UNSTABLE OPERATIONS IN E´TALE AND MOTIVIC COHOMOLOGY 3 BERTGUILLOUANDCHUCKWEIBEL 1 0 2 Abstract. WeclassifyallétalecohomologyoperationsonHent(−,µ⊗ℓi),show- n ing that they were all constructed by Epstein. We also construct operations a Paonthemod-ℓmotiviccohomologygroupsHp,q,differingfromVoevodsky’s J operations;weusethemtoclassifyallmotiviccohomologyoperationsonHp,1 andH1,q andsuggestageneralclassification. 5 ] T In the last decade, several papers have given constructions of cohomology op- K erationsonmotivic andétale cohomology,following the earlierworkof Jardine [J], h. Kriz-May [KM] and Voevodsky [V2, V1]: see [BJ, BJ1, Jo, May1, V3, V4]. The t goal of this paper is to provide, for each n and i, a classification of all such opera- a tions on the étale groups Hn(−,µ⊗i) and the motivic groups Hn,i(−,F ), similar m et ℓ ℓ to Cartan’s classification of operations on singular cohomology Hn (−,F ) in [C]. [ top ℓ We succeed for étale operations and partially succeed for motivic operations. 1 We work over a fixed field k and fix a prime ℓ with 1/ℓ ∈ k. By definition, an v (unstable) étale cohomology operation on Hn(−,µ⊗i) is a natural transformation 2 et ℓ Hn(−,µ⊗i) → Hp(−,µ⊗q) of set-valued functors from the category of (smooth) 7 et ℓ et ℓ 8 simplicialschemesoverk (forsomepandq). Similarly,amotiviccohomologyoper- 0 ationonHn,i is a naturaltransformationHn,i →Hp,q onthis category. By defini- . tion,Hp,q(X)denotestheNisnevichcohomologyHp (X,F (q)), wherethecochain 1 nis ℓ 0 complex Fℓ(q) is defined in [V2] or [MVW]. Note that the set of all cohomology 3 operationsformsaring;theproductofθ andθ isthe operationx7→θ (x)·θ (x). 1 2 1 2 1 Our classification begins with a construction of étale operations Pa, due to Ep- : v stein, as a special case of the operations on sheaf cohomology he described in i his 1966 paper [E]. This is carried out in Theorem 1.1 for odd ℓ; when ℓ = 2, X this was established by Jardine [J]. The coefficient ring H∗(k,µ⊗∗) also acts on r et ℓ a étale cohomology;we prove in Theorem 3.5 that the ring of allétale operations on Hn(−,µ⊗i)isthetwistedringH∗(k,µ⊗∗)⊗H∗ (K ),whereH∗ (K )isCartan’s et ℓ et ℓ top n top n ring of operations on Hn (−,F ). (We give a precise description of Cartan’s ring top ℓ in Definition 0.1 below.) This classifies all étale operations on Hn(−,µ⊗i): they et ℓ are H∗(k,µ⊗∗)-linear combinations of monomials in the operations PI. et ℓ A slightly different approach to cohomology operations was given by May in [May], one which produces operations in the cohomology of any E -algebra. In ∞ Section2, we review this approachin the contextofsheaf cohomologyandshow in Corollary2.3thattheétaleoperationsconstructedinthiswayagreewithEpstein’s. ThisresultallowsustoutilizeMay’streatmentoftheKudoTransgressionTheorem in [May, 3.4]; see Theorem 6.5 below. Date:December11,2013. 