Two-dimensional Id`eles with Cycle Module Coefficients

TW O-DIMENSIONAL ID ` ELES WITH CYCLE MODULE COEFFICIENTS OLIVER BRAUNLING Abstract. W e give a theo ry of id ` eles with coefficients for smo oth surfaces ov er a field. It is an analog ue of Beilinson/Huber’s theory of higher ad ` eles, but handling cycle mo dule shea ve s instead of quasi-coheren t ones. W e prov e that they give a flasque resolution of the cycle module shea ves in the Zaris ki topology . As a tec hnical ingredien t w e show the Gersten property for cycle mo dules on equicharact eristic complete regular lo cal ri ngs, which might be of independent inte rest. 1. Motiv a tion Let K b e a num ber field, A K its ad` eles and I K its id` eles. Then the k ey inv ariants, the group of units a nd the ideal class group a r e given by (1.1) O × K = I 0 K ∩ K × and Cl K = I K / ( I 0 K · K × ), where I 0 K denotes the in tegral id` eles. Many other inv aria n ts can also b e expres sed through ad` eles and id` eles (e.g. L -function, discrimina nt, etc.), but let us stick to the a bove t wo for the moment. Using the num b er field/function field dictionar y , these facts readily c a rry ov er to smo oth curves ov er finite fields, and in fact to smo oth curves X/k ov er arbitra ry fields. Viewing X a s a scheme, we ma y define sheaf versions, na mely (1.2) I 0 ( U ) := Q x ∈ U (1) b O × x and I ( U ) := Q ′ x ∈ U (1) b K × x , where U is a Z a riski o pen, U (1) the set of c losed p o int s in U , b O x the completed lo cal r ing a t x , b K x its field o f fr a ctions a nd the prime sup erscr ipt in Q ′ means that we r estrict to elements such that a ll but finitely many co mponents lie in the subgroup b O × x ⊆ b K × x . The b K x are lo cal fields (in this text a lo c al field refers to a complete discrete v a luation field); here a ll of the s hap e of Laurent se ries fields (1.3) κ (( t )), while Q p , R and their finite extensions are the lo c al fields app earing in the class ical theo ry for num ber fields. One o btains an exa ct sequence of sheav es of ab elian groups, (1.4) 0 − → O × X diag − → I 0 ⊕ K × diff − → I − → 0, where K denotes the sheaf of ra tional functions. In fac t this is a flasque re s olution o f O × X and th us H 0 ( X, O × X ) = I 0 ( X ) ∩ K × and H 1 ( X, O × X ) = I ( X ) / ( I 0 ( X ) · K × ). In view of H 1 ( X, O × X ) = Pic X ∼ = Cl X we have recov ered a p erfect analog ue of eq. 1.1. Although it might seem unnecessa rily fancy , let us write CH 1 ( X ) instead of Cl( X ) fro m now on. Without needing to change definitions, the a bove also works for smo oth s urfaces (indeed in 1991 Mathematics Subj ect Classific ation. Prim ary 11R56 ; Secondary 11G45. Key wor ds and phr ases. ideles, cycle module. This work has been partiall y supported by the DFG SFB/TR45 “P erio ds, mo duli spaces, and ari thmetic of algebraic v arieties”and the Alexander von Humboldt F oundation. 1 2 O. BRA UNLING any dimensio n). That should make us cur ious, since for smo oth surfaces we have the intersection pairing (1.5) CH 1 ( X ) ⊗ Z CH 1 ( X ) / /   CH 2 ( X ) b ?   “Chow s ide” I ( X ) I 0 ( X ) · K × ⊗ Z I ( X ) I 0 ( X ) · K × a ? / / ? “idelic s ide ” It is now very natura l to ask: (1) Is there a lso an id ` ele type ob ject which can b e placed in the lower-right corner? (2) What cohomo logy groups b esides H i ( X, O × X ), CH i ( X ) admit such an “id` ele type” counterpart? These questions rest on the idea that there should exist genera l id ` eles with c o efficients which provide flasque r esolutions for sheaves, th us expr essing c ohomolo gy gr oups thr ough su b quotients of s u itable id` eles . W e prove: Theorem 1. (Main The or em) L et X /k b e a smo oth inte gr al surfac e over an arbitr ary field k , M ∗ a big cycle mo dule (se e § 2 for the definition) and M ∗ the asso ciate d Zariski she af. Then id ` ele she aves with c o efficients in M ∗ (define d in § 4) give a flasque re solution 0 → M ∗ → I 0 M − → I 1 M − → I 2 M − → 0 . As a to ol for the pr o of of the main theor e m, we need the following version of the Gersten conjecture fo r co mplete lo ca l rings : Theorem 2. S upp ose X/k is an e quichar acteristic c omplete r e gular lo c al scheme and let M ∗ b e a big cycle mo dule (se e § 2 for the defin ition). Then t he R ost cycle c omplex C • ( X, M ∗ ) is exact. See § 5 for this res ult, whic h can be read indep endently o f the res t of the text. While one would certainly e x pect its truth, we hav e b een unable to find a liter a ture reference. An applic ation: Once established, this mechanism bridges freely b etw e en the coho mological and the id` ele per sp ective. F or ex ample, supp o se X / F q is a smo oth prop er geometrically integral sur face, now ov er a finite field. Classical work o f Ka to and Sa ito [K S83] shows that there is an unramified class field theory , a morphism (1.6) rec X : CH 0 ( X ) ֒ → π ´ et 1 ( X, x ) ab , where ‘a b’ denotes a belia nization. It is injective with dense image. It s e nds a clo sed p oint to its F robenius, just like in classical class field theory for cur ves. Since CH 0 ( X ) appe a rs as a cohomolog y g roup o f the K -theor y sheaf K 2 , the a bove mech anism automatically pro duces a n id` ele counterpart, which can b e phr ased as (1.7) rec X, id` elic : Q ′′′ x,y ′ i K 2 ( b K x,y ′ i ) Q x K 2 ( b K x ) + Q y K 2 ( b K y ) + Q x,y ′ i K 2 ( b O x,y ′ i ) → π ´ et 1 ( X, x ) ab . Here K n denotes either ordinary K -theor y gro ups o r F esenko’s to po logical Milno r K -gro ups; this do es not affect the quotient. The key p oint is that the lo cal fields Q p , κ (( t )) constituting the classical id` ele s are b eing replaced by 2-lo cal fields b K x,y ′ i ≈ κ (( s ))(( t )) dep ending on flags o f po in ts x lying o n curves y . TW O-DIMENSIONAL ID ` ELES 3 W e do not intend to repr ov e class field theoretic r esults. The p oint o f the pap er is that the equiv ale nc e of the tw o seemingly very different formulations (eq. 1.6 versus eq. 1.7) follows immediately from an id` ele res olution of the K 2 -sheaf. In the same wa y one can complete Fig. 1.5. See [B e i80], [Hub9 1 a], [Hub91b], [O si97], [Gor07], [Gor08] for other ad` ele/id` ele theories resolving particular types of sheaves under v arying assumptions. History: Chev alle y initiated the theory o f id` eles in 19 36 for cla ss field theory . The theory then w as develop ed further by W eil, T ate and Iwasaw a; finiteness theo rems a nd L -functions ent ered the picture. Parshin was the first to formulate a theor y of ad` eles for surfaces [Par76], but alr eady had a far broader range of applications in mind than just cla ss field theory (mostly unpublished [Par83], [P F99]). Beilinson then g av e a genera l theor y o f ad` eles in all dimension with quasi-coher ent coefficients [Bei80], with detailed pr o ofs written down by Hub er [Hub91a], [Hub91b]. Thes e theor ies ar e all ‘finite type over a field’. F esenko developed a theor y ‘over Z ’, also inc o rp orating infinite places, versions o f harmonic a nalysis and L -functions [F es03], [F es08], [F es10]. Osip ov gav e a theor y for K -theor y co efficients [Osi97]. Gorchinskiy develop ed a theor y of ‘not adically co mpleted id` e les’ with very genera l co e fficien ts a nd in all dimensions [Gor07], [Gor08]. A similar ‘adically uncompleted’ theory for quasi- coherent co efficie nts is due to Hub er, aga in [Hub91b]. 