Suslins singular homology and cohomology

On Suslin’s singular homology and cohomology Thomas Geisser ⋆ Universit y of Southern California Summary . W e s tud y prop erties of Suslin homology and cohomology o ver non- algebraicall y cl osed base fields, and t h eir p -part in characteristic p . In the second half w e fo cus on finite fields, and consider finite generation qu estions and connections to tamely ramified class field th eory . 1 Intro duction Suslin and V o evods k y defined Suslin homolo gy (also called singular homo l- ogy) H S i ( X, A ) o f a scheme o f finite type ov er a field k with co efficients in an ab elian gr o up A as T or i (Cor k ( ∆ ∗ , X ) , A ). Here Cor k ( ∆ i , X ) is the free ab elian gr oup generated by in tegra l subschemes Z of ∆ i × X w hich ar e finite and surjective over ∆ i , and the differentials are given by alternating sums of pull-back maps a long face ma ps . Suslin coho mology H i S ( X, A ) is de fined to be Ex t i Ab (Cor k ( ∆ ∗ , X ) , A ). Suslin and V o evo dsky s howed in [19] that ov er a separably closed field in whic h m is invertible, one ha s H i S ( X, Z /m ) ∼ = H i et ( X, Z /m ) (1) (see [2] for the ca se o f fields of characteristic p ). In the first half of this pa p e r we study b oth the situatio n that m is a power of the c hara c teristic of k , and that k is no t algebra ically c losed. In the second half w e focus on finite base fields and discuss a mo dified version of Suslin homolog y , which is closely rela ted to etale cohomolo gy o n the one hand, but is als o exp ected to b e finitely generated. More over, its zero th ho mology is Z π o ( X ) and its firs t homolo g y is exp ected to b e an integral mo del of the ab elianized tame fundamen tal gr oup. W e start by discussing the p -par t of Suslin homology over an a lgebraica lly closed field o f characteristic p . W e s how that assuming resolution of sing ular- ities, the gro ups H S i ( X, Z /p r ) are finite ab elian gro ups, and v anish outside ⋆ Supp orted in part by NSF grant N o.09010 21 2 Thomas Geisser the range 0 ≤ i ≤ dim X . Thus Suslin coho mology with finite co efficients is etale cohomolo g y awa y fro m the characteristic, but better b ehav ed than etale cohomolo gy at the characteristic (for example, H 1 et ( A 1 , Z /p ) is not fi- nite). Moreov er, Suslin homology is a birational inv ar iant in the following strong sense: If X has a resolution of singula rities p : X ′ → X which is a n isomorphism o utside o f the op en subset U , then H S i ( U, Z /p r ) ∼ = H S i ( X, Z /p r ). Next w e examine the situation ov er no n-algebra ically closed fields. W e redefine Suslin homo logy a nd c ohomolog y by imp osing Ga lo is descent. Con- cretely , if G k is the absolute Galois group o f k , then we define Ga lois-Suslin homology to b e H GS i ( X, A ) = H − i RΓ ( G k , Cor ¯ k ( ∆ ∗ ¯ k , ¯ X )) , and Ga lois-Suslin cohomolog y to be H i GS ( X, A ) = Ext i G k (Cor ¯ k ( ∆ ∗ ¯ k , ¯ X ) , A ) . Ideally one would like to define Galois- Suslin homo logy via Galo is homolog y , but we ar e not aw are of such a theo r y . With ra tional co e fficie n ts, the newly defined gr oups ag ree with the or iginal gr oups. On the other hand, with finite co efficients prime to the characteris tic, the pro o f o f (1) in [19] carries ov er to show that H i GS ( X, Z /m ) ∼ = H i et ( X, Z /m ). As a corollary w e obtain an isomorphism betw een H GS 0 ( X, Z /m ) a nd the ab elianized fundamental group π ab 1 ( X ) / m for any separ ated X of finite t yp e ov er a finite field. The s econd half of the pap er fo cuses on the c a se of a finite