and of relative dimension zero over U. The following theorem is proved in [8].

Theorem 3.1 For any A and n, the objects K_{n,A} and K′_{n,A} are isomorphic in the A^{1}-homotopy category of sheaves with transfers.

As a corollary we get the following result.

Corollary 3.2 The pointed sheaves K_{n,A} and K′_{n,A} are isomorphic in H_{•}.

Given a pointed sheaf F define its standard simplicial resolution as the simplicial sheaf G_{•}F with terms of the form

G_{i}F = ( ∐_{X_{0}→...→X_{i};f∈F (X_{i})−{∗}} X_{0})_{+}

where the coproduct is taken over all sequences of morphisms of length i in some small subcategory equivalent to Sm/k (see [2] for more details). One verifies easily that the obvious morphism G_{•}F → F is a weak equivalence of pointed simplicial presheaves.

For a cycle Z on X denote by Supp(Z) the closure of the set of points which appear in Z with non-zero multiplicities. Consider G_{•}K′_{n,A} × A^{n}. For each i let F_{n,i} be the open subset in

G_{i}K′_{n,A} × A^{n} = ( ∐_{X_{0}→...→X_{i};Z∈z(X_{i}×A^{n}/X_{i})−{0}} X_{0} × A^{n})_{+}

whose component corresponding to (X_{0} → … → X_{i}; Z) is the complement to X_{0} ×_{X_{i}} Supp(Z). The following lemma is straightforward.

Lemma 3.3 The collection of subsheaves F_{n,i} forms a simplicial subsheaf in G_{•}K′_{n,A} × A^{n}.

Proposition 3.4 The composition F_{n,•} → G_{•}K′_{n,A} × A^{n} → K′_{n,A} is A^{1}- homotopic to the zero morphism.

Proof: To prove the proposition it is sufficient to construct for any X and any Z in Hom(X, K′_{n,A}) = z(X × A^{n}/X) an A^{1}-homotopy from the map

X × A^{n} − Supp(Z) → X → K′_{n,A}

such that these homotopies are natural in X. Consider the map

h : (X × A^{n} − Supp(Z)) × A^{n} × A^{1} → (X × A^{n} − Supp(Z)) × A^{n}

which sends (x, u, v, t) to (x, u, u(1 − t) + vt. This map is flat over the complement to X × Δ(A^{n}) in the target. Consider the pull-back p^{∗}(Z) of Z along the map p : X × A^{n} − Supp(Z) → X. It is a cycle on the target of h and the support of this cycle does not intersect X × Δ(A^{n}). Therefore, the flat pull-back h^{∗}(p^{∗}(Z)) is defined. One verifies easily that this cycle is equidimensional over (X × A^{n} − Supp(Z)) × A^{1} of relative dimension zero and hence defines a map:

H : (X × A^{n} − Supp(Z)) × A^{1} → K′_{n,A}

The restriction of h^{∗}(p^{∗}(Z)) to t = 0 is p^{∗}(Z) and the restriction to t = 1 is the zero cycle. Therefore, H is a homotopy of the required form. It is clear that our construction is functorial in X.

Corollary 3.5 The object K′_{n,A} is a retract, in H_{•}, of G_{•}K′_{n,A} × A^{n}/F_{n,•}.

We will often use below the following result.

Lemma 3.6 Let k be a field, X be smooth scheme over k and Z a closed subscheme in X everywhere of codimension at least c. Then H^{∗,q}(X/(X − Z)) = 0 for q < c and

H^{p,c}(X/(X − Z), A) = \{ \begin{matrix}⊕_{z∈Z^{c}}A & for p = 2c \\ 0 & for p ≠ 2c \end{matrix}

where Z^{c} is the set of points of Z which are of codimension c in X.

Proposition 3.7 For any n ≥ 0, m < n and any abelian groups A, B one has

H^{∗,m}(K_{n,A}, B) = 0

Proof: Follows immediately from Corollary 3.5 and Lemma 3.6.

Proposition 3.8 For any n > 0 and any abelian groups A, B one has

H^{p,n}(K_{n,A}, B) = \{ \begin{matrix}Hom(A, B) & for p = 2n \\ 0 & for p < 2n \end{matrix}

Proof: The fact that H^{p,n}(K_{n,A}, B) = 0 for p < 2n follows immediately from Corollary 3.5 and Lemma 3.6. Consider the case p = 2n. We have an obvious map from Hom(A, B) to H^{2n,n}(K_{n,A}, B). On the other hand, an element of H^{2n,n}(K_{n,A}, B) considered as an operation defines a map

A = H^{2n,n}(T^{n}, A) → H^{2n,n}(T^{n}, B) = B

Since T = S_{s}^{1} ∧ S_{t}^{1} and n > 0, Corollary 2.7 implies that this map is a homomorphism. Therefore, it is sufficient to show that an element α of H^{2n,n}(K_{n,A}, B) is zero if the corresponding operation is zero on H^{2n,n}(T^{n}, A). By Corollary 3.5 any α is determined by its action on objects of the form X × A^{n}/(X × A^{n} − Z) where Z is a closed subset equidimensional of relative dimension 0 over A^{n}. Let Z_{sing} be the closed subset of singular points of Z. Since k is perfect, Z_{sing} is of codimension at least n + 1 in X × A^{n}. Lemma 3.6 implies that the motivic cohomology of weight n of X ×A^{n}/(X ×A^{n} −Z) map to the motivic cohomology of weight n of (X ×A^{n}−Z_{sing})/(X ×A^{n}−Z) isomorphically. It remains to show that α acts trivially on

H^{2n,n}((X × A^{n} − Z_{sing})/(X × A^{n} − Z), A)

The normal bundle to Z − Z_{sing} in X × A^{n} − Z_{sing} is trivial. Hence, by the homotopy purity theorem [5], we have a weak equivalence

(X × A^{n} − Z_{sing})/(X × A^{n} − Z) = Σ_{T}^{n}((Z − Z_{sing})_{+})

let Z_{i}, i = 1, …, m be the connected components of Z −Z_{sing}. Then we have a map

Σ_{T}^{n}((Z − Z_{sing})_{+}) → ∨_{i=1}^{m}T^{n}

and H^{j,0}(Z_{i}) is non-zero only for j = 0 where it is A, this map defines an isomorphism on H^{2n,n}(−, A). We conclude that α acts by zero since by assumption it acts by zero on H^{2n,n}(T^{n}, A).

4 Thom isomorphism and Euler classes

If E is a vector bundle and P(E) is the projective bundle defined by E then the line bundle O(−1) on P(E) gives a class in H^{2,1}(P(E), Z) which we denote by σ. The following result is proved in [][].

Theorem 4.1 For any smooth scheme X over k and a vector bundle E on X of dimension d, elements 1, σ, …, σ^{d−1} form a basis of the H^{∗,∗}-module H^{∗,∗}(P(E), Z).

Recall that for a vector bundle E on X we denote by T h(E) the pointed sheaf E/E − z(X) where z : X → E is the zero section. Consider the projective bundle P(E ⊕ O). We have two morphisms

X = P(O) → P(E ⊕ O)

P(E) → P(E ⊕ O)

The complement to the image of the second morphism is E and the open excision implies that

T h(E) = P(E ⊕ O)/P(E ⊕ O) − X

On the other hand the map P(E) → P(E ⊕ O) − X is locally of the form P^{n−1} → P^{n} − pt and therefore it is an A^{1}-weak equivalence. We conclude that the morphism

P(E ⊕ O)/P(E) → T h(E)

is a weak equivalence. If f : E → E′ is a monomorphism of vector bundles and P(f) : P(E) → P(E′) is the corresponding morphism of projective bundles then P(f)^{∗}(O(−1)) = O(−1). Together with Theorem 4.1 this implies that the map on motivic cohomology defined by P(E) → P(E ⊕ O) is a split mono and that there is a unique class in H^{2d,d}(T h(E), Z) whose image in the cohomology of P(E ⊕ O) is of the form σ^{d} + ∑_{i<d} a_{i}σ^{i}. This class is called the Thom class of E and denoted t_{E}.

The obvious “diagonal” map T h(E) → X_{+}∧T h(E) defines multiplication (x, y) ↦ x ∧ y with x ∈ H^{∗,∗}(X), y ∈ H^{∗,∗}(T h(E)) and x ∧ y ∈ H^{∗,∗}(T h(E)).

Proposition 4.2 For any pointed simplicial sheaf F_{•} the map a ↦ a ∧ t_{E} from H^{∗,∗}(F_{•} ∧ X_{+}) to H^{∗+2d,∗+d}(F_{•} ∧ T h(E)) is an isomorphism.

Proof: A standard argument shows that it is sufficient to prove the proposition for F_{•} = pt. In this case it follows immediately from out definition of the Thom class and the projective bundle theorem.

Corollary 4.3 The Thom class t_{E} is a unique class in H^{2d,d}(T h(E)) whose restriction to any generic point of X is the tautological class in H^{2d,d}(T^{d}).

For a vector bundle E define the Euler class e(E) in H^{2d,d}(X) as the restriction of t_{E} with respect to the zero section map X_{+} → T h(E).

Lemma 4.4 Let L be a line bundle. Then e(L) coincides with the canonical class of L in H^{2,1}. In particular, for two line bundles L, L′ one has e(L ⊗ L′) = e(L) + e(L′).

Proof: Let σ + c be the image of t_{L} in H^{2,1}(P(L ⊕ O)). The restriction of σ to P(O) is O(−1) for O i.e. zero. Therefore, e(L) = c. On the other hand c is defined by the condition that the restriction of σ + c to P(L) is zero. Since the restriction of σ is the class of −L we conclude that e(L) = L.

Lemma 4.5 Let E, E′ be two vector bundles. Then e(E ⊕ E′) = e(E)e(E′).

Proof: Corollary 4.3 implies that the Thom class for the sum of two bundles is the smash product of Thom classes. In particular, e(E ⊕ E′) = e(E)e(E′).

Lemma 4.6 Let f : E → E′ be a monomorphism of vector bundles on a quasi-projective scheme X such that E′/E is again a vector bundle and th(f) : T h(E) → T h(E′) be the corresponding map of Thom spaces. Then th(f)^{∗}(t′_{E}) = t_{E} ∧ e(E/E′).

Proof: Since X is quasi-projective we can find an affine torsor X′ → X such that X′ is affine. Since motivic cohomology of X and X′ are the same it is sufficients to prove the lemma for an affine X. Over an affine X the sequence E → E′ → E′/E splits and we have E′ = E ⊕ E′/E. Corollary 4.3 implies that t_{E′} = t_{E} ∧ t_{E′/E} which in turn implies the statement of the lemma.

5 Total power operations

Construction 5.1 Let E, L be vector bundles on X and ϕ : E ⊕L → O^{n} an isomorphism. For an cycle Z on E with coefficients in a commuuative ring R which is equidimensional and finite over X consider the cycle on L×_{X} E ×_{X} L whose fiber over a point (x, l) of L is Z_{x} × l. One verifies easily that this is a cycle equidimensional and finite over L. Identifying E ×_{X} L with A_{X}^{n} by means of ϕ we get a section of Z_{tr}(A^{N}) on L. The restriction of this section to L − z(X) where z : X → L is the zero section lies in Z_{tr}(A^{N} − {0}). Therefore, it gives a map of pointed sheaves T h(L) → K_{n,R} which we denote a(Z).

Lemma 5.2 The motivic cohomology class ã(Z) in H^{2dim(E),dim(E)}(X) defined by a(Z) through the Thom isomorphism does not depend on the choice of L and ϕ.

Construction 5.3 Let G be a finite group, r : G → S_{n} a permutational representation of G, U a smooth scheme with free action of G and L a vector bundle on U/G given together with an isomorphism ξ_{n} ⊕ L → O^{N} where ξ_{n} is the vector bundle of dimension n on U/G corresponding to r. Given any such collection and a cycle Z on X × A^{i} equidimensional and finite over X define a map

P̃(Z) : X ∧ T h_{U/G}(L^{i}) → K_{iN,R}

as follows. Let Z^{⊗n} be the external power of Z. It is a cycle on (X × A^{i})^{n}. Let p^{∗}(Z^{⊗n}) be its flat pull-back to (X ×A^{i})^{n} ×U. Since p^{∗}(Z^{⊗n}) is invariant under the action of G and the action of G on U is free there exists a unique cycle Z′ on ((X × A^{i})^{n} × U)/G whose pull-back to (X × A^{i})^{n} × U is Z. One verifies easily that Z′ is finite and equidimensional over (X^{n} × U)/G. Therefore, we can pull it back to a cycle Z′′ on X × (A^{in} × U)/G by means of the diagonal map X → X^{i}. The scheme (A^{in} × U)/G is the vector bundle ξ_{n}^{i} over U/G and we define P̃(Z) as a(Z′′). One verifies immediately that if Z lies in X × (A^{i} − {0}) then P̃(Z) = 0.