2. Cycle modules 2.1. Axi o ms fo r big range of definition. W e will develop a flasque resolutio n in ter ms o f “id` ele sheav es” for sheav es which come from Rost cyc le mo dules (introduced in [Ro s96]), gen- eralizing the mechanism explained in § 1 for cur ves. Actually , we ne e d to work with a broa der range o f definition than in the orig inal pap er, so it is worth stating the axioms w e use in their precise form: Fix a base field k . F or every field extension F /k let K M ∗ ( F ) denote the Z ≥ 0 -graded anticom- m utative Milnor K - theory ring ov er F , K M ∗ ( F ) = ` n ≥ 0 K M n ( F ) = Z ⊕ F × ⊕ K M 2 ( F ) ⊕ . . . . Then a big cycle mo dule is an o b ject function (2.1) M ∗ : { field extensio ns F /k } − → { Z -graded ab elian gro ups } along with so me extra structure D1 - D4 satisfying v arious axioms. The extra structure (called tr ansfers ) is given as fo llows: W e shall tacitly assume that all our fields are extensions of k . D1: F or every field ex tens ion F ֒ → E there is a n ab elian group morphism o f rela tive degree 0, res E F : M ∗ ( F ) → M ∗ ( E ), ca lled r estriction . D2: F or ev ery finite field extension F ֒ → E there is an ab elian group mor phism of relative degree 0, cor E F : M ∗ ( E ) → M ∗ ( F ), ca lled c or estriction (or norm ). D3: F or every field F the group M ∗ ( F ) has the structure of a gra ded left- K M ∗ ( F )-mo dule, K M n ( F ) · M m ( F ) − → M n + m ( F ) . D4: F or every field F and every discrete excellent k -trivial v aluation v on F (see Rmk. 1 for ter minology), there is an ab elian group ho momorphism o f rela tive degr ee − 1, ∂ F v : M ∗ ( F ) − → M ∗− 1 ( κ ( v )) , where κ ( v ) denotes the re s idue fie ld of the v aluation r ing A v ⊆ F cor resp onding to v . Then ∂ F v is called the b oundary map at v . Remark 1. (V aluations) Discr ete mea ns that the v alue g roup is isomorphic to Z . This implies that the r ing of in tegers (= v aluation r ing ) is No etherian. Exc el lent means tha t we dema nd the ring of integers to b e excellent. Finally , k -trivial means that the restr iction of the v aluation to 4 O. BRA UNLING k is trivial. Without this condition the residue field κ ( v ) would not need to b e a n extension o f k . Also, this excludes for example p -adic v aluations . Remark 2. (Changes in the definition) T his differs from the o riginal definition in [Ros96] a s follows: Rost dema nds M ∗ in eq . 2 .1 only to b e defined and hav e transfers on finitely gener ate d field extensions of k . How ever, this ex cludes many interesting maps, e.g. res k (( t )) k ( t ) : M ∗ ( k ( t )) → M ∗ ( k (( t ))) res Q p Q : M ∗ ( Q ) → M ∗ ( Q p ) coming from completions. Such maps a bo und in the cons tr uction of id` eles. They play a key r ole in a rithmetic g eometry , for example in the definition of Selmer and T ate-Shafarev ic h groups or the Brauer -Manin obstr uction. Next, in D4 Rost restr icts to v aluatio ns such that the transcendence degree ov er k from F to κ ( v ) dro ps b y one (“geometric v aluations ” ). Again, this is to o r e strictive for o ur purp os es, e.g. the t -v aluation o f k (( t )) has residue field k and th us is not g eometric. The rela xed co nditio n a dds many v aluations, even to finitely generated fields. F or example, suppo se k = Q and take F := Q ( s, t ). Restrict the s -v aluation coming from F ֒ → Q (( s )), t 7→ se s , to F . It has r ing of integers and residue field A v := Q ( s, se s ) ∩ Q [[ s ]] κ ( v ) = Q , so a drop of transcendence deg ree by t wo. The ring A v is exce lle n t [Rot7 7, Thm. 1, F olgerung 3 (i)]. This v a luation would not b e allow ed in Rost’s original theory . 1 The transfer morphisms from D1 - D4 need to s atisfy a lo ng list of a x ioms, R1a - R 3e (“cycle premo dule a x ioms”) and the tw o crucial a x ioms (“cycle mo dule axioms” FD , C ); they can b e taken ov er literally from [Ros96] (except that field extensio ns need not be finitely g enerated). Example 3. Milnor K - theory , Quillen K -theo ry and Galois co homology groups with ( 6 = c har k ) torsion co efficients (in the sense of [Ros96, Rmk. 1.11]) are big cycle modules. See [Ros 96, Thm. 1.4 and following par agra ph]. In principle, one can fo r m ulate categor ies o f cy cle mo dules C M big ⇄ C M with C M the theory as in Rost’s work, and C M big with the re la xed assumptions here. There is an ob vious restriction functor to the right and one can use the left-adjoint to “prolo ng” class ical cycle mo dules to the present setup. I tha nk F. D´ eglise and S. Kelly for expla ining this to me. W e sha ll now r epea t so me particular ly imp or ta n t constr uctions from [Ros9 6], even though no changes ar e neccess a ry , but we will often refer to them: Let X b e an excellent scheme. W e shall write X ( i ) for the set o f co dimension i points. Supp ose X → Sp ec k , x ∈ X ( i ) for so me i , y ∈ X ( i +1) , y ∈ { x } , i.e. y can b e viewed as a co dimension one p oint in the integral closed subscheme { x } ֒ → X . As in [Ros9 6] define a b oundary map ∂ x y : M ∗ ( κ ( x )) − → M ∗− 1 ( κ ( y )) as follows: The stalk O { x } ,y is a one- dimensional lo cal doma in. By exc e llence, the normaliza tio n (2.2) ϕ : Spec O ′ { x } ,y − → Sp ec O { x } ,y inside its own field of fractions κ ( x ) is a finite mor phism, s o the closed p oint y in Sp ec O { x } ,y has finite preimage { y ′ 1 , . . . , y ′ r } ⊂ Sp ec O ′ { x } ,y . By no rmality the lo calizations ( O ′ { x } ,y ) y ′ i are D VRs. Let v 1 , . . . , v r denote the cor resp onding discrete v aluations of the fraction field κ ( x ). Now one defines (2.3) ∂ x y := P r i =1 cor κ ( v i ) κ ( y ) ◦ ∂ κ ( x ) v i : M ∗ ( κ ( x )) − → M ∗− 1 ( κ ( y )) , 1 Rost already p oints out in his paper that the present r elaxed axioms ar e feasible, see for example [Ros96, Rmk. 1.8, Rmk. 2.8]. TW O-DIMENSIONAL ID ` ELES 5 where κ ( v i ) denotes the r esidue field with r esp ect to the v a luation v i (i.e. the r esidue field κ ( y ′ i ) of the ring O ′ { x } ,y ), ∂ κ ( x ) v i is as in D4 . Note that the finiteness of the normaliza tion mor phis m ϕ , eq . 