Operations in étale and motivic cohomology

Abstract

We classify all étale cohomology operations on H et (−, μ l ), showing that they were all constructed by Epstein. We also construct operations P on the mod-l motivic cohomology groups H , differing from Voevodsky’s operations; we use them to classify all motivic cohomology operations on H and H and suggest a general classification. In the last decade, several papers have given constructions of cohomology operations on motivic and étale cohomology, following the earlier work of Jardine [J], Kriz-May [KM] and Voevodsky [V2, V1]: see [BJ, BJ1, Jo, M1, V3, V4]. The goal of this paper is to provide, for each n and i, a classification of all such operations on the étale groups H et(−, μ ⊗i l ) and the motivic groups H (−,Fl), similar to Cartan’s classification of operations on singular cohomology H top(−,Fl) in [C]. We succeed for étale operations and partially succeed for motivic operations. We work over a fixed field k and fix a prime l with 1/l ∈ k. By definition, an (unstable) étale cohomology operation on H et(−, μ ⊗i l ) over k is a natural transformation H et(−, μ ⊗i l ) → H p et(−, μ ⊗q l ) of set-valued functors from the category of (smooth) simplicial schemes over k (for some p and q). Similarly, an (unstable) motivic cohomology operation on H over k is a natural transformation H → H of functors defined on this category, where H(X) denotes the Nisnevich cohomology H nis(X,Fl(q)), and the cochain complex Fl(q) is defined in [V2] or [MVW]. Fixing k, n and i, the set of all unstable cohomology operations forms a ring; the product of θ1 and θ2 is the operation x 7→ θ1(x) · θ2(x). There are two constructions of étale operations P , independently due to Epstein [E] and May [M]. We will show that the two constructions give the same operations in Corollary 4.7. (A third construction when l = 2 or μl ⊂ k × was given by Jardine [J]; it is easily seen to agree with Epstein’s constuction.) The upshot is that Cartan’s ring H top(Kn) of operations on H n top(−,Fl) embeds into the ring of all étale operations on H et(−, μ ⊗i l ); we refer the reader to Definition 0.1 below for a precise description of Cartan’s ring. Epstein’s construction is more easily accessible to algebraic geometers; see [R]. It uses equivariant sheaf cohomology, and is an application of the method described in his 1966 paper [E]. After stating Epstein’s result in Theorem 1.3, we indicate the key points in his construction that we will need to compare with May’s construction. The classification of étale cohomology operations is given in Sections 2 and 3. Theorem 3.5 gives the general result: the ring of all (unstable) étale operations on H et(−, μ ⊗i l ) over k is the tensor product H ∗ et(k(ζ),Fl) ⊗ H ∗ top(Kn), where ζ is a primitive l root of unity. Thus all (unstable) étale operations on H et(−, μ ⊗i l ) over k are H et(k(ζ),Fl)-linear combinations of monomials in the operations P I . Date: January 26, 2017.