1 2 BERTGUILLOUANDCHUCKWEIBEL InSection4,wecombineEpstein’sconstructionwiththeNormResidueTheorem to define motivic operations Pa (see 4.3). We show they are compatible with the étale operations, and that they are stable under simplicial suspension. The operation P0 is the Frobenius Hn,i →Hn,iℓ on motivic cohomology, induced by the ℓth powermapF (i)→F (iℓ);seeProposition5.4. OnenewresultconcerningVoevod- ℓ ℓ sky’s operations is that for n > i and x ∈ H2n,i we have Pn(x) = [ζ](n−i)(ℓ−1)xℓ V (seeCorollary5.10). ThisextendsLemma9.8of[V1],whichstatesthatPn(x)=xℓ for x∈H2n,n(X). The classification of motivic cohomology operations is complicated by the pres- ence of more operations than those constructed by Voevodsky or via Steenrod- Epstein methods. One example is that an ℓ-torsion element t in the Brauer group of k gives an operation H1,2 →H3,3 by H1,2(X)∼=H1(X,µ⊗2)−∪→t H3(X,µ⊗3)∼=H3,3(X). et ℓ et ℓ Also unexpectedly, we may also use t and the Bockstein β to get an operation H1,2(X) → H4,3(X) (see Example 8.5 below). When k contains a primitive ℓth root of unity ζ, we also have an interesting operation H1,2(X) → H2,1(X) = Pic(X)/ℓ: divide by the Bott element [ζ]∈H0,1(k) and then apply the Bockstein; see Proposition 8.2. InSection7,wedeterminetheringofallunstablemotiviccohomologyoperations on Hn,1. If ℓ 6= 2, it is the twisted ring H∗,∗(k)⊗H∗ (K ), where H∗,∗(k) is the top n motiviccohomologyofk andH∗ (K )isCartan’sring,describedinDefinition0.1 top n below. In Section 8, we determine the ring of unstable cohomology operations on H1,i. When k contains the ℓth roots of unity, this is the graded polynomial ring over H∗,∗(k) on operations γ : H1,i(X) ∼= H1,1(X) and its Bockstein, where γ is given by the Norm Residue Theorem 4.2. For general fields, it is the Galois-invariant subring. The operations on H1,2 referred to above arise in this way. Finally Section 9 contains a conjecture about what the general classification might be for Hn,i when n,i>1. Since it is the topological prototype of our classification theorem, we conclude thisintroductionwithadescriptionoftheringofallsingularcohomologyoperations on Hn (−,F ). Serre observed that the ring of operations from Hn (−,F ) to top ℓ top ℓ H∗ (−,F ) is isomorphic to the cohomology H∗ (K ) of the Eilenberg-MacLane top ℓ top n spaceK =K(F ,n);thestructureofthisringwasdeterminedbySerreandCartan n ℓ in [C] [C1]. The following description is taken from [McC, 6.19]. Definition 0.1. For ℓ > 2, let H∗ (K ) denote the free graded-commutative F - top n ℓ algebra generated by the elements PI(ι ), where I = (ǫ ,s ,ǫ ,...,s ,ǫ ) is an n 0 1 1 k k admissible sequence satisfying either e(I)k, and I is admissible if s ≥ℓs +ǫ for all i1 i For example, every operation on HPt2op(−,Fℓ) is a polynoPmial in id, β, the PIβ and the βPIβ (where PI = Pℓk···PℓP1). This is because the only admissible sequences with excess <2 are 0, (1) and (0,ℓk,0,...,ℓ,0,1,1). UNSTABLE OPERATIONS IN E´TALE AND MOTIVIC COHOMOLOGY 3 1. Epstein’s étale construction Cohomology operations in étale cohomology were constructed by D.Epstein long ago in the 1966 paper [E], and (for constant coefficients) made explicit by M.Raynaud [R, 4.4]. Alternative