2 .2, implies that κ ( v i ) / κ ( y ) is a finite field extens io n, so D2 is indeed av ailable. F or every excellent n -eq uidimensio nal scheme X → Spec k we define (2.4) C p ( X, M q ) := a x ∈ X ( p ) M q − p ( κ ( x )) . (also denoted by C p ( X ; M , q )) Here q is a Z -grading a nd along p ≥ 0 one obtains a co mplex C • ( X, M q ) by defining the q -degree -preserving differential ∂ X : C p ( X, M q ) − → C p +1 ( X, M q ) by using for all pairs x ∈ X ( p ) and y ∈ X ( p +1) , y ∈ { x } (i.e. y is a c o dimens io n o ne p oint of the integral clo sed subscheme { x } ), the bo undary ma p (2.5) ∂ x y : M ∗ ( κ ( x )) − → M ∗− 1 ( κ ( y )) of e q . 2.3 and defining ∂ X := P ∂ x y for all such pairs y ∈ { x } . This is a differential, ∂ 2 X = 0 , see [Ros9 6, Le mma 3.3 ]. In fact this prop erty is essentially equiv a len t to axio m C o f a cycle mo dule. O ne defines (2.6) M ∗ ( X ) := k er  ∂ X : C 0 ( X, M ∗ ) → C 1 ( X, M ∗ )  . Hence, M ∗ ( X ) is a subgro up of C 0 ( X, M ∗ ) and its elements ar e called unr amifie d cycles, s e e [Ros96, p. 3 38]. Letting U 7→ M ∗ ( U ) fo r Za r iski o pens U , we hav e a sheaf M ∗ ( U ) := M ∗ ( U ). Moreov er, we call (2.7) C • ( X , M ∗ ) : 0 − → M ∗ ( X ) − → C 0 ( X , M ∗ ) ∂ X − → C 1 ( X , M ∗ ) ∂ X − → · · · the R ost cycle c omplex o f X with c o efficients in M ∗ . Unlike [Ros 9 6] we prefer to index by co dimension r ather tha n dimension. 2.2. Pul lbac k & Pus hforw ard. Supp o se X , Y are excellent schemes, say they are n X -, n Y - equidimensional, N := n Y − n X . If X , Y are finite type over a field and f : X → Y is prop er, Rost co nstructs a pushforward f ∗ : C • ( X, M ∗ ) → C • + N ( Y , M ∗ ), see [Ros96, § 3.4]. This co nstruction carr ies over to the case of X , Y just equidimensional excellent and f a finite morphism, se e Lemma 4 b elow. If X , Y are finite type over a field and f : X → Y is flat, Rost co nstructs a pullback op era tion. T o prolo ng this to excellen t schemes, w e s ay that a flat morphism of c onstant r elative dimension r is a flat mor phism f : X → Y s uc h that there is some r ∈ Z suc h that for all y ∈ im f and all generic p o int s x ∈ f − 1 ( { y } ) (0) one ha s (1) dim X x = dim Y y + r . (2) or equiv alently dim O X,x ⊗ O Y ,y κ ( y ) = − N − r ≥ 0. See [EK M08, Ch. IX] for this appr oach. Then there is a flat pullback f ∗ : C • ( Y , M ∗ ) → C •− r − N ( X, M ∗ ), see [Ros96, § 3 .5] (Rost als o develops a more gener al pullback op era tion, for which we have no use). Lemma 4 . Supp ose • X , Y , Z ar e n X -, n Y -, n Z -e quidimensiona l exc el lent schemes, N := n Y − n X , • f : X → Y finite, • g : Z → Y is flat of c onstant r elative dimension r , 6 O. BRA UNLING • W := Z × Y X , • f ′ , g ′ ar e the morphisms induc e d by b ase change. Then the diagr ams C p ( X, M ∗ ) ∂ X / / f ∗   C p +1 ( X, M ∗ ) f ∗   C p ( X, M ∗ ) g ′∗ / / f ∗   C p − r − N ( W , M ∗ ) f ′ ∗   C p + N ( Y , M ∗ ) ∂ Y / / C p + N +1 ( Y , M ∗ ) C p + N ( Y , M ∗ ) g ∗ / / C p − r ( Z, M ∗ ) C p ( Y , M ∗ ) ∂ Y / / g ∗   C p +1 ( Y , M ∗ ) g ∗   C p − r − N ( Z, M ∗ ) ∂ Z / / C p − r − N +1 ( Z, M ∗ ) c ommu t e. Mor e over, ( f 1 ◦ f 2 ) ∗ = f 1 ∗ ◦ f 2 ∗ and ( g 1 ◦ g 2 ) ∗ = g ∗ 2 ◦ g ∗ 1 for f 1 , f 2 two finite morphisms X f 2 → Y f 1 → Y ′ , g 1 , g 2 two flat c onstant r elative dimension morphisms Z g 2 → Y g 1 → Y ′ , r ( g 1 ◦ g 2 ) = r ( g 1 ) + r ( g 2 ) for the c onstant r elative dimensions r esp e ctively. See [Ros9 6, Prop. 4.1, 4.6], the pro ofs car ry over verbatim (but note that our f is finite as opp osed to just b eing pro per ). See also [E KM08, Ch. IX] for the case M ∗ = K M ∗ . W e will use the a bove facts witho ut further quoting this lemma. 3. Ad ` eles with q u a si-coherent coefficients Firstly , le t us explain how to find the cor r ect notion of multidimensional ad` eles. The classica l ad` eles ar e built from lo cal fields, which usually can b e written a s an ind-pro limit, fo r example (3.1) Q p = colim − − − → j lim ← − i p − j Z /p i Z κ (( t )) = colim − − − → j lim ← − i t − j k [ t ] /t i , running ov er the system i , j → + ∞ . In the num ber field case, the infinite plac es R , C form an exception to this principle; how ever, such places do not o ccur in the geometric id` ele theory of this text. One may now attempt to ‘globalize’ such an ind-pro per spe ctive to a who le scheme. This leads to Beilinso n’s theory of ad` eles for quasi-co herent sheav es [Bei80]. W e quickly summar ize the key p oints: Let X b e a Noether ian scheme. F or scheme p oints η 0 , η 1 ∈ X write η 0 ≥ η 1 if { η 0 } ∋ η 1 (“sp ecialization” ). Denote by S ( X ) n := { ( η 0 ≥ · · · ≥ η n ) , η i ∈ X } the s e t of chains o f p oints of length n + 1. Let K n ⊆ S ( X ) n be an arbitra ry subset. F or any p oint η 0 ∈ X define η 0 K := { ( η 1 ≥ · · · ≥ η n ) s.t. ( η 0 ≥ · · · ≥ η n ) ∈ K n } , a subset of S ( X ) n − 1 . This can be viewed as the set o f ‘subscheme flags’ b elow η . Let F b e a c oher ent sheaf on X . F or n = 0 and n ≥ 1 define resp ectively (and inductively) A ( K 0 , F ) := Y η ∈ K 0 lim ← − i F ⊗ O X O X,η / m i η (3.2) A ( K n , F ) := Y η ∈ X lim ← − i A ( η K n , F ⊗ O X O X,η / m i η ). The s heav es F ⊗ O X O X,η / m i η are us ually only quasi-cohere nt, so w e complete this definition as follows: F or a quasi-c oher ent sheaf F w e define A ( K n , F ) := co lim − − − → F j A ( K n , F j ), where F j runs through all coherent subsheav es of F (and hereby reducing to eqs. 3.2). Built s uccessively from colimits and Mittag-Leffler limits, A ( K n , − ) is a cov ariant ex act functor from quasi-coher ent sheav es to quasi-coher ent sheav es. TW O-DIMENSIONAL ID ` ELES 7 Example 5 . (Automatic finiteness condition) F or an in tegra l s moo th curve X /k with function field K this recovers the classica l ad` eles for F := O X . Namely , for △ := S ( X ) 1 one computes A ( △ , O X ) = A ( η △ , K ) = colim − − − → D A ( η △ , O X ( D )) = colim − − − → D Y x ∈ X (1) lim ← − i colim − − − → E 6∋ x O X ( D + E ) /x i = colim − − − → D Y x ∈ X (1) b O x ⊗ O x ( D ) ⊆ Y x ∈ X (1) b K x = Y ′ x ∈ X (1) b K x , (3.3) where D runs thro ugh all divisor s so tha t colim − − − → D O X ( D ) = K and E thro ugh all diviso rs whose suppo rt do es not co ntain x so that co lim − − − → E O X ( E ) = O x . The prime sup erscript in eq. 3.3 means that for all but finitely many x the comp onent needs to lie in b O x ⊆ b K x . The key po in t is: This classic al fi niteness c ondition c omes out automatic al ly fr om Beilinson ’s definition. Now, S ( X ) • carries a natura l s tructure of a simplicia l set (omitting the i -th entry in a flag yields fac e s ; duplica ting the i -th entry in a flag degener acies). This turns A • ( U, F ) := A ( S ( U ) • , F ) (for U Zariski o pen) int o a s heaf of cos implicial a belia n gro ups . W e prefer to read it a s a complex of sheav es (via the unr educed Dold-K an cor resp ondence) with its terms called A i F . Theorem 6. (Beilinson [Bei80, § 2] ) F or a No etherian s cheme X and a qu asi-c oher ent she af F on X , the A i ( − , F ) ar e flasque she