base field. W e work under the assumption of r esolution of singular ities in order to see the picture of the pr op erties which can exp ected. The cr itica l reader c a n view our sta tements a s theorems for schemes of dimens io n a t mos t three, and conjectures in ge neral. A theor em of Jannsen- Saito [11] ca n b e generalize d to show that Suslin homolog y a nd cohomolo gy with finite co efficie nts for a n y X ov er a finite fie ld is finite. Rationally , H S 0 ( X, Q ) ∼ = H 0 S ( X, Q ) ∼ = Q π 0 ( X ) . Most other pro p e r ties ar e equiv alent to the following Conjecture P 0 considered in [7]: F or X smoo th and prop er over a finite field, C H 0 ( X, i ) is torsion for i 6 = 0 . This is a par ticular ca se of Parshin’s co njecture that K i ( X ) is torsio n fo r i 6 = 0 . F or ex ample, Conjecture P 0 is equiv alent to the v anis hing o f H S i ( X, Q ) for i 6 = 0 and a ll smo oth X . F or ar bitr ary X o f dimension d , Conjecture P 0 implies the v anishing of H S i ( X, Q ) outside of the rang e 0 ≤ i ≤ d and its finite dimensionality in this r ange. Combining the tors io n and r ational case, we show that H S i ( X, Z ) and H i S ( X, Z ) ar e finitely gener ated for all X if a nd only if Co njecture P 0 holds. Over a finite field and with integral co efficients, it is mor e natura l to im- po se descent b y the W eil g roup G g enerated by the F ro be nius endo mo rphism instead o f the Galois group [14, 3, 4, 7]. W e define arithmetic homology H ar i ( X, A ) = T or G i (Cor ¯ k ( ∆ ∗ ¯ k , ¯ X ) , A ) and a rithmetic cohomolog y On Suslin’s singular homology and cohomology 3 H i ar ( X, Z ) = Ext i G (Cor ¯ k ( ∆ ∗ ¯ k , ¯ X ) , Z ) . W e chose the notation to distinguis h the groups from weight homolo gy and W eil-etale c o homology consider ed elsewhere . W e show that H ar 0 ( X, Z ) ∼ = H 0 ar ( X, Z ) ∼ = Z π 0 ( X ) and that ar ithmetic homolo gy and cohomolo g y lie in long exact seq uences with Galois-Suslin homology and cohomolog y , resp ec- tively . They a re finitely gener a ted ab elian gro ups if and only if Co njecture P 0 holds. The difference be tween ar ithmetic and Suslin homo lo gy is measured by a third theory , which we call Ka to-Suslin homology , and whic h is defined as H K S i ( X, A ) = H i (Cor ¯ k ( ∆ ∗ ¯ k , ¯ X ) G ). By de finitio n there is a lo ng exac t sequence · · · → H S i ( X, A ) → H ar i +1 ( X, A ) → H K S i +1 ( X, A ) → H S i − 1 ( X, A ) → · · · . It fo llows tha t H K S 0 ( X, A ) = Z π 0 ( X ) for any X . As a gene r alization o f the int egr al version [7] o f Kato’s conjecture [12], we prop ose Conjecture 1.1 The gr oups H K S i ( X, A ) vanish for al l i > 0 and al l sm o oth X . Equiv alent ly tha t there are sho rt e x act s equences 0 → H S i +1 ( ¯ X , Z ) G → H S i ( X, Z ) → H S i ( ¯ X , Z ) G → 0 for all i ≥ 0 and all smo oth X . W e show that this conjecture, to o, is equiv alen t to Conjecture P 0 . This leads us to co njecture o n cla s s field theo r y: Conjecture 1.2 F or every X sep ar ate d and of finite typ e over F q , ther e is a c anonic al inje ction H ar 1 ( X, Z ) → π t 1 ( X ) ab with dense image. It mig ht even b e true that the r e la tive g roup H ar 1 ( X, Z ) ◦ := ker( H ar 1 ( X, Z ) → Z π 0 ( X ) ) is isomo rphic to the geometric par t of the ab elia nized fundamental group defined in SGA 3X § 6. T o supp or t our conjecture, we note that the gen- eralized Kato co njecture a b ove implies H S 0 ( X, Z ) ∼ = H ar 1 ( X, Z ) for smo oth X , so that in this case our co njectur e b ecomes a theore m of Schmidt-Spiess [1 7]. In addition, we show (indep endent of any