This construction defines a morphism of pointed sheaves:

P̃ : K_{i,R} ∧ T h_{U/G}(L^{i}) → K_{iN,R}

Since smash products preserve A^{1}-weak equivalences, this morphism defines an operation

P̃ : H^{2i,i}(−, R) → H^{2iN,iN}(− ∧ T h_{U/G}(L^{i}), R)

By Lemma 4.2 the Thom isomorphism defines a morphism in H_{•} of the form

P : K_{i,R} ∧ (U/G)_{+} → K_{in,R}

or, equivalently an operation

P : H^{2i,i}(−, R) → H^{2in,in}(− ∧ (U/G)_{+}, R)

such that

P̃(x) = P (x) ∧ t_{L^{i}}  (5.1)

Lemma 5.2 implies immediately the following result.

Lemma 5.4 The operation P = P_{G,r,U,L,ϕ} does not depend on the choice of L and ϕ.

If U is a quasi-projective scheme with a free action of G then we can find, using the standard trick, an affine smooth scheme Ũ with a free action of G and an equivariant morphism Ũ → U. Since any vector bundle E on an affine scheme is “invertible”, i.e. there is an L such that L⊕E ≅ O^{N}, we can define P for Ũ. Since Ũ → U is an A^{1}-weak equivalence this means that we have a well defined operation P for any G, r : G → S_{n} and any quasi-projective U with a free G-action. The following lemma is straightforward.

Lemma 5.5 Let G be a finite group and r : G → S_{n} a permutational representation of G. Let U, V be smooth quasi-projective schemes with free actions of G and f : U → V an equivariant morphism. Then for any x one has P_{U}(x) = f^{∗}(P_{V} (x)).

Lemma 5.6 Let G, r, U, L, ϕ be as above. Then the following diagram of morphisms of pointed sheaves commutes

\begin{matrix}K_{i} ∧ K_{j} ∧ T h_{U/G}(L^{i+j}) & ⟶ & K_{i} ∧ T h_{U/G}(L^{i}) ∧ K_{j} ∧ T h_{U/G}(L^{j}) \\ ↓ & & ↓ \\ ↓ & & K_{iN} ∧ K_{jN} \\ ↓ & & ↓ \\ K_{i+j} ∧ T h_{U/G}(L^{i+j}) & ⟶ & K_{(i+j)N} \end{matrix} (5.2)

Proof: Direct comparsion.

Lemma 5.7 For a ∈ H^{2i,i}(F_{•}) and b ∈ H^{2j,j}(F′_{•}) one has

P (a ∧ b) = Δ^{∗}(P (a) ∧ P (b))  (5.3)

where Δ : U/G → U/G × U/G is the diagonal.

Proof: By Lemma 5.2 we have P̃(a ∧ b) = δ^{∗}(P̃(a) ∧ P̃(b)) where

δ : T h_{U/G}(L^{i+j}) → T h_{U/G}(L^{i}) ∧ T h_{U/G}(L^{j})

Therefore by (5.1):

P (a ∧ b) ∧ t_{L^{i+j}} = δ^{∗}(P (a) ∧ t_{L^{i}} ∧ P (b) ∧ t_{L^{j}})

Since δ^{∗}(t_{L^{i}} ∧ t_{L^{j}}) = t_{L^{i+j}} the Thom isomorphism theorem implies (5.3).

Lemma 5.8 Let η^{i} ∈ H^{2i,i}(T^{i}) be the tautological class. Then P (η^{i}) = δ^{∗}(t_{ξ_{n}^{i}}) where ξ_{n} is the vector bundle on U/G coresponding to the representation r : G → S_{n} and δ is the map on Thom spaces defined by the embedding of vector bundles O^{i} → ξ_{n}^{i} on U/G.

Proof: Take A^{i}/A^{i} − {0} as a model of T^{i} such that η^{i} is given by the tautological section Z of Z_{tr}(A^{i}) on A^{i}. Applying the construction of P̃ to the corresponding diagonal cycle Z we get the restriction to T h(L^{i}) of the morphism

W : T h(L^{i} ⊕ ξ_{n}^{i}) → Z_{tr}(A^{N}i)

corresponding, by Construction 5.1, to the tautological cycle on ξ_{n}^{2i} over ξ_{n}^{i}. In view of Corollary 4.3 W represents the Thom class of the trivial bundle (L ⊕ ξ_{n})^{i}. Applying Thom isomorphism to get P (Z) we conclude that P (Z) is the restriction of the Thom class of T h(ξ_{n}^{i}) with respect to the morphism δ.

Lemma 5.9 Let E be a vector bundle on X. Then P (t_{E}) = δ^{∗}(t_{E⊗ξ_{n}}) where ξ_{n} is the vector bundle on U/G coresponding to the representation r : G → S_{n} and δ is the map on Thom spaces defined by the embedding of vector bundles E ⊗ O → E ⊗ ξ_{n} on X × (U/G).

Proof: Follows from Corollary 4.3 and Lemma 5.8.

Let ∗ be a k-point of U/G which lifts to a k-point of U. Let i : S^{0} → (U/G)_{+} be the corresponding morphism. The following lemma is straightforward.

Lemma 5.10 The composition

K_{i,R} →^{i∧Id} K_{i,R} ∧ (U/G)_{+} →^{P} K_{in,R}

coincides in H_{•} with the n-th power map.

6 Motivic cohomology of BS_{l}

Let G be a linear algebraic group and G → GL(V) a faithful representation of G. Denote by Ṽ_{n} the open subset in A(V)^{n} where G acts freely. We have a sequence of closed embeddings Ṽ_{n} → Ṽ_{n+1} given by (v_{1}, …, v_{n}) ↦ (v_{1}, …, v_{n}, 0). Set BG = colim_{n}Ṽ_{n}/G where Ṽ_{n}/G is the quotient scheme and the colimit is taken in the category of sheaves. In [5] we used the notation B_{gm}G for BG. As shown there, the homotopy type of BG does not depend on the choice of G → GL(V). We denote by ∗ any k-rational point of BG which lifts to a k-rational point in one of the Ṽ_{n}’s. The goal of this section is to decribe motivic cohomology of BS_{l} with coefficients in Z/l. We start with the following general result.

Proposition 6.1 For any G and V as above the map Ṽ_{n}/G → Ṽ_{n+1}/G defines an isomorphism on motivic cohomology of weigh less than n.

Proof: The morphism Ṽ_{n}/G → Ṽ_{n+1}/G can be represented by the composition

Ṽ_{n}/G → (Ṽ_{n} × V)/G → Ṽ_{n+1}/G

The first of these maps is the zero section of a vector bundle (Ṽ_{n} × V)/G → Ṽ_{n}/G and gives an isomorphism on motivic cohomology by homotopy invariance. The second map is an open embedding and the codimension of the complement is at least n. Therefore, it defines an isomorphism on H^{∗,<n}.

Corollary 6.2 One has

H^{∗,∗}(BG) = lim_{m}H^{∗,∗}(Ṽ_{m}/G)

Proof: Given any sequence of maps of pointed simplicial sheaves F_{•,i} → F_{•,i+1} with the colimit F_{•,∞} we have a long exact sequence of the form

… → ∏ H^{∗−1,∗}(F_{•,i}) → H^{∗,∗}(F_{•,∞}) → ∏ H^{∗,∗}(F_{•,i}) → ∏ H^{∗,∗}(F_{•,i}) → …

The limit lim_{m}H^{∗,∗}(F_{•,i}) is the kernel of the fourth arrow and therefore to prove the corollary it is sufficient to show that the map

∏ H^{∗,∗}(F_{•,i}) → ∏ H^{∗,∗}(Ṽ_{m}/G)

is an epimorphism. This follows from the proposition.

Lemma 6.3 Bμ_{l} = O(−l)_{P^{∞}} − z(P^{∞})

Proof: Take the canonical 1-dimensional representation of μ_{l}. The Ṽ_{n} = A^{n} − {0}. The projection A^{n} − {0} → P^{n−1} is invariant under the action of μ_{l} and therefore gives a map (A^{n} − {0})/μ_{l} → P^{n−1}. One verifies that this map is isomorphic to the complement to the zero section of the line bundle O(−l) on P^{n−1}.

Lemma 6.3 implies that one has a cofibration sequence of the form

(Bμ_{l})_{+} → (O(−l)_{P^{∞}})_{+} → T h(O(−l))  (6.1)

For a vector bundle E of dimension d, the composition of the Thom isomorphism H^{∗,∗}(X) → H^{∗+2d,∗+d}(T h(E)) with the restriction to the zero section z^{∗} : H^{∗,∗}(T h(E)) → H^{∗,∗}(X) is given by

x ↦ z^{∗}(x ∧ t_{E}) = x ∧ z^{∗}(t_{E}) = x ∧ e(E)

By Lemma 4.4, e(O(−l)) = lσ where σ ∈ H^{2,1}(P^{∞}) is the same class as in the projective bundle theorem 4.1. Therefore, the long exact sequence defined by (6.1) is of the form

… → H^{∗−2,∗−1}[[σ]] →^{∧lσ} H^{∗,∗}[[σ]] → H^{∗,∗}(Bμ_{l}) → H^{∗−1,∗−1}[[σ]] → … (6.2)

Lemma 6.4 There exists a unique element u ∈ H^{1,1}(Bμ_{l}, Z/l) such that the restriction of u to ∗ is zero and the image of u in H^{2,1}(Bμ_{l}, Z) is the Euler class e(O(−1)).

Proof: Existence follows from the exact sequence

H^{1,1}(Bμ_{l}, Z) → H^{1,1}(Bμ_{l}, Z/l) → H^{2,1}(Bμ_{l}, Z) →^{l} H^{2,1}(Bμ_{l}, Z)

and the fact that le(O(−1)) = e(O(−l)) = 0 in H^{2,1}(Bμ_{l}, Z). The exact sequence (6.2) around H^{1,1}(Bμ_{l}, Z) shows that H^{1,1}(Bμ_{l}, Z) = H^{1,1}(k). This implies the uniqueness.

Let u be the element defined in Lemma 6.4 and v be the Euler class of O(−1) in H^{2,1}(Bμ_{l}).

Proposition 6.5 For any pointed simplicial sheaf F_{•} the elements v^{i} and uv^{i}, i ≥ 0 form a basis of H^{∗,∗}(F_{•} ∧ (Bμ_{l})_{+}) over H^{∗,∗}(F_{•}).

Proof: The standard argument shows that it is sufficient to consider the case when F_{•} is of the form X_{+} for a smooth scheme X. The same reasoning as we used to establish (6.2) applies to the motivic cohomology groups of X × Bμ_{l} for any smooth scheme X and we get the following result.

Lemma 6.6 For any smooth scheme X there is a long exact sequence of H^{∗,∗}(X)[[σ]]-modules of the form

\begin{matrix}… → H^{∗−2,∗−1}(X)[[σ]] →^{∧lσ} H^{∗,∗}(X)[[σ]] → H^{∗,∗}(X × Bμ_{l}) → \\ → H^{∗−1,∗−1}(X)[[σ]] → … \end{matrix}  (6.3)

For Z/l-coefficients we have lσ = 0 and (6.3) becomes a short exact sequence of H^{∗,∗}(F_{•})[[σ]]-modules of the form

0 → H^{∗,∗}(X)[[σ]] → H^{∗,∗}(X × Bμ_{l}) → H^{∗−1,∗−1}(X)[[σ]] → 0  (6.4)

Let u′ be an element in H^{1,1}(Bμ_{l}, Z/l) whose image in H^{0,0}(Spec(k)) = Z/l is 1. The short exact sequence (6.4) implies that the monomials u′v^{i} and v^{i} form a basis of H^{∗,∗}(X × Bμ_{l}) over H^{∗,∗}(X). On the other hand, the image of u in H^{0,0}(Spec(k)) = Z/l is not zero and hence u = cu′ where c ∈ (Z/l)^{∗}. This implies that the monomials uv^{i}, v^{i} also form a basis.

To describe the multiplicative structure of H^{∗,∗}(F_{•} ∧ (Bμ_{l})_{+}) it is sufficient to find an expression for u^{2}. If l ≠ 2 then u^{2} = 0 since the multiplication in motivic cohomology is graded commutative. Consider the case l = 2. We can clearly assume that F_{•} = Spec(k)_{+}. The element u^{2} lies in H^{2,2} and Lemma 6.5 shows that

H^{2,2}(Bμ_{l}) = H^{0,1}(k)v ⊕ H^{1,1}(k)u ⊕ H^{2,2}(k)

Since u is zero in ∗ the projection of u^{2} to the last factor is zero and we get u^{2} = xv + yu for x ∈ H^{0,1}(k) and y ∈ H^{1,1}(k). To compute y consider the map

A^{1} − {0} = (A^{1} − {0})/μ_{l} → Bμ_{l}

The restriction of u by means of this map is the canonical generator H^{1,1}(A^{1}− {0}, 1). Therefore, the following lemma shows that y is the class of −1 in H^{1,1}.

