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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)<n or e(I)=n and ǫ =1. 0 Here the excessofI isdefinedto be e(I)=2 (s −ℓs −ǫ )+ k ǫ ,where i i+1 i i=0 i s =0 for i>k, and I is admissible if s ≥ℓs +ǫ for all i<k. i i i+1 i When ℓ = 2, H∗ (K ) denotes the free graPded-commutative FP-algebra gen- top n 2 erated by the elements SqI(ι ), with I = (s ,...,s ) admissible (s ≥ 2s ) and n 1 k i i+1 e(I)<n, where the excess is e(I)= (s −2s )=s − s . i i+1 1 i>1 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<bℓ) and PaβPb (a≤bℓ). When ℓ=2, there are Steenrod operations Sqa :Hn(X,µ⊗i)→HPn+a(X,µ⊗2i), et 2 et 2 or Hn(X,Z/2)→Hn+a(X,Z/2), satisfying the usual relations. et et Proof. The existence and basic properties is given in Chapter 7 of [E]; The Adem relationsareestablishedin[E,9.7–8],usingthedictionarythatPaβPb =PaQb. (cid:3) Note that, although the operations multiply the weight by ℓ, the reindexing makes no practical difference because there are canonical isomorphisms µ ∼= µ⊗ℓ ℓ ℓ and µ⊗i ∼= µ⊗iℓ. We have emphasized the twist because of our application to ℓ ℓ motivic operations below. Remark 1.1.1. The operations Pa are natural in X: if f :X →Y is a morphism of simplicial schemes then f∗Pa =Paf∗. This is immediate from the naturality of the construction of Pa with respect to the left exact functor H0(X,−), and also et follows from [E, 11.1(8)]. If Z is a closed simplicial subscheme of X, we get cohomologyoperations Pa on the relativegroupsHn(X,Z;µ⊗i), by replacingH0(X,−)by the leftexactfunctor et ℓ H0(X,Z;−). The same argument shows that Pa is natural in the pair (X,Z). et For later use, we reproduce two key results from [E]. If π is a finite group, we write A[π] for the category of π-equivariant objects of A, i.e., objects A equipped with a homomorphism π →End(A). If A is in A[π] then H0(X,A) is a π-module, andwe define the left exactfunctor H0(X,−) onthe categoryA[π] by the formula π H0(X,A)=H0(X,A)π. We write H∗(X,−) for the derivedfunctors of H0(X,−). π π π 4 BERTGUILLOUANDCHUCKWEIBEL Theorem 1.2. Let Abeaboundedbelow cochain complex of objects of A, on which π acts trivially. Then there is a natural isomorphism H∗(π,F )⊗H∗(X,A)−≃→H∗(X,A). ℓ π Proof. (See [E, 4.4.4].) Let C →Z be the standard periodic Z[π]-resolution [WH, ∗ 6.2.1],withgeneratore ofC ∼=Z[π], andsetC∗=Hom(C ,F ); thusH∗(π,F )is k k ∗ ℓ ℓ thecohomologyof(C∗)π. Chooseaquasi-isomorphismA−∼→I∗ withtheIiinjective inA. Sinceπ actstriviallyonA,wehavequasi-isomorphismsofcomplexesinA[π]: A → I∗ = Z⊗I∗ → Tot(C∗⊗I∗). Since each Cn is a free F [π]-module of finite ℓ rank,Cn⊗Iq ∼=Hom(C ,Iq)isinjectiveinA[π]. ThusTot(C∗⊗I∗)isaninjective n replacement for A in A[π]. By definition, H∗(X,A) is the cohomology of the total complex of π H0(X,C∗⊗I∗)=(C∗)π ⊗I∗(X). π