conjectures ) Prop ositi o n 1 .3 If 1 l ∈ F q , then H ar 1 ( X, Z ) ∧ l ∼ = π t 1 ( X ) ab ( l ) for arbitr ary X . In particula r, the co njectur ed finite genera tion of H ar 1 ( X, Z ) implies the conjecture aw ay fro m the characteristic. W e also give a co nditional r e s ult a t the c har acteristic. Notation: In this pap er , scheme over a field k means s eparated scheme of finite t yp e ov er k . W e thank Uwe Jannsen for interesting dis c ussions re la ted to the sub ject of this paper , a nd Shuj i Saito and T akeshi Saito for helpful comments during a s eries of lectur es I g av e on the topic of this pa per at T okyo University . 4 Thomas Geisser 2 Motivic homology Suslin homo logy H S i ( X, Z ) of a scheme X over a field k is defined as the homology of the globa l sections C X ∗ ( k ) of the c o mplex of etale sheav es C X ∗ ( − ) = Cor k ( − × ∆ ∗ , X ). Here Cor k ( U, X ) is the group o f universal rela tive cycles of U × Y /U [20]. If U is smo o th, then Cor k ( U, X ) is the free ab elian group genera ted by c lo sed irreducible subschemes of U × X which are finite and s ur jective over a c o nnected comp onent of U . More genera lly [1 ], motivic ho mology of weigh t n are the e x tension gr oups in V o evo dsky’s ca tegory of geo metrical mixed motives H i ( X, Z ( n )) = Hom DM − k ( Z ( n )[ i ] , M ( X )) , and a re isomorphic to H i ( X, Z ( n )) = ( H 2 n − i (0) ( A n , C X ∗ ) n ≥ 0 H i − 2 n − 1 ( C ∗  c 0 ( X × ( A n −{ 0 } )) c 0 ( X ×{ 1 } )  ( k )) n < 0 . Here cohomolo g y is taken for the Nisnev ich topo logy . There is an obvious ver- sion with co e fficie nts. Motivic homolog y is a cov ar iant functor on the categ ory of s chemes o f finite type over k , and has the following additional prop erties, see [1 ] (the final three pr op erties r equire resolution o f singularities) a) It is homotopy inv ariant. b) It satisfies a pro jective bundle formula H i ( X × P 1 , Z ( n )) = H i ( X, Z ( n )) ⊕ H i − 2 ( X, Z ( n − 1)) . c) Ther e is a May er-Vietor is long exact sequence fo r o p e n cov ers. d) Given a n abs tract blow-up squa re Z ′ − − − − → X ′   y   y Z − − − − → X there is a lo ng exa ct seque nc e · · · → H i +1 ( X, Z ( n )) → H i ( Z ′ , Z ( n )) → H i ( X ′ , Z ( n )) ⊕ H i ( Z, Z ( n )) → H i ( X, Z ( n )) → · · · (2) e) If X is prop er, then motivic homolo gy ag rees with higher Chow groups, H i ( X, Z ( n )) ∼ = C H n ( X, i − 2 n ). f ) If X is smo oth o f pure dimension d , then motivic ho mo logy agrees with motivic cohomolo gy with compact suppo rt, H i ( X, Z ( n )) ∼ = H 2 d − i c ( X, Z ( d − n )) . On Suslin’s singular homology and cohomology 5 In particular, if Z is a closed subscheme of a smo oth scheme X of pure dimension d , then we have a lo ng exact sequence · · · → H i ( U, Z ( n )) → H i ( X, Z ( n )) → H 2 d − i c ( Z, Z ( d − n )) → · · · . (3) In o r der to remov e the hypo thesis on reso lution o f sing ula rities, it would b e sufficient to find a pro o f of Theorem 5.5 (2 ) of F riedlander-V o evo dsky [1] that do es no t r equire resolution of sing ularities. F o r a ll arguments in this pap er (except the p -part of the Kato co njecture) the sequences (2) and (3) and the existence of a smo oth and pr op er mo del for every function field ar e sufficient. 