Lemma 6.7 Let u be the canonical generator of H^{1,1}(A^{1} − {0}, 1). Then u^{2} = ρu where ρ ∈ H^{1,1}(k) is the class of −1.

To compute x note that H^{0,1}(k, Z/2) = μ_{2}(k). If char(k) = 2 then this group is zero. If char(k) ≠ 2 it is Z/2 and we only need to know whether x is zero or not. The following lemma implies that x = τ is the generator of Z/2.

Lemma 6.8 Let k be a separably closed field of characteristic not equal to 2. Then u^{2} ≠ 0.

We proved the following result.

Theorem 6.9 For any field k and a pointed simplicial sheaf F_{•} over k one has:

H^{∗,∗}(F_{•} ∧ (Bμ_{l})_{+}, Z/l) = H^{∗,∗}(F_{•}, Z/l)[[u, v]]/(u^{2} = τv + ρu)  (6.5)

where:

1. ρ is the class of −1 in H^{1,1}(k)

2. τ is zero if l ≠ 2 or char(k) = 2

3. τ is the generator of H^{0,1}(k, Z/2) = μ_{2}(k) if l = 2 and char(k) ≠ 2.

We also need the following additional fact about cohomology of Bμ_{l}.

Lemma 6.10 Let c ∈ Aut(μ_{l}) = (Z/l)^{∗} and let \underline{c} be the corresponding automorphism of Bμ_{l}. Then \underline{c}^{∗}(u) = cu and \underline{c}^{∗}(v) = cv.

Proof: Let L be the line bundle on Bμ_{l} corresponding to the canonical 1dimensional representation λ of μ_{l}. Then, up to the sign, L is the pull-back of O(−1) from P^{∞}. In particular, v is the Euler class of L and u is the only element in H^{1,1}(−, Z/l) which is zero at ∗ and which maps to e(l) under the map H^{1,1}(−, Z/l) → H^{2,1}(−, Z). The automorphism \underline{c} take the canonical 1-dimensional representation λ to λ^{⊗c} and, therefore, it takes L to L^{⊗c}. Our result follows now from Lemma 4.4.

Our next goal is to compute H^{∗,∗}(F_{•}∧(BS_{l})_{+}, Z/l) where S_{l} is the symmetric group and l is a prime not equal to char(k).

Lemma 6.11 Let G be a finite group and H a subgroup of G. Assume that [G : H] is invertible in the coefficients ring R. Then the map of motivic cohomology

H^{∗,∗}(BG, R) → H^{∗,∗}(BH, R)

is a split mono and its image is contained in the invariants under the action of the normalizer of H in G.

Let d be the Euler class of the (n-1)-dimensional vector bundle ξ_{n}/O on BS_{n} where ξ_{n} is the bundle corresponding to the tautological permutational representation.

Theorem 6.12 Let l be a prime and k a field of characteristic not equal to l. There exists a unique class c ∈ H^{2l−3,l−1}(BS_{l}, Z/l) such that the image of c in H^{2l−2,l−1}(BS_{l}, Z) is d and the restriction of c to ∗ is zero.

For any pointed simplicial sheaf F_{•} over k one has:

H^{∗,∗}(F_{•} ∧ (BS_{l})_{+}, Z/l) = \{ \begin{matrix}H^{∗,∗}(F_{•}, Z/l)[[c, d]]/(c^{2} = τd + ρc) & for l = 2 \\ H^{∗,∗}(F_{•}, Z/l)[[c, d]]/(c^{2} = 0) & for l ≠ 2 \end{matrix}

(6.6)

where:

1. ρ is the class of −1 in H^{1,1}(k)

2. τ is the generator of H^{0,1}(k, Z/2) = μ_{2}(k).

Proof: To simplify the notations we assume F_{•} = (Spec(k))_{+}. For l = 2 we have S_{2} = Z/2 = μ_{2} and our result follows from Theorem 6.9. Assume that l > 2. The transfer argument shows that to prove the theorem for k it is sufficient to prove it for a separable extension of k of degree prime to l. In particular, we may assume that k contains a primitive l-th root of unity ζ. The choice of ζ defines a weak equivalence Bμ_{l} → BZ/l. The inclusion Z/l → S_{l} gives a map BZ/l → BS_{l} and for any ring of coefficients where l is invertible we get, by Lemma 6.11, a split injection:

H^{∗,∗}(BS_{l}) → H^{∗,∗}(Bμ_{l})^{Aut(μ_{l})}  (6.7)

Lemma 6.10 implies that for Z/l-coefficients the right hand side is of the form

H^{∗,∗}(Bμ_{l}, Z/l)^{Aut(μ_{l})} = H^{∗,∗}[[x, y]]/(x^{2} = 0)  (6.8)

where x = vu^{l−2} and y = u^{l−1}.

To prove the existence of c we need to show that d is l-torsion element in H^{∗,∗}(−, Z). Since (6.7) is an injection for coefficients in Z localized at l it is sufficient to prove that the image of d in H^{∗,∗}(Bμ_{l}, Z) is an l-torsion element. This restriction is the Euler class of ξ/O where ξ corresponds to the regular representation of Z/l. Therefore, we have ξ = ⊕_{i=0}^{l−1}L^{i} where L is the line bundle corresponding to the 1-dimensional representation given by ζ. By Lemma 4.5 and Lemma 4.4 we get d = −e(L)^{l−1} and le(L) = e(L^{⊗l}) = 0.

To show that c is unique, it is sufficient to check that the map

H^{2l−3,l−1}(BS_{l}, Z/l) → H^{2l−2,l−1}(BS_{l}, Z)

is injective. Injectivity of (6.7) for Z/l-coefficients implies that it is sufficient to show that the map

H^{2l−3,l−1}(Bμ_{l}, Z/l)^{Aut(μ_{l})} → H^{2l−2,l−1}(Bμ_{l}, Z)^{Aut(μ_{l})}  (6.9)

is injective. Description (6.8) shows that H^{2l−3,l−1}(BS_{l}, Z/l)^{Aut(μ_{l})} is Z/l generated by x. The image of x in H^{2l−2,l−1}(Bμ_{l}, Z) is e(O(−1))^{l−1} and in particular it is non-zero. Therefore, (6.9) is injective.

To prove the second part of the theorem we need to show that the map

H^{∗,∗}[[c, d]]/(c^{2} = 0) → H^{∗,∗}(BS_{l}, Z/l)

is an isomorphism. Since (6.7) is a mono it is sufficient to show that the composition

H^{∗,∗}[[c, d]]/(c^{2} = 0) → H^{∗,∗}(BS_{l}, Z/l) → H^{∗,∗}(Bμ_{l})^{Aut(μ_{l})}  (6.10)

is an isomorphism. One verifies easily that this map takes c to a non-zero multiple of x and d to y. The fact that (6.10) is an isomorphism follows from (6.8).

Lemma 6.13 Let L be a line bundle and l a prime not equal to char(k). Then P_{l}(e(L)) = e(L)^{l} + e(L)d.

Proof: By the transfer argument we may assume that k contains a primitive l-th root of unity. In view of Lemma 6.11 it is sufficient to prove our equality in the motivic cohomology of X × Bμ_{l}. By Lemma 5.9 we have P (e(L)) = e(L ⊗ ξ_{l}). The vector bundle ξ_{l} restricted to Bμ_{l} splits into the sum of line bundles M^{0} ⊕ M^{1} ⊕ ⋯ ⊕ M^{l−1} where M is the line bundle corresponding to the standard 1-dimensional representation of μ_{l}. By Lemmas 4.5 and 4.4 we get:

e(L ⊗ ξ_{l}) = ∏_{i} e(L ⊗ M^{i}) = ∏_{i}(e(L) + ie(M)) = e(L)(e(L)^{l−1} − e(M)^{l−1})

Since the restriction of d to Bμ_{l} is −e(M)^{l−1} this finishes the proof.

7 Symmetry theorem

Let G_{1}, G_{2} be two finite groups acting freely on U_{1} and U_{2} respectively. Let further r_{i} : G_{i} → S_{n_{i}} be permutational representations of G_{i}, i = 1, 2, ξ_{i} the corresponding vector bundles on U_{i}/G_{i} of dimension n_{i} and L_{i} ⊕ ξ_{i} → O^{N_{i}} inverses of ξ_{i}. Consider the action of G_{1} × G_{2} on U_{1} × U_{2} and let ξ_{1} ⊗ ξ_{2} be the vector bundle on (U_{1} × U_{2})/(G_{1} × G_{2}) corresponding to r_{1} × r_{2}. Consider the vector bundle

L_{12} = (L_{1} ⊗ ξ_{2}) ⊕ L_{2}^{N_{1}}

Then

L_{12} ⊕ (ξ_{1} ⊗ ξ_{2}) = ((L_{1} ⊕ ξ_{1}) ⊗ ξ_{2}) ⊕ L_{2}^{N_{1}} = ξ_{2}^{N_{1}} ⊕ L_{2}^{N_{1}} = O^{N_{2}N_{1}}

i.e. L_{12} is an inverse of ξ_{1} ⊗ξ_{2}. The canonical map O → ξ_{1} gives a monomorphism i : L_{1} ⊕ L_{2}^{N_{1}} → L_{12}.

Lemma 7.1 The following diagram of pointed sheaves commutes:

\begin{matrix}K_{i} ∧ T h_{L_{1}^{n}} ∧ T h_{L_{2}^{iN_{1}}} & ⟶^{P̃∧Id} & K_{iN_{1}} ∧ T h_{L_{2}^{iN_{1}}} \\ ^{Id∧T h(i)}↓ & & ↓P̃ \\ K_{i} ∧ T h_{L_{12}} & ⟶^{P̃} & K_{iN_{1}N_{2}} \end{matrix}  (7.1)

Proof: Direct comparison.

Proposition 7.2 Let u be a class in H^{2i,i}(F_{•}). Then:

P_{2}(P_{1}(u))e(ξ_{1}/O)^{N_{1}i} = P_{12}(u)e(ξ_{1}/O)^{N_{1}i}

in H^{2iN_{1}N_{2},iN_{1}N_{2}}(F_{•} ∧ (U_{1}/G_{1} × U_{2}/G_{2})_{+}), where:

P_{i} = P_{G_{i},r_{i},U_{i},L_{i}}
P_{12} = P_{G_{1}×G_{2},r_{1}×r_{2},U_{1}×U_{2},L_{12}}

Proof: By Lemma 7.1 we have

P̃_{2}(P̃_{1}(u)) = T h(i)^{∗}P̃_{12}(u)

or, equivalently,

P_{2}(P_{1}(u) ∧ t_{L_{1}^{i}}) ∧ t_{L_{2}^{iN_{1}}} = T h(i)^{∗}(P_{12}(u) ∧ t_{L_{12}})

By Lemma 4.6 we rewrite it as

P_{2}(P_{1}(u) ∧ t_{L_{1}^{i}}) ∧ t_{L_{2}^{iN_{1}}} = P_{12}(u) ∧ e((L_{1} ⊗ (ξ_{2}/O))^{i}) ∧ t_{L_{1}^{i}} ∧ t_{L_{2}^{iN_{1}}}

By Lemma 5.7 we get

P_{2}P_{1}(u) ∧ P_{2}(t_{L_{1}^{i}}) ∧ t_{L_{2}^{iN_{1}}} = P_{12}(u) ∧ e((L_{1} ⊗ (ξ_{2}/O))^{i}) ∧ t_{L_{1}^{i}} ∧ t_{L_{2}^{iN_{1}}}

By Lemmas 5.9 and 4.6 we get

P_{2}P_{1}(u)∧e((L_{1}⊗(ξ_{2}/O))^{i})∧t_{L_{1}^{i}}∧t_{L_{2}^{iN_{1}}} = P_{12}(u)∧e((L_{1}⊗(ξ_{2}/O))^{i})∧t_{L_{1}^{i}}∧t_{L_{2}^{iN_{1}}}

By Thom isomorphism 4.2, we get

P_{2}P_{1}(u) ∧ e((L_{1} ⊗ (ξ_{2}/O))^{i}) = P_{12}(u) ∧ e((L_{1} ⊗ (ξ_{2}/O))^{i})

Multiplying both sides by e((ξ_{1} ⊗ (ξ_{2}/O))^{i}) we get by Lemma 4.5

P_{2}P_{1}(u) ∧ e(ξ_{2}/O)^{iN_{1}} = P_{12}(u) ∧ e(ξ_{2}/O)^{iN_{1}}

Corollary 7.3 In the notations of the proposition assume that r_{1} = l is a prime diffrent from the characteristic of k. Then

P_{2}(P_{1}(u)) = P_{12}(u)

Proof: Replacing U_{2} by an affine torsor we may assume that it is affine. Let G_{2} → GL(V) be the linear representation of G_{2} corresponding to r_{2} and Ṽ_{m} the open subset of A(V^{⊕m}) where G acts freely. Then, for some m, there exists a G_{2}-equivariant map U_{2} → Ṽ_{m}. By Lemma 5.5 it is sufficient to prove the corollary for U_{2} = Ṽ_{m} and G_{2} = S_{r_{2}}. Proposition 7.2 together with Corrolary 6.2 shows that we have

P_{2}(P_{1}(u))e(ξ_{1}/O)^{N_{1}i} = P_{12}(u)e(ξ_{1}/O)^{N_{1}i}

on F_{•} ∧ (U_{1}/G_{1} × BS_{l}). By Theorem 6.12, multiplication with e(ξ_{1}/O) is injective and we conclude that P_{2}(P_{1}(u)) = P_{12}(u).

