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CYCLIC ADAMS OPERATIONS 6 MICHAELK.BROWN,CLAUDIAMILLER,PEDERTHOMPSON, 1 ANDMARKE.WALKER 0 2 Abstract. Let Q be a commutative, Noetherian ring and Z ⊆ Spec(Q) a b e closedsubset. DefineK0Z(Q)tobetheGrothendieck groupofthosebounded F complexesoffinitelygeneratedprojectiveQ-modulesthathavehomologysup- portedonZ. Wedevelop“cyclic”AdamsoperationsonK0Z(Q)andweprove 2 these operations satisfy the four axioms used by Gillet and Soulé in [GS87]. 2 FromthiswerecoverashorterproofofSerre’sVanishingConjecture. Wealso show our cyclic Adams operations agree with the Adams operations defined ] byGilletandSouléincertaincases. T OurdefinitionofthecyclicAdamsoperatorsisinspiredbyaformuladueto K Atiyah[Ati66]. TheyhavealsobeenintroducedandstudiedbeforebyHaution . [Hau09]. h t a m [ Contents 2 1. Introduction 1 v 2 2. Tensor power operations 4 7 3. Cyclic Adams operations 8 0 4. Commutativity of the cyclic Adams operations 13 5 5. Lambda operations and agreement with those of Gillet-Soulé 15 0 6. Agreement of ψ with ψ 19 . cyc GS 1 References 24 0 6 1 : v 1. Introduction i X In1987,GilletandSoulé[GS87]developedatheoryofAdamsoperationsonthe r a Grothendieckgroupofchaincomplexesoflocallyfreecoherentsheavesonascheme that satisfy a support condition, and they proved these operations satisfy the four keyaxioms(A1)–(A4)listedbelow. Asamajorapplicationofthistheory,Gilletand Soulé proved Serre’s Vanishing Conjecture in full generality. (Serre [Ser65] proved thisinmanycases,andRoberts[Rob85]alsoprovedthegeneralcase,independently andataboutthesametimeasGilletandSoulé.) Thegoalofthispaperistodevelop an alternative, simpler, notion of Adams operations on such Grothendieck groups, incertainimportantcases,onewhichisbasedonanideaduetoAtiyah[Ati66]. As a consequence,we arrive at a proof of the full case of Serre’s Vanishing Conjecture that is considerably shorter than the proofs of Gillet-Soulé or Roberts. ThisworkwaspartiallysupportedbyagrantfromtheSimonsFoundation(#318705forMark Walker)andgrantsfromtheNationalScienceFoundation(NSFAwardDMS-0838463forMichael BrownandPederThompsonandNSFAwardDMS-1003384forClaudiaMiller). 1 2 MICHAELK.BROWN,CLAUDIAMILLER,PEDERTHOMPSON,ANDMARKE.WALKER In more detail, suppose X is a separated, Noetherian scheme and Z ⊆ X is a closed subset; define KZ(X) to be the Grothendieck group of bounded complexes 0 of locally free coherent sheaves on X whose homology is supported on Z. This is theabeliangroupgeneratedbythe isomorphismclassesofsuchcomplexes,modulo relations coming from short exact sequences and quasi-isomorphisms. For each integer k ≥1, Gillet and Soulé define an operator ψk :KZ(X)→KZ(X) GS 0 0 and they prove [GS87, Prop 4.12] that this operator satisfies the following axioms: (A1) ψk is a homomorphism of abelian groups for all X and Z; GS (A2) ψk is multiplicative: if Z,W are closed subsets of X, α ∈ KZ(X), and GS 0 β ∈KW(X), then 0 