2.1 Susl in cohomology Suslin cohomolo gy is b y definition the dual of Suslin homology , i.e. for an ab elian group A it is defined as H i S ( X, A ) := Ext i Ab ( C X ∗ ( k ) , A ) . W e hav e H i S ( X, Q / Z ) ∼ = Hom( H S i ( X, Z ) , Q / Z ), and a short exact sequenc e o f ab elian groups gives a lo ng exact sequence of coho mo logy g roups, in particular long exact sequences · · · → H i S ( X, Z ) → H i S ( X, Z ) → H i S ( X, Z /m ) → H i +1 S ( X, Z ) → · · · . (4) and · · · → H i − 1 S ( X, Q / Z ) → H i S ( X, Z ) → H i S ( X, Q ) → H i S ( X, Q / Z ) → · · · . Consequently , H i S ( X, Z ) Q ∼ = H i S ( X, Q ) if Suslin-homolog y is finitely gener - ated. If A is a r ing, then H i S ( X, A ) ∼ = Ext i A ( C X ∗ ( k ) ⊗ A, A ), and we get a sp ectral sequence E s,t 2 = Ext s A ( H S t ( X, A ) , A ) ⇒ H s + t S ( X, A ) . (5) In particular, there are p erfect pa ir ings H S i ( X, Q ) × H i S ( X, Q ) → Q H S i ( X, Z /m ) × H i S ( X, Z /m ) → Z /m. Lemma 2. 1 Ther e ar e natur al p airings H i S ( X, Z ) / tor × H S i ( X, Z ) / tor → Z and H i S ( X, Z ) tor × H S i − 1 ( X, Z ) tor → Q / Z . 6 Thomas Geisser Pr o of. The sp ectral sequence (5) gives a s hort exact s equence 0 → E x t 1 ( H S i − 1 ( X, Z ) , Z ) → H i S ( X, Z ) → Hom( H S i ( X, Z ) , Z ) → 0 . (6) The r esulting map H i S ( X, Z ) / tor ։ Hom( H S i ( X, Z ) , Z ) induces the first pair- ing. Since Ho m( H S i ( X, Z ) , Z ) is torsion free, we obtain the map H i S ( X, Z ) tor ֒ → Ext 1 ( H S i − 1 ( X, Z ) , Z ) ։ Ext 1 ( H S i − 1 ( X, Z ) tor , Z ) ∼ ← Hom( H S i − 1 ( X, Z ) tor , Q / Z ) for the s econd pairing. ✷ 2.2 Com parison to mo tivic cohomolo g y Recall that in the ca tegory D M − k of b o unded a bove complexe s of homotopy inv a riant Nisnevich sheav es with trans fer s, the motive M ( X ) of X is the complex of presheaves with transfers C X ∗ . Since a field has no higher Nisnevich cohomolog y , tak ing glo bal s ections ov er k induces a canonical ma p Hom DM − k ( M ( X ) , A ) → R Hom Ab ( C X ∗ ( k ) , RΓ ( k , A )) , hence a natural map H i M ( X, A ) → H i S ( X, A ) . (7) Even though the coho mology groups do not dep end o n the base field, the map do es. F or example, if L/k is an extension of degr ee d , then the diagra m of groups isomorphic to Z , H 0 M (Spec k , Z ) H 0 S (Spec k , Z )      y × d H 0 M (Spec L, Z ) − − − − → H 0 S (Spec L, Z ) shows that the lower horizontal map is multiplication by d . W e will see b elow that co njecturally (7) is a map b etw een finitely ge ne r ated g r oups which is rationally an isomor phism, and one might ask if its E uler characteristic has any int erpr etation. 3 The mo d p Suslin homology in characteristic p W e examine the p -part of Suslin homolo gy in c hara cteristic p . W e a ssume that k is p erfect and r e solution of sing ula rities exists ov er k in order to o btain stronger r esults. W e first give an auxiliar y res ult on motivic cohomolog y with compact suppo rt: On Suslin’s singular homology and cohomology 7 Prop ositi o n 3 .1 L et d = dim X . a) We have H i c ( X, Z /p r ( n )) = 0 for n > d . b) If k is algebr aic al ly close d, then H i c ( X, Z /p r ( d )) is fin ite, H i c ( X, Q p / Z p ( d )) is of c ofinite typ e, and t he gr oups vanish u n less d ≤ i ≤ 2 d . Pr o of. By induction on the dimension and the lo calization sequence, the state- men t for X and a dense op en subset of X are equiv a lent. Hence replacing X by a smo oth subscheme and then by a smo oth and prop er mo del, we can a ssume that X is smo o th and pr op er. Then a) follows from [8]. If k is a lgebraica lly closed, then H i ( X, Z /p ( d )) ∼ = H i − d ( X N is , ν d ) ∼ = H i − d ( X et , ν d ) , by [8] a nd [13]. Tha t the latter g roup is finite and of co finite type, resp ectively , can b e der ived from [15, Thm.1.1 1], and it v anishes outside of the g iven rang e by r easons of cohomolog ic a l dimens io n. ✷ Theorem 3.2 L et X b e sep ar ate d and of