Lemma 7.4 Let U be a scheme with a free action of G and let r : G → S_{n} be a permutational representation of G. Consider:

P = P_{G×G,r×r,U×U−Δ(U)} : K_{i} ∧ ((U × U − Δ(U))/(G × G))_{+} → K_{in^{2}}

Then P is invariant under the permutation of two copies of U.

Proof: The action of G × G on U × U − Δ(U) extends to a free action of the semidirect product (G × G) ∝ Z/2. The permutational representation r × r also extends to a permutational representation of (G×G) ∝ Z/2. Therefore, P factors through the map

K_{i} ∧ ((U × U − Δ(U))/(G × G))_{+} → K_{i} ∧ ((U × U − Δ(U))/(G × G) ∝ Z/2)_{+}

which implies that it is symmetric.

Lemma 7.5 Let X be a smooth variety and Z a smooth subvariety in Z of codimension c. Then for any pointed simplicial sheaf F_{•} the map

H^{2i,i}(F_{•} ∧ X_{+}) → H^{2i,i}(F_{•} ∧ (X − Z)_{+})

is an isomorphism for i < c.

Proof: Follows from Lemma 3.6.

Lemma 7.6 Under the assmptions of Lemma 7.4 the morphism

P = P_{G×G,r×r,U×U} : K_{i} ∧ (U × U/G × G)_{+} → K_{in^{2}}

is invariant under the permutation of two copies of U.

Proof: Replacing U by an affine torsor we may assume that it is affine. Let G → GL(V) be a faithful liner representation of G and Ṽ_{m} the open subset of A(V^{⊕m}) where G acts freely. Then, for some m, there exists a G-equivariant map U → Ṽ_{m}. By Lemma 5.5 it is sufficient to show that P is symmetric for U = Ṽ_{m}. This follows from Lemma 7.4 and Lemma 7.5 since we may choose m such that the codimension of Δ(Ṽ_{m}) is larger than in^{2}.

Corollary 6.2 together with Lemma 5.5 implies that there is a well defined morphism:

P_{l} : K_{n} ∧ BS_{l} → K_{nl}

Theorem 7.7 The composition

K_{i} ∧ (BS_{l})_{+} ∧ (BS_{l})_{+} →^{P ∧Id} K_{il} ∧ (BS_{l})_{+} →^{P} K_{il^{2}}

is invariant under the permutation of two copies of BS_{l}.

Proof: Let η be the tautological motivic cohomology class of K_{i}. We need to show that P (P (η)) is invariant under the permutation of two copies of BS_{l}. Corollary 7.3 and Corollary 6.2 imply that P (P (η)) = P_{12}(η) where P_{12} is the power operation corresponding to S_{l} × S_{l} → S_{l^{2}}. We conclude by Lemma 7.6 and, again, Corollary 6.2.

8 Power operations and the Bockstein homomorphism

Let β denote the Bockstein homomorphism H^{∗,∗}(−, Z/l) → H^{∗+1,∗}(−, Z/l) which is defined by the short exact sequence of the coefficients

0 → Z/l → Z/l^{2} → Z/l → 0

It has the same properties as the Bockstein homomorphism in the ordinary cohomology. In particular, we have ββ = 0 and for u ∈ H^{p,∗},

β(u ∧ v) = β(u) ∧ v = (−1)^{p}u ∧ β(v)  (8.1)

The goal of this section is to prove Theorem 8.4. The method we use follows closely the method used to prove an analogous result in [6].

Let U and L be as in Construction 5.3. The proper push-forward of cycles defines a “transfer” map

tr : \underline{Hom}(T h_{U}(L^{i}), K_{n,R}) → \underline{Hom}(T h_{U/G}(L^{n}), K_{n,R})

where \underline{Hom}(−, −) denotes the internal Hom-object in the category of pointed sheaves. Let Z_{(l)} be the local ring of l in Z.Denote by

Φ = \underline{Hom}(T h_{U/G}(L^{n}), K_{nl,Z_{(l)}})/lIm(tr)

the pointed sheaf which corresponds to the quotient of the sheaf of abelian groups \underline{Hom}(T h_{U/G}(L^{n}), K_{nl,Z_{(l)}}) by the subgroup of elements of the form lx where x is in the image of the transfer map.

Lemma 8.1 The map of pointed sheaves

K_{n,Z/l} → \underline{Hom}(T h_{U/G}(L^{n}), K_{nl,Z/l})

adjoint to the power operation P̃_{l} lifts to a map of pointed sheaves

K_{n,Z/l} → Φ

Proof: The pointed sheaf K_{n,Z/l} is a quotient sheaf of the sheaf K_{n,Z_{(l)}} and the power operation for integral coefficients defines a map

K_{n,Z_{(l)}} → \underline{Hom}(T h_{U/G}(L^{n}), K_{nl,Z_{(l)}})

It is sufficient to show that the composition of this map with the projection to Φ factors through K_{n,Z/l} i.e. that for two cycles Z_{1}, Z_{2} be cycles on X × A^{n} with intergral coefficients such that Z_{1} − Z_{2} is divisible by l the cycle P̃_{l}(Z_{1}) − P̃_{l}(Z_{2}) is in lIm(tr). Let Z′_{i} be the cycle on (X^{l} × A^{nl} × U)/S_{l} whose pull-back to X^{l} × A^{nl} × U is (p : X^{l} × A^{nl} × U → X^{l} × A^{nl})^{∗}(Z_{i}^{⊗l}). It is sufficient to show that Z′_{1} − Z′_{2} is in lIm(π_{∗}) where

π : X^{l} × A^{nl} × U → (X^{l} × A^{nl} × U)/S_{l}

is the projection or, equivalently, that

p^{∗}(Z_{1}^{⊗l}) − p^{∗}(Z_{2}^{⊗l}) = lπ^{∗}π_{∗}(Y)

for some Y. We have Z_{1} − Z_{2} = lW and the left hand side can be rewritten as p^{∗}((Z_{2} + lW)^{⊗l}) − p^{∗}(Z_{2}^{⊗l}). Since any S_{l}-invariant cycle with coefficients divisible by l is of the form π^{∗}π_{∗}(−) it is sufficient to consider the summands in this expression with coeffients not divisible by l^{2}. They are of the form l(Z_{2} ⊗ … ⊗ W ⊗ … ⊗ Z_{2}). The sum of all such cycles is, up to multiplication by (l − 1)!, of the form π^{∗}π_{∗}(W ⊗ (Z_{2})^{l−1}).

Lemma 8.2 Let u be the tautological class in H^{2n,n}(K_{n,Z/l}). Then βP̃_{l}(u) lies in the image of the transfer map

H^{2nl,nl}(T h_{U}(L^{n}) ∧ K_{n,Z/l}, Z/l) → H^{2nl,nl}(T h_{U/G}(L^{n}) ∧ K_{n,Z/l}, Z/l)

Proof: Consider the standard simplicial resolution G_{•}K_{n,R} (see Section 3). Since a cycle with Z/l-coefficients on a smooth scheme lifts to an intergral cycle, the map

G_{0}K_{n,Z/l} → \underline{Hom}(T h_{U/G}(L^{n}), K_{n,Z/l})

which represents P̃_{l}(u) lifts to a map with values in \underline{Hom}(T h_{U/G}(L^{n}), K_{n,Z}). Let ϕ be such a lifting. Then ∂_{0}ϕ − ∂_{1}ϕ lands in lK_{n,Z} and (∂_{0}ϕ − ∂_{1}ϕ)/l gives a morphism of complexes G_{∗}K_{i,Z/l} → \underline{Hom}(T h_{U/G}(L^{n}), K_{n,Z/l}) which defines βP̃_{l}(u). Lemma 8.1 implies immediately that (∂_{0}ϕ − ∂_{1}ϕ)/l lifts to a morphism with values in the image of the transfer map which implies the statement of the lemma.

Lemma 8.3 For any F_{•}, the transfer map in cohomology

H^{∗,∗}(F_{•} ∧ T h_{ES_{l}}(L^{n}), Z/l) → H^{∗,∗}(F_{•} ∧ T h_{BS_{l}}(L^{n}), Z/l)

is zero.

Proof: The composition of the transfer map with the restriction map is the multiplication with the degree of the covering, in our case l!. Hence it is sufficient to show that the restriction map

H^{∗,∗}(F_{•} ∧ T h_{BS_{l}}(L^{n}), Z/l) → H^{∗,∗}(F_{•} ∧ T h_{ES_{l}}(L^{n}), Z/l)

is surjective. Since motivic cohomology of ES_{l} are trivial and class on the right can be written as u ∧ t where u is in H^{∗,∗}(F_{•}) and t is the Thom class. Any such u ∧ t is clearly in the image of the restriction map.

Theorem 8.4 For any u ∈ H^{2d,d} one has βP (u) = 0.

Proof: By Lemma 8.2 and Lemma 8.3 we have β(P (u) ∧ t) = βP̃(u) = 0. By (8.1) we get

β(P (u) ∧ t) = β(P (u)) ∧ t

and by the Thom isomorphism theorem we conclude that β(P (u)) = 0.

9 Individual power operations: formulas

In this section we assume that l is a prime different from the characteristic of k. Let w be a class in H^{2d,d}(F_{•}, Z/l). By Theorem 6.12, the class P_{l}(w) can be written uniquely as a linear combination of the form:

P_{l}(u) = ∑_{i≥0} C_{i+1,d}(w)cd^{i} + D_{i,d}(w)d^{i}  (9.1)

This defines cohomological operations:

C_{i,d} : H^{2d,d}(−, Z/l) → H^{2d+2(d−i)(l−1)+1,d+(d−i)(l−1)}(−, Z/l)
D_{i,d} : H^{2d,d}(−, Z/l) → H^{2d+2(d−i)(l−1),d+(d−i)(l−1)}(−, Z/l)

Below we use C_{i} instead of C_{i,d} and D_{i} instead of D_{i,d} when no confusion is possible. Recall that we denote by τ the generator of H^{0,1}(k, Z/2) for char(k) ≠ 2 and by ρ the class of −1 in H^{1,1}(k, Z/2).

Lemma 9.1 Let u ∈ H^{2d,d}, v ∈ H^{2d′,d′}. Then for l odd one has:

D_{i}(u ∧ v) = ∑_{r=0}^{i} D_{r}(u) ∧ D_{i−r}(v)

C_{i+1}(u ∧ v) = ∑_{r=0}^{i} C_{r+1}(u) ∧ D_{i−r}(v) + D_{r}(u) ∧ C_{i−r+1}(v)

and for l = 2 one has

D_{i}(u ∧ v) = ∑_{r=0}^{i} D_{r}(u) ∧ D_{i−r}(v) + τ ∑_{r=0,...,i−1} C_{r+1}(u) ∧ C_{i−r}(v)

C_{i+1}(u∧v) = ∑_{r=0}^{i} C_{r+1}(u)∧D_{i−r}(v)+D_{r}(u)∧C_{i−r+1}(v)+ρC_{r+1}(u)∧C_{i−r+1}(v)

Proof: Follows immediately from Lemma 5.7 and Theorem 6.12.

Lemma 9.2 Let t ∈ H^{2,1}(T, Z/l) be the tautological class. Then one has:

\begin{matrix}C_{i+1}(u ∧ t) = C_{i}(u) ∧ t \\ D_{i}(u ∧ t) = D_{i−1}(u) ∧ t \end{matrix}  (9.2)

Proof: By Lemma 5.8 we have P (t) = δ(T → T h(ξ_{l}/O))(t_{ξ}). In view of Lemma 4.6 and the fact that d = e(ξ_{l}/O) we get P (t) = t∧d i.e. C_{i+1}(t) = 0 for all i ≥ 0, D_{1}(t) = t and D_{i}(t) = 0 for i ≠ 1. Applying Lemma 9.1 we get (9.2).