ψk (α∪β)=ψk (α)∪ψk (β)∈KZ∩W(X), GS GS GS 0 where −∪− is the pairing induced by tensor product of complexes; (A3) ψk is functorial, in the sense that, given a morphism φ : Y → X and GS closed subsets W ⊆ Y and Z ⊆ X such that φ−1(Z) ⊆ W, we have an equality ψk ◦φ∗ =φ∗◦ψk GS GS of maps KZ(X)→KW(Y); and 0 0 (A4) if Q is a commutative Noetherian ring with unit and a ∈ Q is a non-zero- divisor, then ψk ([K(a)])=k[K(a)]∈KV(a)(SpecQ), GS 0 a where K(a) := (··· → 0 → Q −→ Q → 0 → ···) is the Koszul complex on a. Serre’s Vanishing Conjecture follows from the existence of such an operator for any one value of k ≥2; see [GS87, §5]. Gillet and Soulé’s construction of the operator ψk involves first establishing GS λ-operations,λk for allk ≥1,onKZ(X). Thesearedefined usingthe Dold-Puppe 0 construction[Dol58,DP58](seealso[Kan58])ofexteriorpowersofchaincomplexes concentratedinnon-negativedegrees. Indetail,ifE isaboundedcomplexoflocally freecoherentsheavesonX thatissupportedonZ andconcentratedinnon-negative degrees (i.e., E =0 for i<0), we let K(E) be the associatedsimplicial sheaf given i by the Dold-Puppe functor K. Write Λk K(E) for the simplicial sheaf obtained OX by applying Λk (−) degreewise to K(E). Let N(Λk K(E)) be the chain complex OX OX givenbyapplyingthenormalizedchaincomplexfunctorN. Gillet-Soulé[GS87, §4] prove that, for all closed subsets Z of X and all integers k ≥0, there is a function λk :KZ(X)→KZ(X) GS 0 0 such that, if E is concentrated in non-negative degrees, then λk ([E])=[N(Λk K(E))]. GS OX Moreover, they prove that the operations λk , k ≥ 1, make KZ(X) into a GS Z 0 (special) lambda ring. The operator ψk is then defined, as is customary, to be GS L Q (λ1,...,λk) where Q is the k-th Newton polynomial. k k In this paper, we build operations that satisfy the four axioms (A1)–(A4) using asimplerconstruction,albeitonethatexistsonlyinasomewhatrestrictivesetting. Indetail,wefixaprimepandassumeX =Spec(Q),foracommutativeNoetherian CYCLIC ADAMS OPERATIONS 3 ringQsuchthatQcontains 1 andallthe p-throotsofunity. Foranyclosedsubset p Z of Spec(Q), we construct the p-th cyclic Adams operator, which is a function ψp :KZ(Q)→KZ(Q), cyc 0 0 characterized by the following property: if F is a bounded complex of finitely generated projective Q-modules, then ψp ([F])=[Tp(F)(1)]−[Tp(F)(ζ)]. cyc Here, Tp(F) denotes the p-th tensor power of the complex F, equipped with the canonical,signedactionofthesymmetricgroupΣ ,andthesuperscript(w) denotes p the eigenspace of eigenvalue w for the action of the p-cycle (12 ··· p) ∈ Σ . In p particular,the definitionofψp bypassesentirelythe constructionofλ-operations. cyc The idea for the definition of ψp goes back to Atiyah [Ati66] (see also Benson cyc [Ben84] and End [End70]). One of our main results is: Theorem 1. (See Theorem 3.7 for the precise statement.) For any prime p, the p-th cyclic Adams operation ψp satisfies the four Gillet-Soulé axioms (A1)–(A4) cyc