aves and 0 − → F − → A 0 F − → A 1 F − → · · · is a flasque r esolution. See Hub er [Hub91a], [Hub91b], for a detaile d pro of. The purp ose of this pap er is to find an analogo us theorem fo r the sheaves which come from Ros t cycle mo dules. T o get an idea of the problem, let us exp eriment with the K - theo ry shea f K n . Then a lready for a single lo cal field as in eq. 3.1 there are tw o natural candidates for the Q p example: K n (colim − − − → j lim ← − i p − j Z /p i Z ) or colim − − − → j lim ← − i K n ( p − j Z /p i Z ). W e immediately notice that the right-hand side version do es n ot even make sense . While the colimit over the system p − j Z p has a ring structure and eq uals Q p , the individual terms a re just Z p -mo dules, but not r ing s. Although this p oses no pro blem for Beilinson’s ad` eles of qua si- coherent sheaves, it depr ives us from interpreting the terms as schemes, and thus to sp eak o f the K - theory of them. Thus, only the left-hand s ide version r emains as a feasible definition. How ev er, its globa l analog ue is K n ( A i O X ( X )) and Sp ec A i O X ( X ) is a terrible space a nd b ear s no resemblance to cla ssical id` ele-type co nstructions. F or example, the clas s ical Brauer - Hasse- No ether seq ue nce 0 − → Br K − → ` v Br b K v − → Q / Z − → 0 for a num b er field K is sug gestive of the fact that inv aria nts (like the Br a uer group) should be taken for lo ca l fields individually , rather than of the whole ad` eles; “ B r( ` v b K v )” is no t reasona ble. Remark 3. M. Morr ow has recently demonstra ted that the gro ups lim ← − i K n ( Z /p i Z ) play a n int eresting ro le in a co rrected Gersten conjecture for s ing ular schemes. Although no t in this pap er, this sug gests that they might play a role in future id` ele theories [Mor1 4], [Mor1 2]. 8 O. BRA UNLING In order to reso lve the problem, w e need to understand how (higher) lo cal fields o ccur in the Beilinson ad` e les: Definition 7. (Parshin [Par78] ) F or n ≥ 1 an n - lo cal field F with last residue field k is a c omplete discr ete valuation field such that its r esidue field is an ( n − 1) -lo c al field with last r esidue field k . We agr e e on t he c onvent ion that k is the only 0 -lo c al field with last r esidue field k . The datum of a 2-lo cal field ca n b e depicted graphica lly as (3.4) F 2 (level 2 field) ↑ A 2 − → F 1 (level 1 field) ↑ A 1 − → F 0 , (last residue field) with the arrows A i → F i being the inclusions of the rings of int eger s A i int o their fields of fractions, A i → F i − 1 the q uotient maps. Theorem 8 . (Beilinson [Bei8 0] ) Supp ose X is a n -dimensional r e duc e d scheme of finit e typ e over a field k and △ = { ( η 0 ≥ · · · ≥ η n ) } a singleton set such that co dim X η i = i . Then A ( △ , O X ) is a finite dir e ct pr o duct of n -lo c al fi elds Q K j . See [Bei80, p. 2, second pa ragra ph] or a gain Huber [Hub91a], [Hub91b]. Dropping the assumption co dim X η i = i one finds similar s tructures, e.g. fields of fra ctions of complete lo cal rings whose residue fields a re again o f this type − o r even tually a finite extension of k . W e will not go into details . Lemma 9. If F is an n -lo c al field with last r esidue field k and char F = char k , then t her e is a non-c anonic al fi eld isomorphism F ≃ k ′ (( t 1 )) · · · (( t n )) for k ′ /k a finite field ext ension. This follows from a rep eated use of Cohe n’s Structure The o rem in the equicharacteristic case. W e will no t use this lemma, but it might b e instr uctiv e to see that the A ( △ , O X ) for singletons △ have a very uniform structure. See [FK00] for more background. Now we fo cus on the c ase of X/k an inte gr al smo oth su rfac e . In order to pro ceed we need to make A ( △ , O X ) e x plicit for △ any singleton set. Lemma 10. Write η ∈ X (0) for the generic p oint and y ∈ X (1) and x ∈ X (2) with x ∈ { y } ar e arbitr ary. One c omputes the fol lowing: (1) A ( { x } , O X ) = b O x is the c ompletion of the lo c al ring O x . This is a 2 -dimensional c omplete r e gular lo c al domain. (2) A ( { y } , O X ) = b O y is the c ompletion of the lo c al ring O y . This is a 1 -dimensional c omplete r e gular lo c al domain. Henc e, it is a c omplete DVR with r esidue field κ ( y ) . (3) A ( { y ≥ x } , O X ) = Q b O x,y ′ i , wher e • y ′ i runs t hr ough the fi nitely many 2 primes in b O x over y , and 2 They corresp ond to p oints ov er x in the normalization of { y } , see [Die67, § 6, Thm. 6.5 (4)]. In particular there are only finitely m any suc h. TW O-DIMENSIONAL ID ` ELES 9 • b O x,y ′ i denotes t he y ′ i -adic c ompletion of the lo c alization of b O x at y ′ i , one c ould also write \ ( b O x ) y ′ i . This ring is a 1 -dimensional c omplete r e gular lo c al domain. Thus, it is a c omplete DVR and the r esidue field is κ ( y ′ i ) of Sp ec b O x . (4) A ( { η } , O X ) = K := κ ( η ) is t he function field of X . (5) A ( { η ≥ y } , O X ) = b K y := F rac b O y is the field of fr actions of b O y . This is a 1 -lo c al field with last r esidue field κ ( y ) . (6) A ( { η ≥ x } , O X ) = b O x ⊗ K . (7) A ( { η ≥ y ≥ x } , O X ) = Q b K x,y ′ i , wher e F r ac b O x,y ′ i is the field of fr actions of b O x,y ′ i . Each b K x,y ′ i is a 2 -lo c al field with last r esidue field κ ( x ) . Mimicking Fig. 3.4 its structu r e is given by (3.5) b K x,y ′ i ↑ b O x,y ′ i − → κ ( y ′ i ) ↑ ( b O x /y ′ i ) − → κ ( x ) . (Writing b O x /y ′ i we identify the p oint y ′ i with its underlying prime ide al.) (8) b K x := F rac b O x (this do es not app e ar in the ad` eles, but we fix the notation!) This computation is e lemen tary with p ossibly one exception: The fact that there are pro ducts app earing in Q b O x,y ′ i or Q b K x,y ′ i relies on the b ehaviour of completion at singularities. The pro ducts reduce to a single facto r whenever { y } is a normal scheme. In ge ne r al there ar e as many fac tors as there are pr eimages of x in the fiber of the norma lization { y } ′ → { y } ∋ x . This is expla ined in [Die67, § 6, Thm. 6.5 (4 )], [Par83, § 1.1 ]. Example 1 1. Co nsider X = A 2 k = Spec k [ s, t ] and △ = { η ≥ y ≥ x } with η = (0), y = ( s 3 + s 2 − t 2 ), x = ( s, t ) in terms o f prime ideals. The cur ve { y } is singula r at x and the p oint has tw o preimages in the nor malization { y } ′ . Co rresp ondingly , A ( △ , O X ) = b K x,y ′ 1 ⊕ b K x,y ′ 2 . Definition 12. A height one prime in b O x is c al le d a tra nscendental curve if it do es not lie in the fib er of a height one prime under Spec b O x → Spec O x . Example 13. F o r k = Q , A 2 Q = Sp ec Q [ s, t ], x = (0 , 0) and all i ≥ 1 the primes ( s − t i exp t ) constitute an infinite set of tra ns cendent al cur ves in b O x ≃ Q [[ s, t ]]. Lemma 14. A ( { η ≥ x } , O X ) = b O x ⊗ K is a princip al ide al domain with infinitely many maximal ide als. They c orr esp ond bije ctively to the t r ansc endental curves in b O x . Pr o of. b O x is tw o -dimensional. It is re g ular, thus fa ctorial, a nd so a ll height one primes are principal. T ensoring with K kills the maximal ideal (among others), so the r emaining height one primes b ecome max imal. Then us e the Ex a mple (replace ex p t by another transce ndental function if char k > 0). Note that b O x ⊗ K is the fib er of the generic p oint under Sp ec b O x → Spec O x , giving the se cond claim.  