TheKu¨nnethformulatellsusthisisthetensorproductofthecohomologyof(C∗)π and I∗(X), i.e., of H∗(π,F ) and H∗(X,A). (cid:3) ℓ Corollary 1.3. If A is a commutative algebra object, the isomorphism of Theorem 1.2 is an algebra isomorphism. Proof. The verification that a commutative associative product on A induces an algebrastructureonH∗(X,A)andH∗(X,A)is wellknown. We omitthe standard π proof that the isomorphism above commutes with products. (cid:3) Recallthat for any complex C, the symmetric groupS acts on C⊗ℓ by permut- ℓ ing factors with the usual sign change. Now suppose that π is the cyclic Sylow ℓ-subgroup of S . Choosing an injective replacement A⊗ℓ → J∗ in A[π], the com- ℓ parisontheorem [WH, 2.3.7]lifts the equivariantquasi-isomorphismA⊗ℓ →(I∗)⊗ℓ to an equivariant map (I∗)⊗ℓ →J∗, unique up to chain homotopy. Since H∗(X,A) is the cohomology of I∗(X), we can represent any element of Hn(X,A) by an n-cocycle u ∈ In(X). The nℓ-cocycle u⊗···⊗u of I(X)⊗ℓ is π- invariant,because the generatorof π acts as multiplicationby (−1)n(ℓ−1), which is theidentityonanyF -module. ItsimagePuinJnℓ(X)isalsoπ-invariant. Epstein ℓ shows in [E, 5.1.3] that P(u+dv)=Pu+dw for v ∈In−1(X) and w ∈Jnℓ−1(X), so the cohomology class of Pu is independent of the choice of cocycle u. Definition 1.4. The reduced power map is defined to be the resulting map on cohomology: P :Hn(X,A)→Hnℓ(X,A⊗ℓ). π Now suppose that there is a π-equivariant map A⊗ℓ−m→B, and that π acts trivially on B. (When A is a commutative ring, multiplication A⊗ℓ → A is a π- equivariant map.) We write m for the induced map H∗(X,A⊗ℓ) → H∗(X,B). ∗ π π By Theorem 1.2, m P(u) ∈ H∗(X,B) has an expansion w ⊗ D (u), where ∗ π k k w ∈ Hk(π,F ) are the (dual) basis elements of [SE, V.5.2]: if ℓ > 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 n<i. Remark 1.5.1. EpsteinalsodefinesoperationsQa =(−1)a+1ν D (u) n (n−2a)(ℓ−1)−1 in this setting, and establishes Adem relations for them as well. Of course, Epstein’s construction mimicks Steenrod’s construction of D , Pa k andQa (see[SE],VII.3.2andVII.6.1). InSteenrod’ssettingonecanlifttointegral cochains; with this assumption, Steenrod proves that βD = −D and hence 2k 2k+1 that βPa = Qa; see [SE, VII.4.6] and [SErr]. We will show that the formula Qa =βPa also holds in our setting (see Lemma 3.3 and Theorem 5.11.) Lemma 1.6. For all bounded below chain complexes A and B as above, each function Pa :Hn(X,A)→Hn+2a(ℓ−1)(X,B) is a group homomorphism. Proof. Corollary 6.7 of [E] applies in this setting. (cid:3) Lemma 1.7. The Pa and Qa are natural in the map A⊗ℓ−m→B. Proof. Suppose we are given a commutative diagram A⊗ℓ −−−m−→ B 1 1 Ay⊗2ℓ −−−m−→ By2. Applying H∗(X,−) and composing with P, which is natural in A by [E, 5.1.5], π Theorem 1.2 yields the commutative diagram H∗(X,A ) −−−P−→ H∗(X,A⊗ℓ) −−−m−→ H∗(X,B ) −−−∼=−→ H∗(π)⊗H∗(X,A ) 1 π 1 π 1 1 