on the category of affine schemes Spec(Q) with the property that Q contains 1 and p all p-th roots of unity. The hypotheses involving the prime p are not significant restrictions for many purposes. Note that if Q is local, then p is invertible in Q for all primes other thanthe residue characteristic. Moreover,the requirementthat Q containthe p-th roots of unity is a mild one, since adjoining such roots gives an étale extension of Q. Inparticular,Serre’sVanishing Conjecture is a direct consequenceof the above Theorem, via the same argument used by Gillet and Soulé; see Corollary 3.13. Sincethefirstversionofthispaperwasmadepubliclyavailable,wehavelearned that the operator ψp has also been defined previously by Haution in his thesis cyc [Hau09]. Haution works more generally over schemes, but just for schemes defined overa groundfield k notof characteristicp. (We do believe, however,that most of his proofs go through under the more general context of schemes over Z[1,e2pπi].) p In his thesis, Haution establishes the existence, naturality, additivity and multi- plicativity of these operators — i.e., he establishes axioms (A1)–(A3) in the above list. He does not establish (A4), and his approach is different in that he bypasses introducing the power operations that we develop in Section 2. These power operations seem to be necessary for the proof of (A4), and also for our proof of the commutativity of the cyclic Adams operations in Section 4. For these reasons, we havekeptthispapermostlyunchangedfromtheoriginalversion,butwehaveadded careful indications of which results presented here can also be found in [Hau09]. In addition to the results described above, we also address the issue of whether theoperatorψp agreeswiththep-thAdamsoperationofGillet-Soulé. Webelieve cyc that they coincide whenever both are defined, but are only able to prove it in the case that p! is invertible in Q; see Corollary 6.14. In developing the proof of Corollary6.14, we also show that if k! is invertible in Q, then Gillet and Soulé’s operation λk may be defined by taking “naive” exterior powers of complexes; see Theorem 5.10. This fact is a “folklore” result (see the discussion at the beginning of §5), but we provide a careful proof here. In addition to their simplicity, another advantage the operators ψp have over cyc the operators defined by Gillet-Soulé is that their definition ports well to other 4 MICHAELK.BROWN,CLAUDIAMILLER,PEDERTHOMPSON,ANDMARKE.WALKER contexts where the Dold-Puppe functors are unavailable. In a forthcoming paper we establish the existence of analogously defined operators ψp on the K-theory cyc of matrix factorizations, and we prove that the analogues of the four Gillet-Soulé axioms hold. Using these properties, we prove a conjecture of Dao and Kurano [DK12, 3.1 (2)] concerning the vanishing of the θ-invariant. We thank Luchezar Avramov for helpful conversations in preparing this docu- ment,DaveBensonforleadingustohisrelevantpaper[Ben84]aswellasanswering some ofour questions,and PaulRoberts for sharing his unpublished notes [Rob96] describing Hashimoto’s result mentioned in the beginning of §5. We also thank Olivier Haution for drawing our attention to his thesis. 