constructions were later given by L.Breen [Br, III.4] and J.F.Jardine [J, 1.4], [J1, §2]. In Epstein’s approach, one starts with an F -linear tensor abelian category A ℓ (suchassheavesofF -modulesonasite),aleftexactfunctorH0(X,−)(globalsec- ℓ tionsoverX)andacommutativeassociativeringobjectOofA. Epsteinconstructs operations Sqa:Hn(X,O)→Hn+a(X,O) if ℓ=2, and Pa :Hn(X,O)→Hn+2a(ℓ−1)(X,O), ℓ6=2, satisfying the usual relations: Pax = xℓ if n = 2a, Pax = 0 if n < 2a, a Cartan relationforPa(xy)andAdemrelationsforPaPb. Epsteinalsodefinesanoperation Qa for each a, whose degree is one more than that of Pa. One subtlety is that P0 is not the identity but rather the Frobenius map on the F -algebra H0(X,O). ℓ Now suppose that A is the category of étale sheaves (on the big étale site) and that O is the graded étale sheaf ⊕∞ µ⊗i. We will prove in 3.3 below that i=0 ℓ Qa =βPa. With this dictionary, Epstein’s theorem specializes to yield Theorem 1.1. For each odd prime ℓ, there are additive cohomology operations Pa :Hn(X,µ⊗i)→Hn+2a(ℓ−1)(X,µ⊗iℓ) et ℓ et ℓ and a Bockstein β : Hn(X,µ⊗i) → Hn+1(X,µ⊗i) satisfying the usual relations: et ℓ et ℓ Pax=xℓ ifn=2a,Pax=0ifn<2a,theCartanrelationPa(xy)= Pi(x)Pj(y) and Adem relations for both PaPb (a 2 then w = 1, k ℓ 0 P w =βw , w =wi and w =w wi. If ℓ>2 and n≥2a, Epstein defines 2 1 2i 2 2i+1 1 2 (1.5) Pa :Hn(X,A)→Hn+2a(ℓ−1)(X,B), Pau=(−1)aν D (u), n (n−2a)(ℓ−1) where ℓ−1 (ℓ−1)(n2+n) ν =(−1)r !−n and r = . n 2 4 (cid:18) (cid:19) UNSTABLE OPERATIONS IN E´TALE AND MOTIVIC COHOMOLOGY 5 (See [E, 7.1], [SE, VII.6.1] and [SErr].) If n<2a then Epstein defines Pa =0. When ℓ = 2, Epstein defines operations Sqi by: Sqi(u) = D (u) for n ≥ i, n−i and Sqi(u)=0 for n0. Proof. We first consider the case when ζ ∈k, so that µ⊗i ∼= F for all i. Jardine’s ℓ ℓ argumentin[J,pp.108–114]thatEpstein’sSq1 istheBocksteinwhenℓ=2applies when ℓ > 2 as well, and proves that Epstein’s Q0 is the Bockstein operation. The identity Q0 =β in the general case follows from this and the isomorphism (3.2): Hn(X,O) −Q−−0−−→β Hn+1(X,O) et et ∼= ∼= Hn(X(ζ),F ) −Q−−0−−→β Hn+1(X(ζ),F ). et y ℓ et y ℓ Finally,weinvoketheAdemrelationQ0Pb =QbP0[E,9.8(4)]togetβPb =Qb. (cid:3) UsingtheBocksteinandEpstein’soperationsPa,wehaveoperationsPI defined on Hn(−,µ⊗i) for every admissible sequence I in the sense of Definition 0.1. et ℓ In order to classify all operations on Hn, we first consider the case n = 1. In et topology, the ring of operations on H1(−,F ) is H∗ (K ) ∼= F [u,v]/(u2), where ℓ top 1 ℓ u=P0 is indegree1,correspondingto the identityoperation,andv is indegree2, correspondingtotheBocksteinoperation. ByTheorem1.1,thereisacanonicalmap fromF [u,v]/(u2)toétalecohomologyoperationsfromH1(−,µ⊗i)toH∗(−,µ⊗∗), ℓ et ℓ et ℓ sending u to the identity and v to the Bockstein β :H1(−,µ⊗i)→H2(−,µ⊗i). et ℓ et ℓ For any i, the basechange µ⊗i(ζ) of the algebraic group µ⊗i is isomorphic to ℓ ℓ F (ζ), the constant sheaf F on the big étale site of k(ζ). The induced isomor- ℓ ℓ phism(Bµ⊗i)(ζ)∼=(BF )(ζ)inducesanisomorphismofcohomologygroups,which