For u ∈ H^{2d,d} define;

P^{i}(u) = D_{d−i}(u)
B^{i}(u) = C_{d−i}(u)

By (9.2) we have P^{i}(u ∧ t) = P^{i}(u) ∧ t and B^{i}(u ∧ t) = B^{i}(u) ∧ t. As shown in the proof of Proposition 2.4 we can extend P^{i} and B^{i} to operations acting on motivic cohomology groups H^{p,q} for all p, q:

P^{i} : H^{p,q} → H^{p+2i(l−1),q+i(l−1)}

B^{i} : H^{p,q} → H^{p+2i(l−1)+1,q+i(l−1)}

For l = 2 we denote, following the standard convention, Sq^{2i} = P^{i}

Sq^{2i+1} = B^{i}

Theorem 9.3 For any i < 0 one has P^{i} = B^{i} = 0. Proof: This follows from Proposition 3.7. Theorem 9.4 One has P^{0} = Id. Proof: Proposition 3.8 implies that P^{0}(u) = au where a is a constant. Lemma 6.13 applied to the canonical line bundle on P^{1} implies that a = 1. Lemma 9.5 One has βB^{i} = 0 and βP^{i} = B^{i}. Proof: Follows immediately from Theorem 8.4, the fact that β(c) = d and the product formula (8.1) for the Bockstein homomorphism. Proposition 9.6 For u, v ∈ H^{∗,∗} and l ≠ 2 one has:

P^{i}(u ∧ v) = ∑_{r=0}^{i} P^{r}(u) ∧ P^{i−r}(v)

B^{i}(u ∧ v) = ∑_{r=0}^{i}(B^{r}(u) ∧ P^{i−r}(v) + P^{r}(u) ∧ B^{i−r}(v))

For l = 2 we get:

Sq^{2i}(u ∧ v) = ∑_{r=0}^{i} Sq^{2r}(u) ∧ Sq^{2i−2r}(v) + τ ∑_{s=0}^{i−1} Sq^{2s+1}(u) ∧ Sq^{2i−2s−1}(v)

Sq^{2i+1}(u ∧ v) = ∑_{r=0}^{i}(Sq^{2r+1}(u) ∧ Sq^{2i−2r}(v) + Sq^{2r}(u) ∧ Sq^{2i−2r−1}(v))+

+ρ ∑_{s=0}^{i−1} Sq^{2s+1}(u) ∧ Sq^{2i−2s−1}(v)

Proof: Follows immediately from Lemma 9.1 and the vanishing result 9.3.

Lemma 9.7 For u ∈ H^{2n,n} one has P^{n}(u) = u^{l}.

Proof: Follows from Lemma 5.10.

Lemma 9.8 For u ∈ H^{p,q} and n > p − q, n ≥ q one has P^{n}(u) = 0.

Proof: Let i = n + q − p and j = n − q. Then σ_{s}^{i}σ_{t}^{j}(u) is in H^{2n,n}. By Lemma 9.7 we get

σ_{s}^{i}σ_{t}^{j}P^{n}(u) = (σ_{s}^{i}σ_{t}^{j}(u))^{l}

By our assumption i > 0 and the right hand side is zero because the diagonal map S_{s}^{1} → S_{s}^{1} ∧ S_{s}^{1} is zero in H_{•}.

We will also use the total power operation:

R : H^{∗,∗} → H^{∗,∗}[[c, d, d^{−1}]]/(c^{2} = τd + ρc)

where τ = ρ = 0 for l ≠ 2 and for l = 2, τ and ρ are as in Theorem 6.12. We define R by the formula

R(u) = ∑_{i}(B^{i−1}(u)cd^{−i} + P^{i}(u)d^{−i})

For l = 2 this becomes

R(u) = ∑_{i}(Sq^{2i−1}(u)cd^{−i} + Sq^{2i}(u)d^{−i})

For u ∈ H^{2n,n} we have d^{2n}R(u) = P (u). Together with Lemma 5.7 this implies that for any u and v one has

R(u ∧ v) = R(u) ∧ R(v)

where the right hand side is to be computed in the ring H^{∗,∗}[[c, d, d^{−1}]]/(c^{2} = τd + ρc).

10 Adem relations

Lemma 10.1 Let d_{1}, c_{1} be generators of H^{∗,∗}(BS_{l}, Z/l) and c_{2}, d_{2} the generators of the ring appearing in the definition of R(u). Then one has:

R(d_{1}) = d_{1}(1 − d_{1}/d_{2})^{l−1}  (10.1)
R(c_{1}) = (c_{1} + (d_{1}/d_{2})c_{2})(1 − d_{1}/d_{2})^{l−2}  (10.2)

Proof: By transfer argument we may assume that k contains a primitive l-th root of unity. Consider the map ϕ : H^{∗,∗}(BS_{l}) → H^{∗,∗}(Bμ_{l}) defined by a choice of such a root. Then ϕ is a mono (by Lemma 6.11) and one has

ϕ(d_{1}) = −u^{l−1}
ϕ(c_{1}) = −vu^{l−2}

where u and v are the generators from Theorem 6.9. We get:

R(u^{l−1}) = R(u)^{l−1} = d_{2}^{1−l}P (u)^{l−1} = d_{2}^{1−l}(u^{l} + ud_{2})^{l−1}

where the last equality holds by Lemma 6.13. The right hand side equals to −ϕ(d_{1})(1 − ϕ(d_{1})/d_{2})^{l−1} which implies (10.1). For R(vu^{l−2}) we get

R(vu^{l−2}) = R(v)R(u)^{l−2} = d_{2}^{2−l}R(v)P (u) =
= d_{2}^{2−l}(P^{1}(v) + B^{0}(v)c_{2} + P^{0}(v)d_{2})(u^{l} + ud_{2})^{l−2}

By Theorem 9.4 P^{0}(v) = v, by Lemma 9.5 and since β(v) = u, B^{0}(v) = u. By Lemma 9.8, P^{1}(v) = 0 and therefore our expression equals

d_{2}^{1−l}(u^{l−1}c_{2}+vu^{l−2}d_{2})(u^{l−2}+d_{2})^{l−2} = d_{2}^{1−l}(−ϕ(d_{1})c_{2}−ϕ(c_{1})d_{2})(−ϕ(d_{1})+d_{2})^{l−2}

This implies (10.2).

Theorem 10.2 Let l = 2 and 0 < a < 2b. Then for a + b = 0(mod2) one has:

Sq^{a}Sq^{b} = \{ \begin{matrix}∑_{j=0}^{[a/2]} ( \begin{matrix}b − 1 − j \\ a − 2j \end{matrix} ) Sq^{a+b−j}Sq^{j} & for a, b odd \\ ∑_{j=0}^{[a/2]} τ^{jmod2} ( \begin{matrix}b − 1 − j \\ a − 2j \end{matrix} ) Sq^{a+b−j}Sq^{j} & for a, b even \end{matrix}

and for a + b = 1(mod2) one has:

Sq^{a}Sq^{b} = ∑_{j=0}^{[a/2]} ( \begin{matrix}b − 1 − j \\ a − 2j \end{matrix} ) Sq^{a+b−j}Sq^{j} + ρ^{(j+1)mod2}S(a, b)

where:

S(a, b) = \{ \begin{matrix}( \begin{matrix}b − 1 − j \\ a − 2j \end{matrix} ) Sq^{a+b+j}Sq^{j−1} & for a even, b odd \\ ( \begin{matrix}b − 1 − j \\ a − 1 − 2j \end{matrix} ) Sq^{a+b+j−1}Sq^{j} & for a odd, b even \end{matrix}

Proof: Consider the class P (P (u)) for u ∈ H^{2n,n}. Denote by d_{1}, c_{1} the generators of the cohomology of BS_{l} appearing when the first P is applied and by d_{2}, c_{2} the generators of the cohomology of BS_{l} appearing when the second P is applied. According to the symmetry theorem 7.7 the resulting expression is symmetric with respect to the exchange of (d_{1}, c_{1}) and (d_{2}, c_{2}). We have (to simplify the notations we sometimes omit u from our expressions):

P (u) = ∑_{i} Sq^{2n−2i−1}cd^{i} + Sq^{2n−2i}d^{i}

P (P (u)) = d_{2}^{2n}R(P (u)) = d_{2}^{2n} ∑_{i}(R(Sq^{2n−2i−1})R(c_{1}) + R(Sq^{2n−2i}))R(d_{1})^{i} =

= ∑_{i,j} d_{1}^{i}(d_{1}+d_{2})^{i}d_{2}^{2n−j−i}((Sq^{2j−1}Sq^{2n−2i−1}c_{2}+Sq^{2j}Sq^{2n−2i−1})(c_{1}+(d_{1}/d_{2})c_{2})+

+Sq^{2j−1}Sq^{2n−2i}c_{2} + Sq^{2j}Sq^{2n−2i})

Consider the coefficients in this expression at 1, c_{1}, c_{2} and c_{1}c_{2}. At 1 we have:

∑_{i,j} d_{1}^{i}(d_{1} + d_{2})^{i}d_{2}^{2n−j−i}(Sq^{2j}Sq^{2n−2i} + τd_{1}Sq^{2j−1}Sq^{2n−2i−1})

At c_{1}c_{2} we have:

∑_{i,j} d_{1}^{i}(d_{1} + d_{2})^{i}d_{2}^{2n−j−i}Sq^{2j−1}Sq^{2n−2i−1}

At c_{1} we have:

∑_{i,j} d_{1}^{i}(d_{1} + d_{2})^{i}d_{2}^{2n−j−i}Sq^{2j}Sq^{2n−2i−1}

At c_{2} we have:

∑_{i,j} d_{1}^{i}(d_{1} + d_{2})^{i}d_{2}^{2n−j−i}Sq^{2j−1}Sq^{2n−2i}+

+ ∑_{i,j} d_{1}^{i+1}(d_{1} + d_{2})^{i}d_{2}^{2n−j−i−1}(ρSq^{2j−1}Sq^{2n−2i−1} + Sq^{2j}Sq^{2n−2i−1})

Set p = i + r, q = 2n − j − r. Then coefficient at d_{1}^{p}d_{2}^{q} is

∑_{i} ( \begin{matrix}i \\ p − i \end{matrix} ) Sq^{4n−2p−2q+2i}Sq^{2n−2i} + τ ( \begin{matrix}i − 1 \\ p − i \end{matrix} ) Sq^{4n−2p−2q+2i−1}Sq^{2n−2i+1}

Coefficient at c_{1}c_{2}d_{1}^{p}d_{2}^{q} is

∑_{i} ( \begin{matrix}i \\ p − i \end{matrix} ) Sq^{4n−2p−2q+2i−1}Sq^{2n−2i−1}

Coefficient at c_{1}d_{1}^{p}d_{2}^{q} is

∑_{i} ( \begin{matrix}i \\ p − i \end{matrix} ) Sq^{4n−2p−2q+2i}Sq^{2n−2i−1}

Coefficient at c_{2}d_{1}^{p}d_{2}^{q} is

∑_{i} ( \begin{matrix}i \\ p − i \end{matrix} ) Sq^{4n−2p−2q+2i−1}Sq^{2n−2i}+( \begin{matrix}i \\ p − i − 1 \end{matrix} ) Sq^{4n−2p−2q+2i}Sq^{2n−2i−1}

+ρ ∑_{i} ( \begin{matrix}i \\ p − i − 1 \end{matrix} ) Sq^{4n−2p−2q+2i−1}Sq^{2n−2i−1}

Consider the coefficient at c_{1}c_{2}d_{1}^{p}d_{2}^{q} where p = 2^{s} − 1 for sufficiently large s and q = x. For p of this form, the coefficient ( \begin{matrix}i \\ p − i \end{matrix} ) is non-zero if and only if i = p (follows from [6, ]) and we conclude that our coefficient is Sq^{4n−2x−1}Sq^{2n−2^{s+1}+1}. By symmetry it equals to the coefficient at c_{1}c_{2}d_{1}^{q}d_{2}^{p}.