2. Tensor power operations Let Q be a Noetherian, commutative ring, Z ⊆Spec(Q) a closed subset, and G a finite group. Let PZ(Q;G) denote the category of bounded complexes of finitely generatedprojectiveQ-moduleswithhomologysupportedonZ andequippedwith aleftG-action(withGactingviachainmaps). MorphismsareG-equivariantchain maps. Equivalently, PZ(Q;G) consists of bounded complexes of left Q[G]-modules which,uponrestrictingscalarsalongQ⊆Q[G], arecomplexesoffinitely generated projective Q-modules supported on Z. Let KZ(Q;G) denote the Grothendieck group of PZ(Q;G), defined to be the 0 group generated by isomorphism classes of objects modulo the relations [X]=[X′]+[X′′] if there exists an (equivariant) short exact sequence 0→X′ →X →X′′ →0 and [X]=[Y] if there exists an (equivariant) quasi-isomorphism joining X and Y. Observe that the group operation is realized by direct sum of complexes: [X]+[Y]=[X ⊕Y]. We write PZ(Q) and KZ(Q) when G is the trivial group. 0 Remark 2.1. KZ(Q;G) can equivalently be described as the abelian monoid of 0 isomorphism classes of objects of PZ(Q;G), under the operation of direct sum, modulo the two relations above. For observe that for any X ∈PZ(Q;G), we have = the short exact sequence 0 → X → cone(X −→ X) → Σ(X) → 0, where Σ(X) denotes the suspension of X. It follows that [X]+[Σ(X)]=0, and hence that this monoid is an abelian group. In particular,KZ(Q;G) has the following universalmapping property: givenan 0 abelian monoid M and an assignment of an element (X) ∈ M to each object X of PZ(Q;G) such that (0) = 0, (X) = (X′)+(X′′) if there exists a short exact sequence0→X′ →X →X′′ →0,and(X)=(Y)ifX andY arequasi-isomorphic, then there exists a unique group homomorphisms KZ(Q;G)→U(M) sending [X] 0 to (X), where U(M) denotes the group of units of M. Tensor product over Q, with the group action given by the diagonal action, induces a pairing on KZ(Q;G) making it into a non-unital ring. If Z = Spec(Q), 0 then KSpecQ(Q;G) is a unital ring, with 1=[Q], and there is a ring isomorphism 0 R (G)−∼=→KSpecQ(Q;G), Q 0 where R (G) denotes the representation ring of G with Q coefficients: By defini- Q tion, R (G) is the abelian group generated by isomorphism classes of projective Q CYCLIC ADAMS OPERATIONS 5 Q-modules equipped with a G-action, modulo relations coming from short exact sequences. The isomorphism sends the class of a representation ρ : G →Aut (P) Q totheclassoftheevidentcomplexconcentratedindegree0. Theinversemapsends the class of a complex to the alternating sum of the classes of its components. As a special case of this, we have K (Q)∼=KSpecQ(Q). 