W e shall need the following general “Gersten c o njecture”-type prop erty for big cycle mo dules: Prop ositio n 15. (R ost [Ros96, Thm. 6.1] ) Supp ose X := Sp ec O Y ,x is the sp e ctru m of a lo c al ring of a smo oth scheme Y /k and let M ∗ b e a (big) cycle mo dule. Then the cycle c omplex C • ( X , M ∗ ) as in e q. 2.7 is exact. 10 O. BRA UNLING This shows that the sheafification of eq. 2.7 provides a fla sque reso lution of M ∗ , [Ros96, Cor. 6.5]. Mo reov er, we need the a nalogous s tatemen t for co mplete r ing s. Such a statement is no t prov en in [Ros9 6] (and would no t fit in with the axioms in lo c. cit. ), but the general principle of pro of is well-known since Q uillen’s breakthroug h pap er o n algebra ic K -theory [Qui73] and generalizes to cycle mo dules : Prop ositio n 16. Su pp ose X/ k is an inte gr al e quichar acteristic c omplete r e gular lo c al scheme and let M ∗ b e a big cycle m o dule. Then the cycle c omplex C • ( X, M ∗ ) as in e q. 2.7 is exact. W e defer the pro of to § 5. As a result: Prop ositio n 17. The c omplex C • (Sp ec A ( △ , O X ) , M ∗ ) is exact for al l rings A ( △ , O X ) ap- p e aring in the list in L emma 10. Pr o of. F o r the A ( △ , O X ) which are fields this is obvious; for the complete lo cal rings use Prop. 16. O nly A ( { η ≥ x } , O X ) = b O x ⊗ K r equires an arg umen t. The diag r am (3.6) M ∗ ( b K x ) / / ` b y M ∗− 1 ( κ ( b y )) / / 0 M ∗ ( b K x ) / / ∼ = O O ` y ′ M ∗− 1 ( κ ( y ′ )) / / O O M ∗− 2 ( κ ( x )) / / 0 M ∗ ( K ) / / O O ` y M ∗− 1 ( κ ( y )) / / O O M ∗− 2 ( κ ( x )) / / ∼ = O O 0 . commutes, wher e the the to p ro w is the c ycle complex in question, so b y Lemma 14 the v ariable ˆ y runs through the tra ns cendent al curves of b O x ; the middle is the cycle c o mplex of b O x (so that y ′ runs thr ough a ll height one primes) and exact by Pr op. 16; the b ottom r ow the one o f O x and exact by P r op. 15 . The vertical arrows ar e functorially induced from the fla t pullbacks along Spec b O x ⊗ K → Sp ec b O x → Spec O x . Only surjectivity in the to p row needs a n arg umen t. Ma p the elemen ts ( α ˆ y ) ∈ M ∗− 1 ( κ ( b y )) to the middle ro w − the up ward ar row is a pro jector on a subset of s ummands; so it is canonica l split by the cor r esp onding inclusion. If they map to a non-zero element in M ∗− 2 ( κ ( x )), use s urjectivity in the last row to find a lift to ` y M ∗− 1 ( κ ( y )). Change the e lemen t in the middle row by this lift so that it maps to zer o in M ∗− 2 ( κ ( x )). E xactness of the middle row yields a lift in M ∗ ( b K x ), proving surjectivity in the top row by c omm utativity of the dia gram.  4. Id ` eles with cycl e module coefficients As in the pre vious s e ction X/k is an integral smo oth sur fa ce and M ∗ a big cy cle mo dule a s in § 2. Contrary to § 3 we will sp eak of id ` eles instead of ad` eles fro m now on. The conv en tions for this in the literatur e a r e blur ry . F o r a curve ov er a field the classical ad` eles arise from the theory of § 3, whic h is a g o o d motiv ation to spe a k of ‘a d ` eles’. The classic a l id` eles could b e seen as the degree one pa rt of K - theory id` eles in the sense of the pres en t theo ry , so we call them ‘id` eles’. As b efore, write η ∈ X (0) for the generic p oint and y ∈ X (1) and x ∈ X (2) with x ∈ { y } a re arbitrar y . Notation. W e shall write (4.1) Y y M ∗ ( − ) as a sho rthand for F : U → Y y ∈ U (1) M ∗ ( − ), TW O-DIMENSIONAL ID ` ELES 11 where the latter is a (flasque) Zar iski sheaf of Z -gra ded ab elian g roups. Analogously fo r Q x , where x runs through all closed p oints U (2) , e.g . U → Q x ∈ U (2) M ∗ ( − ). W riting Q x,y ′ i , the second par ameter y ′ i runs thr ough the heigh t one primes in b O x except the transcendental cur ves, e.g. U → Q x ∈ U (2) Q y ′ i M ∗ ( − ) with y ′ i any non-transcendental curve in b O x . Example 18. Q x,y ′ i M ∗ ( b O x,y ′ i ) deno tes the flasque sheaf U → Q x ∈ U (2) Q y ′ i ∈ (Sp ec b O x ) (1) , non-transc . M ∗ ( b O x,y ′ i ). Recall that M ∗ ( − ) for a ring is defined by eq. 2.6. These pro ducts are supp osed to mimick the pro duct app earing in eq. 3.2. W e hav e seen in Ex ample 5 that the Beilinson ad` eles automatically imp ose finiteness conditions as they o c cur for example in eq. 1.2 and eq. 3.3. As explained in § 3, a literal tra nslation of Beilinso n’s ad` eles for quas i- coherent sheav es to K -theory sheav es do es not work. The s ame applies to more gener al cycle mo dule co efficients. Thus, we will use the cycle mo dule as so ciated to the lo cal comp onents A ( △ , O X ) for △ a single to n s et, but patch them together manually to imitate (v aguely!) the finiteness conditions a s they would come from Beilinson’s constr uc tio n. Na turally , carry ing o ut this pro cess ‘by hand’ requires a slightly unpleasant amount of individua l definitions: Notation. The symbo l ˜ ∀ x will mean: for all x with p os sibly finitely many exceptions. W e also say “ for a lmost all x ”. Definition 19. F or the she aves as intr o duc e d in e q. 4.1, we int r o duc e the fol lowing variations which imp ose m anu al finiteness c onditions: (1) The prime su p erscript in Q ′ y M ∗ ( b K y ) me ans that we c onsider only those elements ( α y ) y such t hat ˜ ∀ y ∈ X (1) : α y ∈ M ∗ ( b O y ) . (2) The triple prime sup erscript in Q ′′′ x,y ′ i M ∗ ( b K x,y ′ i ) me ans that we c onsider only those elements ( α x,y ′ i ) x,y ′ i such t hat (a): ∀ x ˜ ∀ y ′ i : α x,y ′ i ∈ M ∗ ( b O x,y ′ i ) . (b): ˜ ∀ x : P y ′ i ∂ y ′ i x ∂ b K x,y ′ i y ′ i ( α x,y ′ i ) = 0 ∈ M ∗− 2 ( κ ( x )) . The condition (a) ensures that the sum in (b) is finite. Example 20. Note that (a) still allows α x,y ′ i / ∈ M ∗ ( b O x,y ′ i ) for infinitely many y ′ i as shown in this example: The lines represe nt p oints y ′ i with α x,y ′ i / ∈ M ∗ ( b O x,y ′ i ). In this particular ex ample infinitely many closed p oints x ar e inv olved, so the condition (b) is non-trivial. T ric omplexes: F or e very ab elian categ ory P le t C P deno te the ab elian categor y of bo unded complexes in P . Then C C P can be identified with the category o f bounded bicomplexes. There is the standard total co mplex functor C C P → C P . Applying this to C P , C C C P can be viewed as the ab elian category of tric omplexes and each tricomplex has a natura l total complex given by the