Setting a = 4n − 2x − 1, b = 2n − 2^{s+1} + 1 and j = 2n − 2i − 1 and using the fact that

( \begin{matrix}u \\ v \end{matrix} ) = ( \begin{matrix}2u \\ 2v \end{matrix} ) mod 2  (10.3)

we can write the later as

∑_{j=1 mod 2} ( \begin{matrix}2n − 1 − j \\ a − 2j − 1 \end{matrix} ) Sq^{a+b−j}Sq^{j}

From the standard relation

( \begin{matrix}u \\ v − 1 \end{matrix} ) = ( \begin{matrix}u + 1 \\ v \end{matrix} ) + ( \begin{matrix}u \\ v \end{matrix} )  (10.4)

and the fact that

( \begin{matrix}u \\ v \end{matrix} ) = 0 mod 2

if u is even and v is odd we get the first of the equalities stated in the theorem. A very similar argument starting with the equality between the coefficients at c_{1}d_{1}^{p}d_{2}^{q} and c_{2}d_{1}^{q}d_{2}^{p} gives the third equality - the case of even a and odd b. To prove the case when both a and b are even consider the coefficient at d_{1}^{p}d_{2}^{q}. Consider the second part of this coefficient i.e. the sum

∑_{i} ( \begin{matrix}i − 1 \\ p − i \end{matrix} ) Sq^{4n−2p−2q+2i−1}Sq^{2n−2i+1} =

= ∑_{j=i−1} ( \begin{matrix}j \\ p − 1 − j \end{matrix} ) Sq^{4n−2(p−1)−2q+2j−1}Sq^{2n−2j−1}

It is the coefficient at c_{1}c_{2}d_{1}^{p−1}d_{2}^{q} which equals to the coefficient at c_{1}c_{2}d_{1}^{q}d_{2}^{p−1} i.e. to

∑_{j=i−1} ( \begin{matrix}j \\ q − j \end{matrix} ) Sq^{4n−2(p−1)−2q+2j−1}Sq^{2n−2j−1} =

= ∑_{i} ( \begin{matrix}i − 1 \\ q − i + 1 \end{matrix} ) Sq^{4n−2p−2q+2i−1}Sq^{2n−2i+1}

Equating our new expression for the coefficient at d_{1}^{p}d_{2}^{q} with the old expression for the coefficient at d_{1}^{q}d_{2}^{p} we get

∑_{i} ( \begin{matrix}i \\ p − i \end{matrix} ) Sq^{4n−2p−2q+2i}Sq^{2n−2i} =

= ∑_{i} ( \begin{matrix}i \\ q − i \end{matrix} ) Sq^{4n−2p−2q+2i}Sq^{2n−2i}+

+τ ∑_{i} (( \begin{matrix}i − 1 \\ q − i + 1 \end{matrix} ) + ( \begin{matrix}i − 1 \\ q − i \end{matrix} )) Sq^{4n−2p−2q+2i−1}Sq^{2n−2i+1}

Setting again p = 2^{s} − 1, a = 2n − 2q and b = 2n − 2p and using the standard relations between the binomial coefficients one recovers the identity for Sq^{a}Sq^{b} when both a and b are even. Finally, to get the identity in the case when a is odd and b is even one uses Lemma 9.5, the identity for a and b even and the fact that β(τ) = ρ.

The proof of the following theorem which provides Adem relations for odd l follows the same line of arguments as the proof of the corresponding topological fact given in [6, ].

Theorem 10.3 For l > 2 and 0 < a < lb one has:

P^{a}P^{b} = ∑_{t=0}^{[a/l]}(−1)^{a+t} ( \begin{matrix}(l − 1)(b − t) − 1 \\ a − lt \end{matrix} ) P^{a+b−t}P^{t}

And for 0 ≥ a ≥ lb one has:

P^{a}B^{b} = ∑_{t=0}^{[a/l]}(−1)^{a+t} ( \begin{matrix}(l − 1)(b − t) \\ a − lt \end{matrix} ) B^{a+b−t}P^{t}+

+ ∑_{t=0}^{[(a−1)/l]}(−1)^{a+t−1} ( \begin{matrix}(l − 1)(b − t) − 1 \\ a − lt − 1 \end{matrix} ) P^{a+b−t}B^{t}+

Proof: These relations are exactly the same as the Adem relations in the topological Steenrod algebra for odd coefficients. The proof of these relations given in [6, Theorem VIII.1.6] works in exactly the same way in the motivic context as in the topological one if one replaces the reference to [6, Corollary VIII.1.2] with the refernce to our symmetry theorem 7.7. The apparent difference in signs bewteen our situation and the situaution of [6] is explained by the fact that the image of d in the cohomology of BZ/l is −u^{l−1} = −w_{2l−2}.

11 Motivic Steenrod algebra

Define the motivic Steenrod algebra A^{∗,∗}(k, Z/l) as the subalgebra in the algebra of bistable cohomological operations in the motivic cohomology with Z/l coefficients over k generated by operations P^{i}, B^{i}, i ≥ 0 and operations of the form u ↦ au where a ∈ H^{∗,∗}(k, Z/l).

Let I = (ϵ_{0}, s_{1}, ϵ_{1}, s_{2}, …, s_{k}, ϵ_{k}) be a sequence where ϵ_{i} ∈ {0, 1} and s_{i} are non-negative integers. Denote by P^{I} the product

P^{I} = β^{ϵ_{0}}P^{s_{1}} … P^{s_{k}}β^{ϵ_{k}}

A sequence I is called admissible if s_{i} ≥ ps_{i+1} + ϵ_{i}. Monomials P^{I} corresponding to admissible sequences are called admissible monomials.

Lemma 11.1 Admissible monomials generate A^{∗,∗}(k, Z/l) as a left H^{∗,∗}- module.

Proof: This follows from Adem relations and the Cartan formula 9.6.

Our next goal is to show that the admissible monomials are linearly independent with respect to the left H^{∗,∗}-module structure on A^{∗,∗} and, therefore, form a basis of this module. Consider the submodule H^{∗,>0}A^{∗,∗} in A^{∗,∗}. The Cartan formulas 9.6 imply that its is a two-sided ideal in A^{∗,∗}. Set

A_{rig}^{∗,∗} = A^{∗,∗}/H^{∗,>0}A^{∗,∗}

Using again the Cartan formula one observes that the action of A^{∗,∗} on H^{∗,∗}(X) defines an action of A_{rig}^{∗,∗} on H^{∗,∗}(X)/H^{∗,>0}H^{∗,∗}(X). Theorem 6.9 immediately implies the following result.

Lemma 11.2 For any l and any k one has:

H^{∗,∗}(Bμ_{l})/H^{∗,>0}H^{∗,∗}(Bμ_{l}) = Z/l[u, v]/(v^{2} = 0)

Lemma 10.1 implies the following.

Lemma 11.3 Let u and v be as in Lemma 11.2. Then one has the following equalities in H^{∗,∗}(Bμ_{l})/H^{∗,>0}H^{∗,∗}(Bμ_{l}):

\begin{matrix}β(v) = u & P^{i}(v) = 0 for i > 0 \\ β(u^{k}) = 0 & P^{i}(u^{k}) = ( \begin{matrix}k \\ i \end{matrix} ) u^{k+i(l−1)} \end{matrix}

Let d(I) be the degree of an admissible monomial P^{I} i.e. the integer such that P^{I} is an operation from H^{∗,∗} to H^{∗+d(I),∗+q}.

Proposition 11.4 For any n ≥ 0 there exists N ≥ 0 and an element w ∈ H^{∗,∗}((Bμ_{L})^{N}) such that the elements P^{I}(w), for all I such that d(I) ≤ n, are linearly independent with respect to the left H^{∗}, ∗-module structure on H^{∗,∗}((Bμ_{L})^{N}).

Proof: It is sufficient to show that there exists w such that P^{I}(w) are linerly independent in H^{∗,∗}((Bμ_{L})^{N})/H^{∗,>0}H^{∗,∗}((Bμ_{L})^{N}) with respect to the Z/lmodule structure. One starts with Lemma 11.3 and uses exactly the same reasoning as in the proof of [6, Proposition VI.2.4]

The following is an immediate corollary of the proof of Proposition 11.5.

Corollary 11.5 The admissible monomials are linearly independent with respect to the left H^{∗,∗}-module structure on A^{∗,∗}.

Let l be an odd prime. Denote by A_{top}^{∗,∗} the Z/l-submodule of A^{∗,∗} generated by the admissible monomials. The Adem relations (Theorem 10.3) show that A_{top}^{∗,∗} is a subring of A^{∗,∗}. Together with Corollary 11.5 they imply that A_{top}^{∗,∗} is isomorphic to the usual topological Steenrod algebra with the second grading given by assigning the weight (l − 1)i to P^{i} and the weight 0 to β. The Cartan formula (Proposition 9.6) shows that the action of elements of A_{top}^{∗,∗} on products of motivic cohomology classes has the same expansion as in topology. Taken together these observations show that all the standard results aboput the topological Steenrod algebra and its dual in the case of odd coefficients translate without change to the motivic context. In what follows we consider both the odd and vene coefficients cases but give the proofs only in the even case where the motivic Steenrod algebra has a much more complicated structure than its topological counterpart.

Below we denote by A^{∗,∗}⊗_{H^{∗,∗}} A^{∗,∗} the tensor product of left H^{∗,∗}-modules A^{∗,∗}. The action of A^{∗,∗} on H^{∗,∗}(X) is not, in general, H^{∗,∗}-linear. Hence, we can not speak of the action of A^{∗,∗} ⊗_{H^{∗,∗}} A^{∗,∗} on H^{∗,∗}(X) ⊗_{H^{∗,∗}} H^{∗,∗}(Y). However, since the action of A^{∗,∗} is Z/l-linear it defines an action of A^{∗,∗} ⊗_{Z/l} A^{∗,∗} on H^{∗,∗}(X) ⊗_{Z/l} H^{∗,∗}(Y). If x, y are two elements of A^{∗,∗} ⊗_{Z/l} A^{∗,∗} which become equal in the tensor product over H^{∗,∗} then for any u in H^{∗,∗}(X) ⊗_{Z/l} H^{∗,∗}(Y) we have x(u) = x(v) in H^{∗,∗}(X) ⊗_{H^{∗,∗}} H^{∗,∗}(Y). Therefore, for x in A^{∗,∗} ⊗_{Z/l} A^{∗,∗} and u in H^{∗,∗}(X) ⊗_{Z/l} H^{∗,∗}(Y) there is a well defined element x(u) in H^{∗,∗}(X) ⊗_{H^{∗,∗}} H^{∗,∗}(Y).

Lemma 11.6 For any element x of A^{∗,∗} there exists a unique element

ψ^{∗}(x) = ∑ x′_{i} ⊗ x′′_{i}

of A^{∗,∗} ⊗_{H^{∗,∗}} A^{∗,∗} such that for any X and any u ∈ H^{p,∗}(X), v ∈ H^{∗,∗}(X) one has

x(u ∧ v) = ∑(−1)^{dim(x′′_{i})p}x′_{i}(u) ∧ x′′_{i}(v)

Proof: Exactly paralell to the proof of [3, Lemma 1, p.154] where Proposition 11.4 is used to prove uniqueness.

We will need the following lemma below.

Lemma 11.7 Let x be an element of A^{∗,∗} and ψ^{∗}(x) = ∑ x′_{i} ⊗ x′′_{i}. Then for a ∈ H^{∗,∗} one has:

ψ^{∗}(xa) = ∑ x′_{i}a ⊗ x′′_{i} = ∑ x′_{i} ⊗ x′′_{i}a

Proof: By uniqueness part of Lemma 11.6 it is enough to check that for any u, v ∈ H^{∗,∗}(X) one has:

(∑ x′_{i}a ⊗ x′′_{i})(u ⊗ v) = (∑ x′_{i} ⊗ x′′_{i}a)(u ⊗ v) = (xa)(u ∧ v)

This follows immediately from definitions.

Since H^{∗,∗} is not in the center of A^{∗,∗}, the ring structure on A^{∗,∗}⊗_{Z/l} A^{∗,∗} does not define a ring structure on A^{∗,∗} ⊗_{H^{∗,∗}} A^{∗,∗}. The best we can get in general is an action of A^{∗,∗} ⊗_{H^{∗,∗}} A^{∗,∗} on A^{∗,∗} ⊗_{Z/l} A^{∗,∗} with values in A^{∗,∗} ⊗_{H^{∗,∗}} A^{∗,∗} given by

(u ⊗ v)(u′ ⊗ v′) = uu′ ⊗ vv′

We say that an element f of A^{∗,∗} ⊗_{H^{∗,∗}} A^{∗,∗} is an operator-like element if for any two elements x, y of A^{∗,∗} ⊗_{Z/l} A^{∗,∗} which belome equal in the tensor product over H^{∗,∗} one has fx = fy. For an operator-line element f and any other element x the product fx is well defined. If f and g are two operator-like elements the product fg is again operator-line. This shows that operator-like elements form a ring which we denote by (A^{∗,∗} ⊗_{H^{∗,∗}} A^{∗,∗})_{r}.