0 0 For any n ≥ 1, let Σ denote the group of permutations of the set {1,...,n}. n For n ≥ 1 and X ∈ PZ(Q;G), let Tn(X) ∈ PZ(Q;G × Σ ) be the complex n n X⊗ ···⊗ X equipped with the diagonal G-action, Q Q z }| { g(x1⊗···⊗xn)=g(x1)⊗···⊗g(xn), and equipped with a Σ -action given by n σ(x ⊗···⊗x )=±x ⊗···⊗x , 1 n σ(1) σ(n) where the sign is uniquely determined by the following rule: if σ is the adjacent transposition (ii+1) for some 1≤i≤n−1, then σ(x ⊗···⊗x )=(−1)|xi||xi+1|x ⊗···⊗x ⊗x ⊗x ⊗x ⊗···⊗x . 1 n 1 i−1 i+1 i i+2 n For 0 ≤i ≤n, let Σ denote the subgroup of Σ consisting of permutations i,n−i n that stabilize the subsets {1,2,...,i} and {i+1,i+2,...,n}. We identify Σ i,n−i withΣ ×Σ intheobviousway. IfX ∈PZ(Q;Σ ×G)thenQ[Σ ]⊗ X i n−i i,n−i n Q[Σi,n−i] is in PZ(Q;Σ ×G). n Theorem 2.2. For any Q, Z, G, and n≥1 as above, there is a function tn :KZ(Q;G)→KZ(Q;G×Σ ) Σ 0 0 n such that tn([X])=[Tn(X)] Σ for any object X of PZ(Q;G). Moreover, if 0 → X′ → X → X′′ → 0 is a short exact sequence of objects of PZ(Q;G), then n tn([X])= Q[Σ ]⊗ Ti(X′)⊗ Tn−i(X′′) . Σ n Q[Σi,n−i] Q i=0 X(cid:2) (cid:3) Remark 2.3. See[Hau09,II.3.4andII.3.8]forasimilarresultinvolvingdirectsums of complexes and for a method of reducing the case of a short exact sequence to a direct sum. The proof of the Theorem occupies the remainder of this section. Lemma 2.4. The bi-functor PZ(Q;Σ ×G)×PZ(Q;Σ ×G)→PZ(Q;Σ ×G) i j i+j sending (X,Y) to Q[Σ ]⊗ X ⊗ Y, equipped with the diagonal G-action, i+j Q[Σi,j] Q induces a bilinear pairing ⋆=⋆ :KZ(Q;Σ ×G)×KZ(Q;Σ ×G)→KZ(Q;Σ ×G). i,j 0 i 0 j 0 i+j The pairing is associative and commutative, in the sense that (a⋆ b)⋆ c=a⋆ (b⋆ c) i,j i+j,k i,j+k j,k and a⋆ b=b⋆ a. i,j j,i 6 MICHAELK.BROWN,CLAUDIAMILLER,PEDERTHOMPSON,ANDMARKE.WALKER Remark 2.5. This“starpairing”is relatedtopairingsconsideredbyAtiyah[Ati66, §1] and Knutson [Knu73, p. 127]. See the discussion in §6. Proof of Lemma 2.4. Note that Q[Σ ] is a flat Q[Σ ]-module, and hence this i+j i,j functor preservesshort exact sequences and quasi-isomorphismsin eachargument. It thus induces a bilinear pairing on Grothendieck groups as indicated. Associativity holds since there is an isomorphism in PZ(Q;Σ ×G) from i+j+k Q[Σ ]⊗ (Q[Σ ]⊗ X ⊗ Y)⊗ Z i+j+k Q[Σi+j,k] i+j Q[Σi,j] Q Q to Q[Σ ]⊗ X⊗ (Q[Σ ]⊗ Y ⊗ Z) i+j+k Q[Σi,j+k] Q j+k Q[Σj,k] Q given by σ⊗ω⊗x⊗y⊗z 7→σω⊗x⊗1⊗y⊗z. As for commutativity,let τ :=(12 ··· i+j)j ∈Σ , andlet h denote the auto- i+j morphism of Σ given by σ 7→τστ−1. Notice that h restricts to an isomorphism i+j Σ −∼=→Σ , i,j j,i and, moreover,this isomorphism coincides with the map given by the composition of evident isomorphisms Σ −∼=→Σ ×Σ −∼=→Σ ×Σ −∼=→Σ . i,j i j j i j,i It follows that one has an isomorphism in PZ(Q;Σ ×G) i+j Q[Σ ]⊗ X ⊗ Y −∼=→Q[Σ ]⊗ Y ⊗ X i+j Q[Σi,j] Q i+j Q[Σj,i] Q that sends elements of the form σ⊗x⊗y, where σ ∈Σ , to στ−1⊗y⊗x. (cid:3) i+j Lemma 2.6. Given a short exact sequence 0→X′ →X →X′′ →0 in PZ(Q;G), for any n≥1 there is a filtration 0=F ⊆F ⊆F ⊆···⊆F =Tn(X) −1 0 1 n in PZ(Q;Σ ×G) such that n F /F ∼=Q[Σ ]⊗ Tn−i(X′)⊗ Ti(X′′). i i−1 n Q[Σn−i,i] Q Consequently, in KZ(Q;Σ ×G) we have 0 p [Tn(X)]= [Tn−i(X′)]⋆ [Ti(X′′)]. n−i,i i X Proof. We identify X′ as a subcomplex of X. Define F as the image of i α :Q[Σ ]⊗ Tn−i(X′)⊗ Ti(X)→Tn(X) i n Q Q sending σ⊗x ⊗···⊗x to σ(x ⊗···⊗x ), where x ,...,x ∈ X′. In other 1 n 1 n 1 n−i words, F is the closure under the action of Σ of the image of the canonical map i n Tn−i(X′)⊗ Ti(X)→Tn(X). Q The map α factors as i Q[Σ ]⊗ Tn−i(X′)⊗ Ti(X)։Q[Σ ]⊗ Tn−i(X′)⊗ Ti(X)−α→i Tn(X). n Q Q n Q[Σn−i,i] Q Also, the restriction of α to the subcomplex Q[Σ ]⊗ Tn−i+1(X′)⊗ Ti−1(X) i n Q Q coincides with α . We have a right exact sequence i−1 Q[Σ ]⊗ Tn−i+1(X′)⊗ Ti−1(X)→Q[Σ ]⊗ Tn−i(X′)⊗ Ti(X) n Q Q n Q[Σn−i,i] Q →Q[Σ ]⊗ Tn−i(X′)⊗ Ti(X′′)→0. n Q[Σn−i,i] Q CYCLIC ADAMS OPERATIONS 7 These facts imply the existence of a surjective map (2.7) Q[Σ ]⊗ Tn−i(X′)⊗ Ti(X′′)։F /F n Q[Σn−i,i] Q i i−1 and it remains to prove it is injective too. We mayassumeSpec(Q)isconnected,sothateachcomplexX′,X,X′′ haswell- defined total rank r′,r,r′′, respectively, where we define total rank to be to be the sumoftheranksofallthecomponentsofacomplex. Moreover,wehaver =r′+r′′. Then the total rank of Q[Σ ]⊗ Tn−i(X′)⊗ Ti(X′′) n Q[Σn−i,i] Q is ((r′)n−i(r′′)i) n . Observe that i (cid:0) (cid:1) n ((r′)n−i(r′′)i) =(r′+r′′)n =rn. i i (cid:18) (cid:19) X But the sum of the ranks of the complexes F /F is also rn, since they are the i i−1 associatedgradedmodules associatedto a filtration of Tn(X). It follows that each map (2.7) must be injective too. (cid:3) We define a multiplicative abelian monoid M as follows. As a set, M is ∞ {1}× KZ(Q;Σ ×G)zi, 0 i i=1 Y the collection of power series in z of the form 1+ α z + α z2 + ··· with α ∈ 1 2 i KZ(Q;Σ ×G) for alli. We define amultiplication ruleonM using the ⋆ pairings: 0 i α zi ⋆ β zj := (α ⋆ β )zl, i j i i,j j !   i j l i+j=l X X X X   wherebyconventionα =1,β =1,α ⋆β =β ,andα ⋆β =α . Theassociative 0 0 0 j j i 0 i andcommutativepropertiesof⋆giveninLemma2.4implythat(M,⋆)isanabelian monoid. For X ∈PZ(Q;G), define t(X)∈M by t(X)=1+[X]z+[T2(X)]z2+··· By Lemma 2.6, t(X) = t(X′)⋆t(X′′) if 0 → X′ → X → X′′ → 0 is a short exact sequence in PZ(Q;G). If X −∼→ X′ is a quasi-isomophism in PZ(Q;G), then the induced map Ti(X) → Ti(X′) is also a quasi-isomorphism for all i, and hence t(X)=t(X′). By Remark 2.1, we get an induced group homomorphism t:KZ(Q;G)→U(M) 0 landing in the group of units of M. The functiontn is definedtobe the compositionoftwiththe functionU(M)→ Σ KZ(Q;Σ ×G) sending a power series to its zn coefficient. The first assertion of 0 n Theorem 2.2 follows, and the second is a consequence of Lemma 2.6. 