H∗(Xy,A2) −−−P−→ Hπ∗(Xy,A⊗2ℓ) −−−m−→ Hπ∗(Xy,B2) −−−∼=−→ H∗(π)⊗Hy∗(X,A2). The result now follows from the definition (1.5) of Pa and Qa. (cid:3) RecallthatthesimplicialsuspensionSX ofasimplicialschemeX isagainasim- plicial scheme. There is a canonical isomorphism Hn(X,µ⊗i)−∼=→Hn+1(SX,µ⊗i). et ℓ et ℓ Proposition 1.8. The operations Pa are simplicially stable in the sense that they commute with simplicial suspension: there are commutative diagrams for all X, n and i, with N =n+2a(ℓ−1): Hn(X,µ⊗i) −−P−−a→ HN(X,µ⊗iℓ) et ℓ et ℓ ∼= ∼= Hn+1(SX,µ⊗i) −−P−−a→ HN+1(SX,µ⊗iℓ). et y ℓ et y ℓ Proof. The proofs of Lemmas 1.2 and 2.1 of [SE] go through, using homotopy invariance ofétale cohomology and excision. (cid:3) 6 BERTGUILLOUANDCHUCKWEIBEL 2. May’s adjoint construction Asomewhatdifferentapproachtoconstructingcohomologyoperationswasgiven byPeterMayin[May]. BecausewewillneedMay’sversionofKudo’sTheorem(in 6.5 below), we need to know how the two constructions compare. First, we need a chain level version of the Steenrod-Epstein function m P :Hn(X,A)→Hnℓ(X,A) ∗ π usedin(1.5)todefinePa. Wesawin1.4thatthemultiplicationmapm:A⊗ℓ →A lifts to an equivariant map J∗(X) → C∗ ⊗ I∗(X) of their injective resolutions, inducing an equivariant map mˆ :I⊗ℓ(X)→J∗(X)→C∗⊗I∗(X). The cohomologyfunction m P is induced by the chain-levelfunction u7→mˆ(u⊗ℓ). ∗ The expansionmˆ(u⊗ℓ)= w ⊗D (u)inC∗⊗I∗(X)effectivelydefines functions k k D :In(X)→Inℓ−k(X). k Consider the isomorphiPsm φ:C∗⊗I∗(X)→Hom(C ,I∗(X)), defined by ∗ φ(f ⊗x)(c)=(−1)|x||c|f(c)x. Asaspecialcase,φ(w ⊗x)(e )=(−1)k|x|δ x.Thecompositionφmˆ sendsI⊗ℓ(X) j k jk to Hom(C ,I∗(X)). It is the (signed) adjoint of the map φmˆ, ∗ (2.1) θ :C ⊗I⊗ℓ(X)→I∗(X), ∗ whichformsthe basisfor May’sapproach;see[May,2.1]. May defines the function DM :In(X)→Inℓ−k(X) by the formula k DM(u)=θ(e ⊗u⊗ℓ). k k May defines Pa and Qa by M M Pa (u)=(−1)aν DM (u) and Qa (u)=(−1)aν DM (u) M n (n−2a)(ℓ−1) M n (n−2a)(ℓ−1)−1 (see [May, pp.162,182]; his ν(−n) is our ν ). The sign differences in the formulas n for Pa and Pa (and for Qa and Qa ) are explained by the following calculation. M M Proposition 2.2. For u in Hn(X,A), DM =(−1)kD . k k Proof. Considertheisomorphismφ:C∗⊗I∗(X)→Hom(C ,I∗(X)),definedabove. ∗ The adjoint θ of φmˆ is the composite C ⊗I(X)⊗ℓ ∼=I(X)⊗ℓ⊗C φ−mˆ→⊗1Hom(C ,I(X))⊗C −η→I(X), ∗ ∗ ∗ ∗ where the first map is the signed symmetry isomorphism and η is evaluation. We now compute that DM(u)=θ(e ⊗u⊗ℓ)=(−1)knℓη φ[mˆ(u⊗ℓ)]⊗e k k k =(−1)knℓη(cid:0)φ w ⊗D (cid:1)(u) ⊗e j j k =(−1)knℓ (cid:16) φhX[w ⊗D (u)](ei) (cid:17) j j k j X =(−1)knℓ(−1)k(nℓ−k)D (u) k =(−1)kD (u). (cid:3) k Corollary 2.3. May’s operations Pa and Qa coincide with the Pa and Qa of M M (1.5) and 1.5.1. UNSTABLE OPERATIONS IN E´TALE AND MOTIVIC COHOMOLOGY 7 Lemma 2.4. Set m=(ℓ−1)/2. Then for each u∈In(X): (i) dPa(u)=Pa(du) and dQa(u)=−Qa(du), and (ii) if u is a cocycle representing x∈Hn(X,A) then Pa(u) and Qa(u) are cocycles representing Pa(x) and Qa(x), respectively. Proof. In Theorem 3.1 of [May], May shows that (i) dPa (u) = Pa (du) and M M dQa (u) = −Qa (du), and (ii) if u is a cocycle representing x ∈ Hn(X,A) then M M Pa (u) and Qa (u) are cocycles representing Pa (x) and Qa (x). The result is M M M M immediate from Corollary 2.3. (cid:3) Remark 2.5. In [May1], May gave a different approach to power operations in sheaf cohomology. If A is any sheaf of F -algebras, May shows (in 3.12) that ℓ the sections over X of the Godement resolution F•A of A yield an algebra C• = Hom (Λ,F•A)overtheEilenberg-ZilberoperadIoncochainsofF -modules,which ∆ ℓ therefore inherits the structure of an E -algebra. Since the cohomology of C• ∞ is the sheaf cohomology H∗(X,A), the technique in [May] produces cohomology operations. 8 BERTGUILLOUANDCHUCKWEIBEL 3. The étale Steenrod algebra In this section we determine the algebra of all étale cohomology operations Hn(−,µ⊗i)→H∗(−,µ⊗j) over a field k containing 1/ℓ. et ℓ et ℓ RecallfromSGA 4(V.2.1.2in[Ver])thatif M is a(simplicial)étalesheafofF - ℓ modulesthenthesheafcohomologygroupsH∗(X,M)areisomorphictothe(hyper) et Ext-groups Ext∗(F [X],M) in the category of étale sheaves of F -modules. (Here ℓ ℓ we regard M as a cochain complex using Dold-Kan.) If K is a second simplicial étale sheaf of F -modules, one writes H∗(K,M) for Ext∗(K,M). ℓ et It is well known that cohomology operations Hn(−,L) → H∗(−,M) are in et et 1–1 correspondence with elements of H∗(K,M), where K denotes the standard et simplicial Eilenberg-MacLane sheaf K(L,n) associated to L. We first discuss the case of constant coefficients (M =F ), which is known, and ℓ duetoBreen[Br,4.3–4]andJardine[J]. Thegradedringofallunstableétalecoho- mology operations from Hn(−,F ) to H∗(−,F ) is isomorphic to the cohomology et ℓ et ℓ ring H∗(K ,F ), where K = K(F ,n) is the constant simplicial sheaf classifying et n ℓ n ℓ elements of Hn(−,F ). By Theorem 1.1, there is a ring homomorphism from the et ℓ classical unstable Steenrod algebra H∗ (K ) of Definition 0.1 to H∗(K ,F ). top n et n ℓ There is also a ring homomorphism from H∗(k,F ) to H∗(K ,F ). These two et ℓ et n ℓ ringhomomorphismsdonotcommute, becausethe Bocksteinandotheroperations PI may be nontrivial on H∗(k,F ). If c ∈ H∗(k,F ) then cPI and PIc send x et ℓ et ℓ to c · PI(x) and to PI(cx), respectively. Nevertheless, we can define a twisted multiplication on He∗t(k,Fℓ) ⊗Fℓ Ht∗op(Kn) using the Cartan relations Pa ◦ α = Pi(α)Pj, α∈H∗(k,F ). We shall refer to this non-commutative algebra as the et ℓ twisted tensor algebra. It is free as a left H∗(k,F )-module, and a basis is given P et ℓ by the monomials in the Steenrod operations PI and βPI where I has excess <n, exactly as in the topological case. We