Lemma 11.8 For any x in A^{∗,∗}, ψ^{∗}(x) is an operator-like element. The map ψ^{∗} : A^{∗,∗} → (A^{∗,∗} ⊗_{H^{∗,∗}} A^{∗,∗})_{r} is a ring homomorphism.

Proof: Let y, z be two elements of A^{∗,∗} ⊗_{Z/l} A^{∗,∗} which become equal modulo H^{∗,∗}. To check that ψ^{∗}(x)y = ψ^{∗}(x)z it is sufficient, in view of Proposition 11.4, to check that for any X and any w_{1}, w_{2} ∈ H^{∗,∗}(X) one has ψ^{∗}(x)y(w_{1} ⊗ w_{2}) = ψ^{∗}(x)z(w_{1} ⊗ w_{2}). Let c : H^{∗,∗}(X) ⊗ H^{∗,∗}(X) → H^{∗,∗}(X) be the cup product. Then by definition of ψ we have

ψ^{∗}(x)y(w_{1} ⊗ w_{2}) = x(c(y(w_{1} ⊗ w_{2})))
ψ^{∗}(x)z(w_{1} ⊗ w_{2}) = x(c(z(w_{1} ⊗ w_{2})))

Our assumption on y, z implies that c(y(w_{1} ⊗ w_{2})) = c(z(w_{1} ⊗ w_{2})).

To prove that ψ^{∗} is a ring homomorphism we have to check that for x, y ∈ A^{∗,∗} and w_{1}, w_{2} ∈ H^{∗,∗}(X) we have

ψ(xy)(w_{1} ⊗ w_{2}) = ψ(x)(ψ(y)(w_{1} ⊗ w_{2}))

This follows immediately from definitions.

Lemma 11.9 The comultiplication map ψ^{∗} is associative and commutative.

Proof: The associativity follows immediately from the definition. To verify commutativity it is sufficient, in view of Lemma 11.8, to verify it on generators of the algebra A^{∗,∗} i.e. on operations P^{i} and B^{i}. For this operations commutativity follows directly from the Cartan formulas (Proposition 9.6).

12 Structure of the dual to the motivic Steenrod algebra

Let A_{∗,∗} be the dual to A^{∗,∗} i.e. the group of left module maps from A^{∗,∗} to H^{∗,∗}. Since A^{∗,∗} is free, A_{∗,∗} is a free H^{∗,∗} module. The homomorphism ψ^{∗} defines a homomorphism ϕ_{∗} : A_{∗,∗} ⊗ A_{∗,∗} → A_{∗,∗}. Lemma 11.9 immediately implies the following result.

Proposition 12.1 The homomorphism ϕ_{∗} makes A_{∗,∗} into an associative ring which is graded commuuative with respect to the first grading.

Let e^{i} ∈ A^{p_{i},q_{i}} be a basis of A^{∗,∗} over H^{∗,∗} such that:

1. e_{0} = 1 and q_{i} > 0 for i > 0

2. p_{i} ≥ 2q_{i}

3. for any q there are only finitely many i with q_{i} ≤ 1.

An example of such a bais is given by the basis of admissible monomials. Let e_{i} be the dual basis of A_{∗,∗}. Let X be a smooth scheme over k. Then H^{p,q}(X) = 0 for p > q + dim(X) and therefore for any w ∈ H^{∗,∗}(X) there is only finitely many i’s such that e^{i}(w) ≠ 0. We can define a map

λ^{∗,∗} : H^{∗,∗}(X) → H^{∗,∗}(X) ⊗_{H^{∗,∗}} A_{∗,∗}

by the formula

λ^{∗,∗}(w) = ∑ e^{i}(w) ⊗ e_{i}

The following lemma is straightforward.

Lemma 12.2 The map λ^{∗,∗} is a ring homomorphism which does not depend on the choice of e^{i}.

Note that in the case when X = Spec(k) this homomorphism does not coincide with the canonical homomorphism H^{∗,∗} → A_{∗,∗} which we used to define the H^{∗,∗}-module structure on A_{∗,∗} since the action of e_{i}’s on H^{∗,∗} may be nontrivial. In particular λ^{∗,∗} is not a H^{∗,∗}-module homomorphism.

If we consider Bμ_{l} as a colimit of smooth schemes we can write a formal analog of λ^{∗,∗}. In particular for the canonical generators u and v we get:

λ^{∗,∗}(v) = ∑_{i=0}^{∞}(vu^{i} ⊗ x_{i} + u^{i} ⊗ y_{i})

λ^{∗,∗}(u) = ∑_{i=0}^{∞}(vu^{i} ⊗ x′_{i} + u^{i} ⊗ y′_{i})

where x_{i}, y_{i}, x′_{i}, y′_{i} are some well defined elements in A_{∗,∗}(k, Z/l). We denote:

ξ_{i} = y′_{l^{i}} ∈ A_{2(l^{i}−1),l^{i}−1}(k, Z/l)
τ_{i} = y_{l^{i}} ∈ A_{2l^{i}−1,l^{i}−1}(k, Z/l)

Since for any basis of A^{∗,∗} such that e^{0} = 1 we have dim(e^{i}) > 0 for i ≠ 0 we conclude that ξ_{0} = 1. Denote by M_{k} the monomial P^{l^{k−1}}P^{l^{k−2}} … P^{1}.

Lemma 12.3 We have in H^{∗,∗}(Bμ_{l}):

M_{k}(u) = u^{l^{k}} M_{k}β(v) = u^{l^{k}}  (12.1)

If f is any monomial in P^{i} and β other than M_{k} (resp. other than M_{k}β) then f(u) = 0 (resp. f(v) = 0).

Proof: The equations (12.1) follow from Lemma 9.7 and the fact that β(v) = u. The other statement follows from Lemma 9.8, Lemma 6.13 and multiplicativity of P.

Taking the basis of admissible monomials to compute λ^{∗,∗} and using Lemma 12.3 we conclude that

λ^{∗,∗}(v) = v ⊗ 1 + ∑_{i=0}^{∞} u^{l^{i}} ⊗ τ_{i}  (12.2)

λ^{∗,∗}(u) = u ⊗ 1 + ∑_{i=0}^{∞} u^{l^{i}} ⊗ ξ_{i}.

For an element ϕ in A^{∗,∗} and an element ψ in A_{∗,∗} let ⟨ψ, ϕ⟩ ∈ H^{∗,∗} be the value of ψ on ϕ. Then we have:

ϕ(v) = ⟨ξ_{0}, ϕ⟩v + ∑_{i}⟨τ_{i}, ϕ⟩u^{l^{i}}  (12.3)

ϕ(u) = ∑_{i}⟨ξ_{i}, ϕ⟩u^{l^{i}}

Let I be a sequence of the form (ϵ_{0}, r_{1}, ϵ_{1}, r_{2}, …,) where ϵ_{i} ∈ {0, 1}, r_{i} ≥ 0 are nonnegative integers and I has only finitely many nontrivial terms. Followig [3] we set:

ω(I) = τ_{0}^{ϵ_{0}}ξ_{1}^{r_{1}}τ_{1}^{ϵ_{1}}ξ_{2}^{r_{2}} …

in A_{p(I),q(I)}(k, Z/l) where

p(I) = ϵ_{0} + ∑_{i≥1}(ϵ_{i}(2l^{i} − 1) + 2r_{i}(l^{i} − 1))

q(I) = ∑_{i≥1}(ϵ_{i} + r_{i})(l^{i} − 1).

and

θ(I) = β^{ϵ_{0}}P^{s_{1}}β^{ϵ_{1}}P^{s_{2}} …

where

s_{n} = ∑_{i≥n}(ϵ_{i} + r_{i})l^{i−n}.

Simple computation shows that θ(I) belongs to A^{p(I),q(I)}(k, Z/l).

In the following theorem we consider, following [4], the lexicographical order on the set of sequences I such that (1, 2, 0, …) < (0, 0, 1, …).

Theorem 12.4 Then, for I < J one has ⟨θ(I), ω(J)⟩ = 0 and ⟨θ(I), ω(I)⟩ = ±1

Proof: The value of ⟨θ(I), ω(J)⟩ is a homogenious element of H^{∗,∗} of degree zero. Hence, it is sufficient to show that the image of ⟨θ(I), ω(J)⟩ in H^{∗,∗}/H^{∗,>0} is 0 or ±1 depending on whether I < J or I = J. This is done using the action of A^{∗,∗}/A^{∗,>0} on H^{∗,∗}(Bμ_{l})/H^{∗,>0}H^{∗,∗}(Bμ_{l}) in exactly the same way as in the proof of [3, Lemma 8, p.160].

Corollary 12.5 Elements ω(I) (resp. θ(I)) form a basis for H^{∗,∗}-module A_{∗,∗}(k, Z/l) (resp. A^{∗,∗}(k, Z/l)).

Proof: Elements θ(I) are exactly the admissible monomials. They form a basis of A^{∗,∗} by Lemma 11.1 and Corollary 11.5. The fact that elements ω(I) form a basis of A_{∗,∗} follows now from Theorem 12.4.

Theorem 12.6 The (graded commutative) algebra A^{∗,∗}(k, Z/l) over H^{∗,∗} is canonically isomorphic to the (graded commutative) algebra with generators τ_{i}, ξ_{i}, i ≥ 0 and relations

1. ξ_{0} = 1

2. τ_{i}^{2} = \{ \begin{matrix}0 & for l ≠ 2 \\ τξ_{i+1} + ρτ_{i+1} + ρτ_{0}ξ_{i+1} & for l = 2 \end{matrix}

Proof: We already know by Corollary 12.5 that A^{∗,∗}(k, Z/l) has a basis which consits of monomials in ξ_{i}, τ_{i} which are of degree ≤ 1 in each τ_{i}. The realtion τ_{i}^{2} = 0 for odd l is a corollary of graded commutativity. Thus we may assume that l = 2 in which case we have only to show that τ_{i}^{2} = τξ_{i+1}+ρτ_{i+1}+ρτ_{0}ξ_{i+1}. The required relation follows immediately from (12.2), the multiplicativity of λ^{∗,∗} and the realtion v^{2} = τu + ρv in the motivic cohomology ring of Bμ_{2}.

Lemma 12.7 For any ϕ ∈ A^{∗,∗} one has:

ϕ(u^{l^{j}}) = ∑_{i}⟨ξ_{i}^{l^{j}}, ϕ⟩u^{l^{i+j}}  (12.4)

Proof: Let x_{1}, x_{2} be any elements of A_{∗,∗}. If ψ^{∗}(ϕ) = ∑ ϕ′_{k} ⊗ ϕ′′_{k} we have, by definition of product in A_{∗,∗}:

⟨x_{1}x_{2}, ϕ⟩ = ∑⟨x_{1}, ϕ′_{k}⟩⟨x_{2}, ϕ′′_{k}⟩

This implies by induction starting with (12.3) that for any n one has

ϕ(u^{n}) = ∑_{(j_{1},...,j_{n})}⟨ξ_{j_{1}} … ξ_{j_{n}}, ϕ⟩u^{l^{j_{1}+···+jn}}

For n = l^{j} all the terms except for the ones which show up in the right hand side of (12.4) cancell out since we work with Z/l-coefficients.

Proposition 12.8 Let ϕ, ψ be elements of A^{∗,∗} such that ⟨ξ_{k}, ψ⟩, ⟨τ_{k}, ψ⟩ are in Z/l ⊂ H^{∗,∗}. Then one has:

⟨ξ_{k}, ϕψ⟩ = ∑_{i}⟨ξ_{k−i}^{l^{i}}, ϕ⟩⟨ξ_{i}, ψ⟩

⟨τ_{k}, ϕψ⟩ = ⟨τ_{k}, ϕ⟩⟨ξ_{0}, ψ⟩ + ∑_{i}⟨ξ_{k−i}^{l^{i}}, ϕ⟩⟨τ_{i}, ψ⟩

Proof: We have by (12.3)

ϕψ(u) = ∑⟨ξ_{i}, ϕψ⟩u^{l^{i}}

On the other hand by (12.3) and (12.4) we get:

ϕψ(u) = ϕ(∑⟨ξ_{i}, ψ⟩u^{l^{i}}) = ∑_{i,j}⟨ξ_{j}^{l^{i}}, ϕ⟩⟨ξ_{i}, ψ⟩u^{l^{i+j}}

Comparing coefficients at powers of u we get the first of the required equalities. To get the second one we write

ϕψ(v) = ⟨ξ_{0}, ϕψ⟩v + ∑_{i}⟨τ_{i}, ϕψ⟩u^{l^{i}}

by (12.3). And by (12.3) and (12.4) we get:

ϕψ(v) = ⟨ξ_{0}, ψ⟩ϕ(v) + ∑_{i}⟨τ_{i}, ψ⟩ϕ(u^{l^{i}}) =

= ⟨ξ_{0}, ϕ⟩⟨ξ_{0}, ψ⟩v + ∑_{i}⟨τ_{i}, ϕ⟩⟨ξ_{0}, ψ⟩u^{l^{i}} + ∑_{i,j}⟨ξ_{j}^{l^{i}}, ϕ⟩⟨τ_{i}, ψ⟩u^{l^{i+j}}

Comparing coefficients we get the second equality.