8 MICHAELK.BROWN,CLAUDIAMILLER,PEDERTHOMPSON,ANDMARKE.WALKER 3. Cyclic Adams operations We define a “cyclic” Adams operation, ψp , on KZ(Q) for each prime p. The cyc 0 definition is motivated by an observation of Atiyah [Ati66, 2.7]; see also Benson [Ben84] and End [End70]. In the case p = 2, the operator ψ2 was defined and cyc developed in unpublished work of P. Roberts [Rob96], who in turn credited the idea to unpublished work of M. Hashimoto and M. Nori. Finally, as mentioned in the introduction,these operatorshavealsobeen definedanddevelopedby Haution [Hau09] when Q contains a field of characteristic different than p. Throughout this section, assume p is a prime and Q is an A -algebra,where A p p is the subring of C defined by Ap =Z p1,e2pπi , h i Define C to be the subgroup of Σ generated by the p-cycle σ := (12 ··· p). p p For a p-th root of unity ζ (including the case ζ = 1), let Q denote the Q[C ]- ζ p moduleQequippedwiththe C -actionσq =ζq. Since QisanA -algebra,wehave p p Q[C ]∼= Q as Q[C ]-modules. p ζ ζ p For Y ∈PZ(Q;C ), define L p Y(ζ) :=Hom (Q ,Y)=ker(σ−ζ :Y →Y). Q[Cp] ζ Since Q is a projective Q[C ]-module, Y 7→Y(ζ) is an exact functor, and hence it ζ p induces a map φp :KZ(Q;C )→KZ(Q). ζ 0 p 0 Proposition 3.1. Assume Q is an A -algebra. For each p-th root of unity ζ, there p is a function tp :KZ(Q)→KZ(Q) ζ 0 0 with tp([X])=[Tp(X)(ζ)]. ζ Proof. Restriction along the inclusion C ֒→Σ determines a map p p KZ(Q;Σ )−r−e→s KZ(Q;C ). 0 p 0 p We define tp to be the composition of ζ KZ(Q)−t→pΣ KZ(Q;Σ )−r−e→s KZ(Q;C )−φ→pζ KZ(Q). 0 0 p 0 p 0 (cid:3) Definition 3.2. For an A -algebra Q and closed subset Z of Spec(Q), define the p function ψcpyc :K0Z(Q)→K0Z(Q)⊗ZZ[e2πpi] by ψp := ζtp cyc ζ ζ X where the sum is taken over all p-th roots of unity ζ. Thus for X ∈PZ(Q), ψp ([X])= ζ[Tp(X)(ζ)]. cyc ζ X CYCLIC ADAMS OPERATIONS 9 In view of the following lemma, the map ψp is independent of the generator cyc chosen for C . The lemma is proven in [Hau09, II.3.6], but we include the details p here. Lemma 3.3. Assume Q is an A -algebra. If ζ and ζ′ are both primitive p-th roots p of unity, then [Tp(X)(ζ)]=[Tp(X)(ζ′)]∈KZ(Q) 0 for all X ∈PZ(Q). Proof. We show res(Y)(ζ) and res(Y)(ζ′) are isomorphic objects of PZ(Q) for any Y ∈PZ(Q;Σ ), where res:PZ(Q;Σ )→PZ(Q;C ) is the restriction functor. p p p Note that ζ′ = ζi for some 1 ≤ i ≤ p−1. Let τ ∈ Σ be a permutation such p that τ−1στ = σi. (Recall σ = (12 ··· p).) Then τ determines an isomorphism fromres(Y) to res′(Y), where res′ is restrictionalong C −σ−7→−−σ→i C ⊆Σ . We have p p p res′(Y)(ζ) =res(Y)(ζi). (cid:3) Remark 3.4. More generally,for any integer n≥1, [Tn(X)(ζ)]=[Tn(X)(ζ′)] holds in KZ(Q), as long as ζ and ζ′ are n-th roots of unity of the same order. 