summarize this: Theorem3.1. Theringofétalecohomology operationsonHn(−,F )isthetwisted et ℓ tensor product H∗(k,F )⊗H∗ (K ): every operation is a polynomial in the oper- et ℓ top n ations PI with coefficients in H∗(k,F ). et ℓ Example 3.1.1. When k =R and ℓ=2, the ring of étale cohomology operations overR isthe gradedpolynomialringH∗ (K )[σ]withgeneratorσ indegree1and top n all SqI(ι ) with I admissible and e(I)<n. This is because H∗(R,F )=F [σ]. n et 2 2 Remark3.1.2. Whenk =C,theactionofthePI iscompatiblewiththecanonical comparison isomorphism H∗(X,F ) ∼= H∗ (X(C),F ). This is clear from the et ℓ top ℓ constructions in [E] and [J]. Whenkcontainsaprimitiveℓth rootofunity,thesheavesµ⊗i areallisomorphic. ℓ ThustheringofoperationsHn(−,µ⊗i)→H∗(−,µ⊗j)isisomorphictoH∗(k,F )⊗ et ℓ et ℓ et ℓ H∗ (K ) as a left H∗(k,F )-module. Since this is alwaysthe case when ℓ=2, we top n et ℓ shall restrict to the case of an odd prime ℓ. Now fix i and consider cohomology operations Hn(−,µ⊗i) → H∗(−,µ⊗j). As et ℓ et ℓ observedabove,theyarein1–1correspondencewithelementsofH∗(K,µ⊗j),where et ℓ K denotes the simplicial Eilenberg-MacLane scheme K(µ⊗i,n). For example, the ℓ identity operation on Hn(−,µ⊗i) corresponds to ι ∈ Hn(K,µ⊗i), and the étale et ℓ n ℓ Bockstein β :Hn(X,µ⊗i)→Hn+1(X,µ⊗i) corresponds to β(ι )∈Hn+1(K,µ⊗i). et ℓ et ℓ n ℓ Fix a field k with 1/ℓ ∈ k, and let G be the Galois group of the extension k(ζ)/k, where ζ denotes a primitive ℓth root of unity. Then G is cyclic of order UNSTABLE OPERATIONS IN E´TALE AND MOTIVIC COHOMOLOGY 9 d=[k(ζ):k],d|ℓ−1,andµ⊗d ∼=F . Sincei≡j (mod d)impliesthatµ⊗i ∼=µ⊗j, ℓ ℓ ℓ ℓ we are led to consider the Z/d-graded étale sheaf of rings O = d−1µ⊗i. Thus i=0 ℓ our problem is to determine the ring H∗(K,O)=⊕d−1H∗(K,µ⊗j). et j=0 et ℓL Since ℓ does not divide |G|, Maschke’s Theorem gives an identification of the étale sheaf F [G] with the direct sum of the sheaves of irreducible F [G]-modules ℓ ℓ µ⊗i, i.e., with O. For any X, Shapiro’s Lemma provides an isomorphism ℓ (3.2) H∗(X,O)∼=H∗(X(ζ),F ), et et ℓ whereX(ζ)denotesX× Spec(k(ζ)). Infact,O=Ri F ,wherei:Spec(k(ζ)) → k ∗ ℓ et Spec(k) . et Any F [G]-module M is the sumof its isotypicalsummands,the isotypicalsum- ℓ mandforµ⊗i being Hom (µ⊗i,M). Inparticular,the actionofGonX(ζ)decom- ℓ G ℓ poses H∗(X(ζ),F ) into its isotypical pieces. Because the µ⊗i are the isotypical et ℓ ℓ summands of O =Ri F , the summand H∗(X,µ⊗i) in (3.2) is the isotypical sum- ∗ ℓ et ℓ mand of H∗(X(ζ),F ) for µ⊗i. et ℓ ℓ This is the context in which Epstein defines operations Pa and Qa via (1.5). Because the Frobenius is the identity on O, P0 is the identity operation. We can now show that Epstein’s operationQ0 is theétale Bocksteinβ, and his Qa is βPa. Lemma 3.3. In étale cohomology, Q0 =β and Qa =βPa for a>0. 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