Now we can describe the dual to the ring structure on A^{∗,∗}. We have two H_{∗,∗} module structures on A_{∗,∗}. The first one is the left module structure which we used all the time is given by

a ∗_{l} ϕ(x) = ϕ(ax) = aϕ(x)

where ϕ ∈ A_{∗,∗}, a ∈ H^{∗,∗} and x ∈ A^{∗,∗}. The other one is the right module structure given by

ϕ ∗_{r} a(x) = ϕ(xa)

Lemma 11.7 implies that (ϕϕ′) ∗_{r} a = ϕ(ϕ′ ∗_{r} a) and, in particular, that ϕ ∗_{r} a = ϕ(1 ∗_{r} a). The map a ↦ 1 ∗_{r} a coincides with the map λ^{∗,∗} for X = Spec(k) and we denote it by λ. Therefore, the two module structures are defines by two ring homomorphisms a ↦ aξ_{0} and a ↦ λ(a) from H^{∗,∗} to A_{∗,∗}.

Denote by A_{∗,∗} ⊗_{r,l} A_{∗,∗} the tensor product with the property ϕ ∗_{r} a ⊗ ψ = ϕ ⊗ a ∗_{r} ψ. Similarly, denote by A^{∗,∗} ⊗_{r,l} A^{∗,∗} the tensor product with the property xa ⊗ y = x ⊗ ay. The following lemma is essentially taken from [1, Lemma 3.3].

Lemma 12.9 Let f, g be elments of A_{∗,∗} and x, y elements of A^{∗,∗}. The formula:

⟨θ(f ⊗ g), x ⊗ y⟩ = (−1)^{deg(g)deg(m)}⟨f, x⟨g, y⟩⟩  (12.5)

defines an isomorphism

θ : A_{∗,∗} ⊗_{r,l} A_{∗,∗} → (A^{∗,∗} ⊗_{r,l} A^{∗,∗})^{∗}

where the upper star on the right hand side denotes the left H^{∗,∗}-module of left H^{∗,∗}-module maps from A^{∗,∗} ⊗_{H^{∗,∗}} A^{∗,∗} to H^{∗,∗}.

Proof: One verifies easily that θ is indeed well defined by (12.5). To prove that it is an isomorphism consider the basis ω(I) in A_{∗,∗} and let ω(I)^{∗} be the dual basis in A^{∗,∗}. The elements ω(I)⊗ω(J) clearly generate A_{∗,∗}⊗_{r,l}A_{∗,∗} as a left H^{∗,∗}-module. The image of ω(I)⊗ω(J) with respect to θ is the functional which equals one on ω(I)^{∗} ⊗ ω(J)^{∗} and zero on all other elements of the form ω(I′)^{∗} ⊗ ω(J′)^{∗}. This implies that ω(I) ⊗ ω(J) are linearly independent in A_{∗,∗} ⊗_{r,l} A_{∗,∗} and hence form a basis of this left H^{∗,∗}-module. It also implies that θ maps this basis to a basis of (A^{∗,∗} ⊗_{H^{∗,∗}} A^{∗,∗})∗ and hence it is an isomorphism.

Composing the dual to the multiplication map A^{∗,∗} ⊗ A^{∗,∗} → A^{∗,∗} with θ we get a map

ψ_{∗} : A_{∗,∗} → A_{∗,∗} ⊗_{r,l} A_{∗,∗}  (12.6)

By construction, the map ψ_{∗} is defined by the property that ψ_{∗}(f) = ∑ f′_{i} ⊗ f′′_{i} and for any x, y ∈ A^{∗,∗} one has:

∑⟨f′_{i}, x⟨f′′_{i}, y⟩⟩ = ⟨f, xy⟩  (12.7)

Lemma 12.10 The map (12.6) is a ring homomorphism.

Proof: It follows from a direct computation and Lemma 11.7.

In view of Lemma 12.10 and Theorem 12.6, the map ψ_{∗} is completely determined by its values on the generators ξ_{i}, τ_{i}.

Lemma 12.11 One has:

ψ_{∗}(ξ_{k}) = ∑_{i=0}^{k} ξ_{k−i}^{l^{i}} ⊗ ξ_{i}

ψ_{∗}(τ_{k}) = ∑_{i=0}^{k} ξ_{k−i}^{l^{i}} ⊗ τ_{i} + τ_{k} ⊗ 1

Proof: Follows from Proposition 12.8 and (refcopr3).

13 Operations Q_{i} and their properties

Denote by Q_{i} the element of A^{2l^{i}−1,l^{i}−1}(k, Z/l) which is dual to τ_{i} with respect to the basis of A_{∗,∗}(k, Z/l) which consists of monomials ω(I) introduced in the previous section. More generally, for a sequence \underline{ϵ} = (ϵ_{0}, …, ϵ_{n}) where ϵ_{i} ∈ \{0, 1\} denote by Q(\underline{ϵ}) the element in A^{∗,∗} dual to τ_{0}^{ϵ_{0}} … τ_{n}^{ϵ_{n}} with respect to the basis ω(I). If l is odd these operations have the same properties as their topological counterparts. In what follows we assume that l = 2. Let ρ be as before the class of −1 in H^{1,1}. denote by B^{∗,∗} the Z/2[ρ]-submodule in A^{∗,∗} geberated by elements of the form Q(\underline{ϵ}). Let B_{∗,∗} be the dual of B^{∗,∗}. Then

B_{∗,∗} ⊗_{Z/2[ρ]} H^{∗,∗} = A^{∗,∗}/{ξ_{i} = 0 i > 0}

Lemma 12.11 implies that for f in the ideal I generated by ξ_{i} for i > 0, ψ_{∗}(f) is in I ⊗ A_{∗,∗} + A_{∗,∗} ⊗ I and therefore ψ_{∗} defines a comultiplication on B_{∗,∗} ⊗_{Z/2[ρ]} H^{∗,∗} given by

ψ_{∗}(τ_{k}) = τ_{k} ⊗ 1 + 1 ⊗ τ_{k}

Form this one gets easily the following result.

Proposition 13.1 The Z/2[ρ]-algebra B_{∗,∗} is generated by elements τ_{k}, k ≥ 0 with the relations τ_{k}^{2} = τ_{k+1}. Its has a Hopf algebra structure compatible with the Hopf algebroid structure of A_{∗,∗} given by ψ_{∗}(τ_{k}) = τ_{k} ⊗ 1 + 1 ⊗ τ_{k}.

Dualizing we get the following theorem on the structure of B^{∗,∗}.

Proposition 13.2 The Z/2[ρ] algebra B^{∗,∗} is the exterior algebra with generators Q_{i}. For \underline{ϵ} = (ϵ_{0}, …, ϵ_{n}) one has Q(\underline{ϵ}) = ∏ Q_{i}^{ϵ_{i}}. The coproduct is given on Q_{i}’ by

ψ^{∗}(Q_{i}) = 1 ⊗ Q_{i} + Q_{i} ⊗ 1 + ρ ∑_{\underline{ϵ},\underline{ϵ}′} c_{\underline{ϵ},\underline{ϵ}′}Q(\underline{ϵ}) ⊗ Q(\underline{ϵ}′)

for l = 2, where \underline{ϵ}, \underline{ϵ}′ run through sequences of the form (ϵ_{0}, …, ϵ_{i−1}) and c_{\underline{ϵ},\underline{ϵ}′} are elements of H^{∗,∗}.

For R = (i_{1}, …, i_{n}) where i_{j} ≥ 0 denote by P^{R} the element in A^{∗,∗} dual to ξ^{R} = ξ_{1}^{i_{1}} … ξ_{n}^{i_{n}} in the basis of monomials in τ_{i}, ξ_{j}.

Lemma 13.3 P^{(n,0,...,0,...)} = ±P^{n}

Proof: Follows immediately from Theorem 12.5.

Proposition 13.4 Let \underline{ϵ}, R = (i_{1}, …, i_{n}) be as above. Then the element in A^{∗,∗} dual to τ\underline{ϵ}ξ^{R} in the basis of monomials is Q(\underline{ϵ})P^{R}.

Proof: We have to compute the pairing of Q(\underline{ϵ})P^{R} with τ^{\underline{ϵ}′}ξ^{R′} and show that this pairing is 1 for \underline{ϵ} = \underline{ϵ}′, R = R′ and zero otherwise. By (12.7) we have:

⟨τ^{\underline{ϵ}′}ξ^{R′}, Q(\underline{ϵ})P^{R}⟩ = ∑⟨f′_{i}, Q(\underline{ϵ})⟨f′′_{i}, P^{R}⟩⟩  (13.1)

where

∑ f′_{i} ⊗ f′′_{i} = ψ_{∗}(τ^{\underline{ϵ}′}ξ^{R′}) = ψ_{∗}(τ^{\underline{ϵ}′})ψ_{∗}(ξ^{R′})

We can choose our representation ∑ f′_{i} ⊗ f′′_{i} such that f′′_{i} are the dual basis elements and in particular ⟨f′′_{i}, P^{R}⟩ are in Z/2. Then, the expression (13.1) depends only on the class of ψ_{∗}(τ^{\underline{ϵ}′}ξ^{R′}) in A_{∗,∗}/I ⊗_{r,l} A^{∗,∗} where I is the ideal generated by ξ_{i} for i > 0. In this quotient ring we have:

ψ_{∗}(ξ_{k}) = 1 ⊗ ξ_{k}
ψ_{∗}(τ_{k}) = 1 ⊗ τ_{k} + τ_{k} ⊗ 1

This immediately implies that (13.1) is non-zero if and only if \underline{ϵ} = \underline{ϵ}′ and R = R′.

The following two results complete the proof of all the properties of operations Q_{i} used in [7].

Proposition 13.5 Let q_{i} be the dual in the monomials basis to ξ_{i}. Then for i ≥ 1 one has

Q_{i} = [Q_{0}, q_{i}]

Proof: We have to show that q_{i}Q_{0} = Q_{0}q_{i} + Q_{i} i.e. that the only monomials which pair non-trivially with q_{i}Q_{0} are τ_{0}ξ_{i} and ξ_{i} and that for those monomials the pairing gives 1. Using (12.7) we see that it is sufficient to show that the only monomials M = τ\underline{ϵ}ξ^{R} such that ξ_{i} ⊗τ_{0} appears in the decomposition of ψ_{∗}(M) are τ_{0}ξ_{i} and τ_{i} and that for those monomials ξ_{i} ⊗ τ_{0} appears with coefficient 1. The later follows immediately from Lemma 12.11. To prove the former note that the question of whether or not ξ_{i} ⊗τ_{0} appears in the expression for ψ_{∗}(M) depend only on the class of ψ_{∗}(M) in A_{∗,∗} ⊗_{r,l} A_{∗,∗}/J where J is generated by τ_{k} for k > 0. In this quotient ψ_{∗}(τ_{k}) = ξ_{k} ⊗ τ_{0} + τ_{k} ⊗ 1 and τ_{0}^{2} = τξ_{1} + ρτ_{0}ξ_{1}. This shows that the only way to get ξ_{i} ⊗ τ_{0} is to consider ψ_{∗}(τ_{i}) or ψ_{∗}(τ_{0}ξ_{i}).

The following example shows that not all of the standard topological formulas for Q_{i}’s hold in the motivic context.

Example 13.6 In topology, one can define operations Q_{i} in terms of the Steenrod squares inductively by the formula Q_{0} = Sq^{1}, Q_{i+1} = [Q_{i}, Sq^{2^{i+1}}]. Let us show that in the motivic Steenrod algebra Q_{2} ≠ [Q_{1}, Sq^{4}] if ρ ≠ 0 i.e. if k does not contain the square root of −1. Using (12.7) and Lemma 12.11 we can compute Sq^{4}Q_{1} in terms of the basis dual to τ(\underline{ϵ})ξ(R). We get:

Sq^{4}Q_{1} − Q_{1}Sq^{4} = Q_{2} + ρQ_{0}Q_{1}Sq^{2}

References

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[3] John Milnor. The Steenrod algebra and its dual. Annals of Math., 67(1):150–171, 1958.

[4] John Milnor. Algebraic K-theory and quadratic forms. Inv. Math., 9:318– 344, 1970.

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[8] Vladimir Voevodsky. Motivic cohomology are isomorphic to higher Chow groups. www.math.uiuc.edu/K-theory/378, 1999.