0 Since ζ =−1, we deduce from the Lemma: ζ6=1 X Corollary 3.5. Assume Q is an A -algebra and let ζ be a primitive p-th root of p unity. We have ψp ([X])=[Tp(X)(1)]−[Tp(X)(ζ)]. cyc The corollaryshows, in particular, that ψp takes values in KZ(Q) viewed as a cyc 0 subgroup of K0Z(Q)⊗ZZ[e2pπi], and we will henceforth view ψcpyc as a function of the form (3.6) ψp :KZ(Q)→KZ(Q). cyc 0 0 Theorem 3.7. Fix a prime p. The operation ψp satisfies the Gillet-Soulé axioms cyc ofbeingan“Adamsoperation ofdegreep”onthecategoryofcommutative,Noether- ian A -algebras. That is, letting Q and R be commutative, Noetherian A -algebras, p p we have: (1) ψp isagroupendomorphismofKZ(Q)forallclosedsubsetsZ ofSpec(Q). cyc 0 (2) Given α∈KZ(Q)andβ ∈KW(Q)for closedsubsetsZ andW ofSpec(Q), 0 0 we have ψp (α∪β)=ψp (α)∪ψp (β)∈KZ∩W(Q), cyc cyc cyc 0 where −∪− is the pairing determined by tensor product over Q. (3) Given a morphism of affine schemes φ:Spec(R)→Spec(Q) over Spec(A ) and given closed subsets W ⊆ Spec(R) and Z ⊆ Spec(Q) p such that φ−1(Z)⊆W, we have an equality ψp ◦φ∗ =φ∗◦ψp cyc cyc of maps KZ(Q)→KW(R). 0 0 10MICHAELK.BROWN,CLAUDIAMILLER,PEDERTHOMPSON,ANDMARKE.WALKER (4) If a=(a ,...,a ) is any sequence of elements in Q and K(a) is the asso- 1 n ciated Koszul complex, viewed as an object of PV(a1,...,an)(Q), we have ψp ([K(a)])=pn[K(a)]∈KV(a1,...,an)(Q). cyc 0 Remark 3.8. TheGillet-Souléaxiomsinclude non-affineschemestoo,but wewon’t require that level of generality. Also, their fourth axiom assumes a is a regular sequence, but the property holds more generally for any such sequence, both for our operators and theirs. Remark 3.9. Proofsof(1)–(3)ofthetheoremcanbefoundinHaution’sworkunder the additionalassumptionthatQ containsa field of characteristicdifferent than p. Specifically [Hau09, II.3.8] proves ψp is additive and [Hau09, II.3.10] proves (2) cyc and (3). We believe his proofs apply verbatim to the slightly more general setting of this paper. Nevertheless, for the sake of making this paper self-contained, we will include proofs of (1)–(3). Proof. By construction, ψp factors as cyc KZ(Q)−t→pΣ KZ(Q;Σ )−r−e→s KZ(Q;C )−φ→p KZ(Q) 0 0 p 0 p 0 whereφp = ζφp. ForY ∈PZ(Q;C ),letussayY isextendedifY ∼=Y′⊗ Q[C ] ζ ζ p Q p for some Y′ ∈PZ(Q). P The following result may also be found in [Hau09, II.3.7]. Lemma 3.10. If Y ∈PZ(Q;C ) is extended, then φp([Y])=0. p Proof. If Y is extended, Y(ζ) ∼=Hom (Q ,Y′⊗ Q[C ])∼=Hom (Q ,Q[C ])⊗ Y′ ∼=Y′ Q[Cp] ζ Q p Q[Cp] ζ p Q as objects of PZ(Q), since Q[Cp] = ζQζ and HomQ[Cp](Qζ,Qζ′) is 0 for ζ 6= ζ′ and Q for ζ =ζ′. Thus L φp([Y])= ζ [Y′]=0. (cid:3)   ζ X   We claim that for each 1≤i≤p−1 and X,Y ∈PZ(Q), Q[Σ ]⊗ Ti(X)⊗ Tp−i(Y) p Q[Σi,p−i] Q is an extended complex of Q[C ]-modules. Granting this claim, by Theorem 2.2 p tp([X]+[Y])= Q[Σ ]⊗ Ti(X)⊗ Tp−i(Y) , Σ p Q[Σi,p−i] Q i X(cid:2) (cid:3) and thus Lemma 3.10 shows that ψp ([X]+[Y])=ψp ([X])+ψp ([Y]), cyc cyc cyc and part (1) of the Theorem follows. To prove the claim, we show more generally that for any 1≤i≤p−1 and any left Q[Σ ]-module M, the Q[C ]-module Q[Σ ]⊗ M is extended. Let i,p−i p p Q[Σi,p−i] C τ Σ ,...,C τ Σ be a set of double coset representatives in Σ . Since p p 1 i,p−i p m i,p−i p is prime, τΣ τ−1 intersects C trivially for all τ ∈Σ . It follows that, for each i,p−i p p j, Q[C τ Σ ]∼=Q[C τ ]⊗ Q[Σ ] p j i,p−i p j Q i,p−i