compo sition C C ( C P ) → C C P → C P . The sign correction of the differen tial in the total complex then induces the corre c t s ign for the given complex. Dep ending o n o ne’s preferred sign conv en tion, the ov erall sign in explicit formulas might come out o ppo site. 12 O. BRA UNLING Definition 2 1. We define a c omplex of she aves of Z -gr ade d ab elian gr oups (4.2) 0 − → M ∗ − → I 0 M − → I 1 M − → I 2 M − → 0 as the total c omplex of the fol lowing tric omplex: (4.3) Q x M ∗ ( b O x ⊗ K ) u u ❦ ❦ ❦ ❦ Q x M ∗ ( b O x ) , u u ❦ ❦ ❦ ❦ o o Q ′′′ x,y ′ i M ∗ ( b K x,y ′ i ) Q x,y ′ i M ∗ ( b O x,y ′ i ) o o ` η M ∗ ( K ) u u ❦ ❦ ❦ ❦ O O M ∗ u u ❦ ❦ ❦ ❦ ❦ ❦ ❦ o o O O Q ′ y M ∗ ( b K y ) O O Q y M ∗ ( b O y ) O O o o The arr ows ar e induc e d fr om the flat pul lb acks along the re sp e ctive c ompletio n and lo c alization maps. As X has only a single generic point η , Q η M ∗ ( K ) is a redundant for m ulation, but it stresses the g eneral patter n. Remark 4. The ring s b O x , b O x,y ′ i ,. . . appea ring in the tricomplex are precisely the lo ca l co mpo - nent s A ( − , O X ) of the Beilinson ad` eles with c o e fficie n ts in the structure shea f O X , see Lemma 10 (a lb eit with the degenera te simplices r emov ed). Mo r eov er, the complex A • O X of Thm. 6 can − (not q uite, but) ro ughly − b e written down as the total co mplex of a trico mplex resembling the o ne in Fig. 4.3. Lemma 2 2. The morphisms in the tric omplex in e q. 4.3 ar e wel l-define d. Pr o of. The cla im is obvious, except p oss ibly for chec king that the individual finiteness condi- tions a r e b eing res pected. But this is s traightforw ard.  This cons truction b oils down to a co mplex M ∗ → Y η M ∗ ( K ) ⊕ Y y M ∗ ( b O y ) ⊕ Y x M ∗ ( b O x ) → Y ′ y M ∗ ( b K y ) ⊕ Y x M ∗ ( b O x ⊗ K ) ⊕ Y x,y ′ i M ∗ ( b O x,y ′ i ) (4.4) → Y ′′′ x,y ′ i M ∗ ( b K x,y ′ i ) → 0. Except M ∗ the sheav es are all flasque. If M ∗ is under sto o d, we write I i instead of I i M . W e shall also write I 0 = I (0) ⊕ I (1) ⊕ I (2) ; I 1 = I (01) ⊕ I (02) ⊕ I (12) ; I 2 = I (012) according to the direct sum decomp o sitions of the three terms in eq. 4.4 (here the s uper scripts indicate the co dimensions of the points a ppea ring in the resp ective flag s). This is co n venien t to stres s what comp onent s ections of these sheaves ar e a s so ciated to; we will usually use sup erscr ipts using this indexing as in ( f 0 , f 1 y , f 2 x ); ( f 01 y , f 02 x , f 12 x,y ′ i ); ( f 012 x,y ′ i ) r e spec tiv ely . The maps all turn out to be induced from s igned diagonal maps a nd the appr opriate restrictions, e.g . the first arrow lo cally unwinds as f 7→ ( f , Y y res b K y K f , Y x res b K x K f ), the la st as ( f 01 y , f 02 x , f 12 x,y ′ i ) 7→ ( f 12 x,y ′ i − res b K x,y ′ i b K x f 02 x + res b K x,y ′ i b K y f 01 y ). W r ite A i ( − , M ∗ ) := H i ( C • ( − , M ∗ )) to deno te the co homology of the Rost cycle co mplex. TW O-DIMENSIONAL ID ` ELES 13 Theorem 23. (Main The or em) L et X/k b e a smo oth inte gr al surfac e, M ∗ a big cycle mo du le and M ∗ the asso ciate d Zariski she af. (1) Then M ∗ − → I 0 M − → I 1 M − → I 2 M − → 0 is a flasque r esolution. (2) The isomorphisms α i : H i ( X, I • M ) → A i ( X, M ∗ ) ar e explicitly given by α 0 : ( f 0 , f 1 y , f 2 x ) 7→ f 0 ∈ M ∗ ( κ ( η )) α 1 : ( f 01 y , f 02 x , f 12 x,y ′ i ) 7→ ∂ b K y y ( f 01 y ) ∈ M ∗− 1 ( κ ( y )) (4.5) α 2 : ( f 012 x,y ′ i ) 7→ X y ′ i ( ∂ y ′ i x ◦ ∂ b K x,y ′ i y ′ i )( f 012 x,y ′ i ) ∈ M ∗− 2 ( κ ( x )) (4.6) Part 1 is the analo gue of Beilinson’s resolution of qua si-coherent sheav es in Thm. 6 . The rest o f the sectio n is devoted to the pr o of. W e fix M ∗ and write I i instead of I i M . Firstly , we define ’id` eles with a r e cipro city constraint’. These she aves are defined as kernels: 0 − → Y ′ +recip. y M ∗ ( b K y ) − → Y ′ y M ∗ ( b K y ) ∂ y x ∂ y − → a x M ∗− 2 ( κ ( x )) 0 − → Y ′′′ +recip. x,y ′ i M ∗ ( b K x,y ′ i ) − → Y ′′′ x,y ′ i M ∗ ( b K x,y ′ i ) ∂ y ′ i x ∂ y ′ i − → a x M ∗− 2 ( κ ( x )) W e shall keep this notation for later use below. Here ‘recipr o city constraint’ refers to the fact that a n element in M ∗ ( K ), when b eing mapp ed dia gonally to Q ′ y M ∗ ( b K y ), auto matically satisfies P y ∂ y x ∂ y x = 0. As this sum r uns over a ll curves y thro ug h a fixed clo sed p oint x , this is so metimes c a lled a ‘recipro city law around a p oint’. Now we focus on the bicomplex which constitutes the b ottom face of the tricomplex of Def. 21. The total complex of this bicomplex turns out to be homologica lly concentrated in a single degree: Lemma 2 4. The se quenc e of she aves 0 − → M ∗ − → K ⊕ Y y M ∗ ( b O y ) − → Y ′ + r e cip. y M ∗ ( b K y ) − → 0 is exact. W e may rephrase this as follows: the b ottom face of our trico mplex is qua si-isomor phic to the s heaf ` x M ∗− 2 ( κ ( x )), placed in degr ee tw o. Pr o of. The injectivity is clear since fo r each op en set U the g roup of sections M ∗ ( U ) is defined as a subgroup of M ∗ ( K ). Exactness in the middle is ea sy: Let x b e a p oint in X , U x ∋ x some op en neighbourho o d. Suppose lo cal sections α ∈ M ∗ ( K ) and α y ∈ M ∗ ( b O y ) ov er U x are given and go to zer o on the rig h t. Then for a ll co dimension o ne p oints y ∈ U (1) x we hav e res b K y K α − α y = 0 ∈ M ∗ ( b K y ). Thus, ta king the b oundary at y in b O y we obtain 0 = ∂ b K y y res b K y K α − ∂ b K y y α y = res κ ( y ) κ ( y ) ∂ K y α − ∂ b K y y α y by the compatibility o f flat pullbacks (here a lready made explicit as res b K y K ) with the differential in the c y cle complex , axiom R3a . W e also hav e ∂ b K y y α y = 0 since α y ∈ M ∗ ( b O y ) and so we 14 O. BRA UNLING deduce ∂ K y α = 0. How ever, this shows that α ∈ M ∗ ( U x ). F or the surjectivity s uppo s e we ar e given ( α y ) y on the right-hand side. The cycle complex of Sp ec O x is exact by Pr op. 15: 0 → M ∗ ( O x ) → M ∗ ( K η ) → a y ∈ ( O x ) (1) M ∗− 1 ( κ ( y )) → M ∗− 2 ( κ ( x )) → 0, The element ( ∂ b K y y α y ), pla c e d in the third term of this sequence, a dmits a preimage f ∈ M ∗ ( K ) ( Pr o of: by the finitenes s condition Q ′ this is no n-zero only for finitely many p oints y and the recipro city cons tr aint shows that the element go es to zero on the right). Secondly , no te that β y := res b K y K f − α y satisfies ∂ b K y y β y = 0 fo r all y in the neighbour ho o d, so β y ∈ M ∗ ( b O y ). It is now clear that ( f , β y ) is a preimage of α y .  