ℓ ℓ immediately yields the following calculation. Proposition 3.4. The graded algebra of cohomology operations from H1(−,µ⊗i) et ℓ to H∗(−,O)=⊕d−1H∗(−,µ⊗j) is isomorphic to the H∗(k(ζ),F )-module et j=0 et ℓ et ℓ H∗(Bµ⊗i,O)∼=H∗(Bµ⊗i(ζ),F )∼=H∗(k(ζ),F )⊗F [u,v]/(u2), β(u)=v. ℓ ℓ ℓ ℓ ℓ 10 BERTGUILLOUANDCHUCKWEIBEL Every operation on H1(−,µ⊗i) is uniquely a sum of operations φ(x) =cxεβ(x)m, et ℓ where c∈H∗(k,µ⊗j) and ε∈{0,1}. et ℓ Proposition 3.4 is the case n=1 of the following result. Theorem 3.5. For each i and n ≥ 1, the ring of all étale cohomology operations from Hn(−,µ⊗i) to H∗(k,O) is the free left H∗(k(ζ),F )-module H∗(k(ζ),F )⊗ et ℓ et et ℓ et ℓ H˜∗ (K ). top n The operations PI send Hn(−,µ⊗i) to H∗(−,µ⊗i). Thus the operations from et ℓ et ℓ Hn(−,µ⊗i) to H∗(−,µ⊗i+j) are isomorphic to H∗(−,µ⊗j)⊗H˜∗ (K ). et ℓ et ℓ et ℓ top n Proof. We first show that the basechange K(µ⊗i,n) × Spec(k(ζ)) is the space ℓ k K(µ⊗i,n) over k(ζ). This is clear for n = 0, and follows inductively from the ℓ constructionofK(A,n+1)via the barconstructiononK(A,n), togetherwith the observation that (X × Y)× Spec(k(ζ)) is X(ζ)× Y(ζ). k k k(ζ) By (3.2), the cohomology of K(µ⊗i,n) with coefficients in O is the same as the ℓ cohomology of K(µ⊗i,n)× Spec(k(ζ)) with coefficients in F . The Breen-Jardine ℓ k ℓ result, Theorem 3.1, shows that this is H∗(k(ζ),F )⊗H∗ (K ). (cid:3) et ℓ top n 4. Motivic Steenrod operations In this section we construct operations Pa on the motivic cohomology groups Hn,i(X)= Hn,i(X,F ), n ≥ 2a, compatible with the operations Pa in étale coho- ℓ mology in the sense that there are commutative diagrams Hn,i(X,F ) −−P−−a→ Hn+2a(ℓ−1),iℓ(X,F ) ℓ ℓ (4.1) Hn(X,µ⊗i) −−P−−a→ Hn+2a(ℓ−1)(X,µ⊗iℓ). et y ℓ et y ℓ Let α denote the direct image functor from theétale site to the Nisnevich site. ∗ If F is any étale sheaf then we may regard A = Rα F as a complex of Nisnevich ∗ sheaves such that H∗ (X,Rα F)∼=H∗(X,F). nis ∗ et The following theorem, due to Voevodsky and Rost, is sometimes known as the BeilinsonConjecture;itisequivalenttotheNormResidueTheorem;see[SV],[V4], [W]), [HW]). Let τ≤iA denote the good truncation of A in cohomological degrees at most i; Hn(τ≤iA) is Hn(A) for i≤n, and zero for n>i. (cf. [WH, 1.2.7]). Norm Residue Theorem 4.2. For any X smooth over a field of characteristic 6=ℓ, the map F (i)→τ≤iRα µ⊗i is a quasi-isomorphism, and hence ℓ ∗ ℓ Hn,i(X,F )∼=Hn (X,τ≤iRα µ⊗i). ℓ nis ∗ ℓ In particular, if n≤i then Hn,i(X,F )∼=Hn(X,µ⊗i). ℓ et ℓ Explicitly, if F is an étale sheaf and F → I∗ is an injective resolution, then τ≤iRα F isrepresentedbyτ≤iα (I∗). Itisquasi-isomorphictoachaincomplexI∗ ∗ ∗ nis of injective Nisnevich sheaves on X with In = α Ii for n ≤ i, because each α In nis ∗ ∗ isaninjectiveNisnevichsheafandZn(α I)injectsintoα In. TakingF =µ⊗i,the ∗ ∗ ℓ theoremstates that Hn,i(X,F )=Hn (X,F (i)) is the nth cohomologyof I∗ (X). ℓ nis ℓ nis

Unstable operations in étale and motivic cohomology

Similar Unstable operations in étale and motivic cohomology