Now we re p ea t this analy sis with the top face of the tricomplex . Again, the to tal co mplex of this bicomplex turns out to be ho mologically concentrated in degr ee tw o: Lemma 2 5. The se quenc e of she aves 0 → Y x M ∗ ( b O x ) → Y x M ∗ ( b O x ⊗ K ) ⊕ Y x,y ′ i M ∗ ( b O x,y ′ i ) · · · · · · → Y ′′′ + r e cip. x,y ′ i M ∗ ( b K x,y ′ i ) → 0 is exact. In o ther w ords, the top fac e of our tr icomplex is quas i-isomorphic to the sheaf ` x M ∗− 2 ( κ ( x )), placed in degr ee tw o. Pr o of. The pro o f is very simila r to the one of the previous lemma. Injectivity is clear since b O x and b O x ⊗ K b o th have field of fractions b K x . F or exa ctness in the middle, s upp os e we are given α x ∈ M ∗ ( b O x ⊗ K ) and α x,y ′ i ∈ M ∗ ( b O x,y ′ i ) g oing to zero on the rig ht . This unwin ds as the co cycle c o ndition (4.7) ∀ x, y ′ i : res b K x,y ′ i b K x α x − α x,y ′ i = 0 ∈ M ∗ ( b K x,y ′ i ) and a pplying the b oundar y ∂ b K x,y ′ i y ′ i yields r es κ ( y ′ ) κ ( y ′ ) ∂ b K x y ′ i α x = ∂ b K x,y ′ i y ′ i α x,y ′ i . W e co nc lude ∂ b K x y ′ i α x = 0. The height one pr imes of b O x disjointly decomp ose into transcendental b y and non- transcendental y ′ i curves. Since α x ∈ M ∗ ( b O x ⊗ K ) we therefore know that the bounda ries at al l heig ht one primes of b O x v anish and thus α x ∈ M ∗ ( b O x ). Finally , eq. 4.7 shows that α x maps to α x,y ′ i under the first ar r ow, showing tha t we hav e found a preimag e. T o s e e surjectivity , suppo se we ar e given α x,y ′ i ∈ M ∗ ( b K x,y ′ i ). By the r e c ipro city cons tr aint a nd the exa ctness of the cycle complex of Spec b O x , b C • : M ∗ ( b O x ) / / M ∗ ( b K x ) / / ` ˜ y M ∗− 1 ( κ ( ˜ y )) ∂ y ′ i α x,y ′ i ∈ / / M ∗− 2 ( κ ( x )) / / 0 we obtain a preimage f ∈ M ∗ ( b K x ). Since b y construction ∂ b y f = 0 at tra nscendental curves b y in b O x , P rop. 17 tells us that f ∈ M ∗ ( b O x ⊗ K ). It is now obvious that β x,y ′ i := res b K x,y ′ i b K x f − α x,y ′ i ∈ M ∗ ( b O x,y ′ i ) a nd that f ⊕ β x,y ′ i provides a preimage.  TW O-DIMENSIONAL ID ` ELES 15 Combining the tw o lemmata , we may quasi-iso morphically replace the top and b ottom fa c e of our tricomplex so that we arrive at 0 u u ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ 0 u u ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ o o ` x M ∗− 2 ( κ ( x )) 0 o o 0 u u ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ O O 0 u u ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ o o O O ` x M ∗− 2 ( κ ( x )) ∼ = O O 0 O O o o A direct inspection r eveals that the morphism along the fro n t left edge is indeed a n isomorphism. How ev er, this immediately implies that the total complex of the tricomplex is acy clic. This finishes the pro of of Thm. 23.1. The existence of some formula a s in the cla im of Thm. 23.2 is clear since the flas q ue reso lution prop erty alrea dy implies abstr actly that H i ( X, I • M ) ∼ = − → A i ( X, M ∗ ). One then finds the concrete formula by explicitly mak ing the dia g ram chase underlying this isomorphism (one just ne e ds to go thro ugh the pro ofs of Lemma 2 4 and Lemma 25 ag ain and keep track of the individual steps). Remark 5. (Compa rison with Gorchinskiy’s theory) Statement a nd compa rison ma ps in eq. 4.5 are entirely analogo us to Gorchinskiy’s comparison maps ν ∗ [Gor07], [Go r08, Thm. 1.1 and map in Pro p. 2.16 ] for unco mpleted ad` eles. In lo c. cit. the Gersten complex is called ‘Cous in complex’ and the b oundary maps ar e c a lled ‘r esidue maps’ as in the analog ue o f the theory for quasi-coher ent sheav es [Har66]. Remark 6. The s heav es I • M are different from the I , I 0 app earing in the int ro duction in § 1. Using K -theor y a s a cycle mo dule the ab ov e yields a c anonical isomor phis m CH 1 ( X ) ∼ = H 1 ( X, K 1 ) ∼ = H 1 ( I • K , 1 ) (the subscript ‘1 ’ refers to taking degr e e o ne in the gr ading of the cycle mo dule). There is no har m using this isomorphism instead o f the o ne in § 1. Ther e is no way the isomor phis m in § 1 g eneralizes to arbitrar y cycle mo dules since eq. 1.4 is a r esolution concentrated in degr ees [0 , 1] and if it generaliz e d this would imply H 2 ( X, M ∗ ) = 0 for all M ∗ , which is just false. As in § 4, let X/k b e an integral smo oth sur face ov er an arbitr ary field. Let M ∗ be a big cycle mo dule with a pro duct, a pa iring on itself as defined in [Ros9 6, Def. 2 .1]. Such a pairing consists of a bilinea r pair ing of ab elian g roups (4.8) · : M p ( F ) × M q ( F ) − → M p + q ( F ) . Certain axioms P1 - P3 [Ros96] need to b e fulfilled. Such a pair ing induces an a sso ciated pair ing of cycle co homology gr o ups. Example 26 . (Pro ducts) M ∗ = K M ∗ has such a pairing, just g iven by the ordinary pr o duct in the Milnor K -theor y ring . Similarly , Quillen K -theo ry a nd Galois cohomolog y with Z /ℓ co efficients with its cup pro duct, ℓ coprime to char k ; M i ( F ) := H i ( F, Z /ℓ ( i )). Given any such pr o duct, we obtain a graded-co mm utative product o n the cycle cohomo logy groups A i ( X, M q ) ⊗ A j ( X, M r ) − → A i + j ( X, M q + r ). Using Milno r or Q uillen K - theory as a cycle mo dule, one has A i ( X, K i ) = CH i ( X ) and the bidegree ( i, i ) exce r pt o f the ab ov e pro duct agrees with the usual co mm utative pro duct of the 16 O. BRA UNLING Chow ring . F eeding this into the ma in theorem, Thm. 1, we obtain a version o f the commutativ e square in Fig. 1.5 in the intro duction § 1. CH 1 ( X ) ⊗ Z CH 1 ( X ) / /   CH 2 ( X )   H 1 ( X, I • M ) ⊗ Z H 1 ( X, I • M ) ∗ / / H 2 ( X, I • M ) Note that it is actually slightly different from the one in § 1. The cohomolog y gro up H 1 ( X, I • K ) comes from a mo r e c o mplicated presentation than in § 1, but as a lready ex plained in Rmk. 6 this cannot b e av oided if one wan ts a uniform id` ele reso lutio n for a ll cycle mo dule sheaves. Example 27. (S. Go rchinskiy) I thank S. Go rchinskiy for co mm unicating this example to me. It would be des irable to lift the pro duct ( ∗ ), which we have essentially av oided to co nstruct from scr a tch by tr ansp orting it from A ∗ ( X, M ∗ ), to a pro duct defined on the id` ele resolution itself. The mos t straig h tforward definition would b e (4.9) I p ⊗ Z I q − → I p + q ; ( e ⊗ f ) z 0 ...z p + q := res ∗ ∗ e z 0 ...,z p · res ∗ ∗ f z p ...z p + q where r e s ∗ ∗ comes fro m the flat pullback a long the resp ective ar r ow in Fig. 4.3. How ever, this definition do es not work: Consider affine 2-space X := Sp ec k [ s, t ] a nd take Milnor K -theo ry as the cyc le mo dule. Then define e 01 y :=  s for y = ( s ) 1 otherwise f 12 x,y :=    t − α for x = ( s, t − α ), α ∈ k and y = ( s ). 1 otherwise in degree one (i.e. K M 1 ( F ) = F × ). These id` eles meet the finiteness conditions of § 4. Howev er, e ⊗ f do es not meet the finiteness conditio n o f Def. 19.2. And indeed an ev aluation of the compariso n map in eq. 4.6 yields X y ( ∂ y x ◦ ∂ b K x,y y )( e ⊗ f ) x,y = ∂ ( t − α ) ∂ ( s ) { s, t − a α } = +1 for al l p oints x = ( s, t − α ) ( α a rbitrary). This do e s not lie in ` x ∈ X (2) Z (unless k is finite). W e may even allow all α ∈ k [ s ]. It is still r easonable to b elieve that there exists a subcomplex of I • M on which eq. 4.9 exhibits a well-defined pr o duct. Gorchinskiy constructed such a sub complex in his theory of adically non-co mpleted id` eles [Gor07], [Gor08]. 5. Appendix: Gersten prope r ty for co mplete rings Finally , we need to prov e Pro p. 1 6, which we rep eat for conv enience: Prop ositio n 28. Su pp ose X/ k is an inte gr al e quichar acteristic c omplete r e gular lo c al scheme and let M ∗ b e a big cycle m o dule as in § 2. Then the c omplex C • ( X, M ∗ ) is exact. This cr uc ia l technical fact is independent of the re s t o f the pap er. W e s hall use the following lemma, v ar iations of which o ccur in most pro ofs of No ether normalizatio n. Lemma 29 . (Nagata) L et F b e a field. Su pp ose f ∈ F [[ w 1 , . . . , w n ]] is non-zer o. Then t her e is a ring automorphism σ (fixing F ) of the shap e w n 7→ w n w i 7→ w i + w c i n for suitable c i ∈ Z > 0 and i = 1 , . . . , n − 1 such that σ f ∈ F [[ w 1 , . . . , w n ]] is w n -distinguishe d (i.e. the c o efficients as a p ower series in R [[ w n ]] with R = F [[ w 1 , . . . , w n − 1 ]] do not al l lie in the ide al ( w 1 , . . . , w n − 1 ) of R ). TW O-DIMENSIONAL ID ` ELES 17 Pr o of. [BGR84, s ee pro o f of Pro p. 1 in § 5.2 .4].  Pr o of. (roughly fo llows the metho d o f [Qui73, Thm. 5.13 ]) W rite X = Sp ec R with ( R, m ) an equicharacteristic complete reg ular lo cal doma in o f dimensio n n . W e can ass ume n ≥ 1 as the case n = 0 is trivial. The injectivit y M ∗ ( X ) ֒ → C 0 ( X , M ∗ ) follows from the very definition o f M ∗ ( X ) as a kernel, so we o nly ne e d to show that given some α = ( α y ) y ∈ X ( p +1) ( p ≥ 0) as in the middle ter m o f · · · − → a x ∈ X ( p ) M ∗ ( κ ( x )) ∂ X − → a y ∈ X ( p +1) M ∗− 1 ( κ ( y )) ∂ X − → a z ∈ X ( p +2) M ∗− 2 ( κ ( z )) such that ∂ X α = 0, there e xists some β = ( β x ) x ∈ X ( p ) with ∂ X β = α . W e will do this b y picking a suitable closed subs cheme Y on which α is supp orted a nd s how that the pushforward i ∗ along Y ֒ → X is homotopic to zero: F o r given α pick a closed subscheme Y of X o f pure co dimension one containing the (finitely many) clo sed subsets { y } such that α y 6 = 0. As R is reg ula r, Y is cut o ut from X by a principal divisor, say ω ∈ R , Y = Sp ec R/ ( ω ). By Cohen’s Structure Theorem there is a (non-canonica l) ring isomorphism R ≃ κ ( m ) [[ w 1 , . . . , w n ]]. After po ssibly changing the isomorphism by Lemma 2 9 w e can assume tha t ω is distinguished. Then by the W eier straß Pre pa ration Theorem [BGR84, Thm. 1 in § 5.2.2] there is a unit u ∈ R × and a W eierstra ß p olynomial f ∈ κ ( m ) [[ w 1 , . . . , w n − 1 ]][ ˜ w n ] such that ω = u · f . Hence, w.l.o.g. ω = f as this ge ne r ates the same ideal. Next, we mimick the dia g ram of [Ro s96, pr o of of Pro p. 6.4.] Z π ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ g ~ ~ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ Y i / / ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ X ~ ~ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ A, where in o ur setup • Y := Sp ec R/ ( ω ) = Spec R/ ( f ); • X := Sp ec R = Sp ec κ ( m ) [[ w 1 , . . . , w n ]]; • A := Sp ec κ ( m ) [[ w 1 , . . . , w n − 1 ]]; • Z := Y × A X . Here g , π deno te the pro duct pro jections. The arr ow X → A is not s mo o th (unlike its counterpart in [Ro s96, pr o of of Prop. 6.4 .]), but still flat o f consta n t relative dimension one. The arrow Y → A is a finite morphism since by our W eierstraß P reparatio n we have O Y = κ ( m ) [[ w 1 , . . . , w n − 1 ]][ ˜ w n ] / ( f ( ˜ w n )) . The arrow i : Y → X is a close d immersion. Finally , no te that O Z = κ ( m ) [[ w 1 , . . . , w n − 1 ]][ ˜ w n ] / ( f ( ˜ w n )) . . . (5.1) . . . ⊗ κ ( m )[[ w 1 ,...,w n − 1 ]] κ ( m ) [[ w 1 , . . . , w n − 1 ]][[ w n ]] = κ ( m ) [[ w 1 , . . . , w n − 1 , w n ]][ ˜ w n ] / ( f ( ˜ w n )) . W e get a clo s ed immersion σ coming from the diagona l ideal ( w n − ˜ w n ) in O Z . This r ealizes Y as the underlying close d subscheme of Z o f a principa l divisor . W e let V be the op en 18 O. BRA UNLING complement of the clos ed subset σ ( Y ) in Z so that Z = V ∪ σ ( Y ) disjointly . Algebra ic ally , O V = κ ( m ) [[ w 1 , . . . , w n − 1 , w n ]][[ ˜ w n ]] / ( f ( ˜ w n )) [ 1 w n − ˜ w n ]. Since f is a nonzero divis or, dim Z = n , and since w n − ˜ w n ∈ m , the max imal ideal gets killed after inv erting w n − ˜ w n . The resulting maximal primes co rresp ond bijectiv ely to those primes of O Z maximal with the prop erty no t to contain w n − ˜ w n ; a nd these are all one- dimensional. Hence, dim V = n − 1, and V is usually not lo c al ! Note that the open subscheme V ha s strictly lower dimensio n than Z (this is one of the peculia rities o f lo ca l ring s). Let g ′ : V → Y b e the restriction of the pr o jection Z → Y to the op en subscheme V . This morphism is flat of constant r elative dimens io n zero (whereas the full g : Z → Y has co nstant re la tive dimension one). Define H := π ∗ ◦ j ∗ ◦ { w n − ˜ w n } ◦ g ′∗ C p ( Y , M ∗ ) → C p ( V , M ∗ ) → C p ( V , M ∗ +1 ) → C p ( Z , M ∗ +1 ) → C p ( X, M ∗ +1 ) , where • g ′∗ is the flat pullback of constant rela tive dimension one; • { w n − ˜ w n } deno tes the left-multiplication by units on C • ; • j : V → Z is the op en immersion (and flat of co ns tant r e lative dimension zero) and j ∗ is a non -prop er pushforward along this o pen immersio n. Bewar e: As j is not prop er , ∂ ◦ j ∗ 6 = j ∗ ◦ ∂ . • π : Z → X is the pro duct pr o jection. I t is a finite mor phis m, one sees this by direct insp e ction of eq. 5.1 or a bstractly b y base change from the finiteness of Y → A . As for the pro o f of [Ro s96, P r op. 6.4 ] one co mputes that H is a chain homotopy b etw een the c losed immers ion pushfor w ard i ∗ and the zero mor phis m: (5.2) ∂ X ◦ H + H ◦ ∂ Y = i ∗ − 0 : C p ( Y , M ∗ ) → C p +1 ( X, M ∗ +1 ). This completes the pr o o f.  One ca n proba bly prove a version for all equicharacteristic regular lo cal r ings us ing Panin’s metho d via N ´ er on-Popescu des ingularization [P an03]. Howev er, then one needs the co c ontin uity prop erty M ∗ (colim F i ) ∼ = colim M ∗ ( F i ) which holds for K -theor y , but need not hold for big c y cle mo dules without adding further axio ms. There a re int eresting big cycle mo dules without this prop erty: Define ” K top n ( F ) ” := K M n ( F ) / (divisible elements) . This is a big cycle mo dule. The name is inspired from F esenk o’s top olo g ical Milno r K - g roups [F e s 01]. They a ppea r in lo cal c la ss field theory , but are only defined for higher lo cal fields. The link stems fro m: Theorem 30. (F esenko [F es01, Thm. 4.7 (iv)] ) L et F b e an n -lo c al field with char F = p > 0 and last r esidue field algebr aic over F p . Then K top n ( F ) ∼ = K n ( F ) / ( divisibl e elements ) as abstr act gr oups. 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