fin ite typ e over k . a) The gr oups H i ( X, Z /p r ( n )) vanish for al l n < 0 . b) If k is algebr aic al ly close d, t hen the gr oups H S i ( X, Z /p r ) ar e finite, the gr oups H S i ( X, Q p / Z p ) ar e of c ofinite typ e, and b oth vanish unless 0 ≤ i ≤ d . Pr o of. I f X is smo oth, then H i ( X, Z /p r ( n )) ∼ = H 2 d − i c ( X, Z /p r ( d − n )) and we conclude by the Pr o p osition. In g eneral, w e can as sume by (2) and induc- tion on the num ber of irreducible co mpo nents that U is integral. P ro ceeding by induction on the dimension, we choose a resolution of singularities X ′ of X , let Z b e the clos e d subscheme of X where the map X ′ → X is not an isomorphism, and le t Z ′ = Z × X X ′ . Then w e conclude by the s equence (2). ✷ Example. If X ′ is the blow up of a smo oth scheme X in a smo oth subscheme Z , then the strict transfor m Z ′ = X ′ × X Z is a pro jective bundle ov er Z , hence by the pro jective bundle formula H S i ( Z, Z /p r ) ∼ = H S i ( Z ′ , Z /p r ) and H S i ( X, Z /p r ) ∼ = H S i ( X ′ , Z /p r ). More generally , we have Prop ositi o n 3 .3 Assu me X has a desingularization p : X ′ → X which is an isomorphism outside of the dense op en su bset U . Then H S i ( U, Z /p r ) ∼ = H S i ( X, Z /p r ) . In p articular, the p - p art of S uslin homolo gy is a bir ational in- variant. The hypothesis is satisfied if X is s mo oth, or if U co nt ains a ll singular po int s of X a nd a resolution of singularities exists which is an is omorphism outside o f the singular p oints. Pr o of. If X is smo oth, then this follows from P rop osition 3.1 and the lo cal- ization se q uence (3). In gener a l, let Z b e the s e t o f p oints where p is not an isomorphism, and consider the cartesian diagram 8 Thomas Geisser Z ′ − − − − → U ′ − − − − → X ′   y   y   y Z − − − − → U − − − − → X . Comparing long e xact sequence (2 ) of the left a nd outer square s , → H S i ( Z ′ , Z /p r ) − − − − → H S i ( U ′ , Z /p r ) ⊕ H S i ( Z, Z /p r ) − − − − → H S i ( U, Z /p r ) →         y → H S i ( Z ′ , Z /p r ) − − − − → H S i ( X ′ , Z /p r ) ⊕ H S i ( Z, Z /p r ) − − − − → H S i ( X, Z /p r ) → we see that H S i ( U ′ , Z /p r ) ∼ = H S i ( X ′ , Z /p r ) implies H S i ( U, Z /p r ) ∼ = H S i ( X, Z /p r ). ✷ Example. If X is a no de, then the blow-up s equence gives H S i ( X, Z /p r ) ∼ = H S i − 1 ( k , Z /p r ) ⊕ H S i ( k , Z /p r ), which is Z /p r for i = 0 , 1 and v anishes o therwise. Reid constructed a nor mal surface with a singular po int who se blow-up is a no de, s howing that the p -pa rt of Suslin homo logy is no t a birationa l inv ar iant for normal schemes. Corollary 3. 4 The higher Chow gr oups C H 0 ( X, i, Z /p r ) and the lo garithmic de Rham-Witt c oho molo gy gr oups H i ( X et , ν d r ) , for d = dim X , ar e bir ational invariants. Pr o of. Suslin ho mology a g rees with higher Chow g roups for pro p er X , and with motivic cohomolog y for smo o th and pr op er X . ✷ Note that integrally C H 0 ( X ) is a birationa l inv ariant, but the higher Chow groups C H 0 ( X, i ) a re generally no t. Suslin a nd V o evods ky [19, Thm.3.1 ] s how that for a smo oth compactifica - tion ¯ X of the smo oth curve X , H S 0 ( X, Z ) is isomor phic to the relative Pic a rd group Pic( ¯ X , Y ) and that all higher Suslin ho mology gro ups v anish. Prop o- sition 3.3 implies tha t the kernel and cokernel of Pic( ¯ X , Y ) → P ic( ¯ X ) ar e uniquely p -divisible. Given U with compa ctification j : U → X , the normal- ization X ∼ of X in U is the affine bundle defined by the integral closure of O X in j ∗ O U . W e ca ll X normal in U if X ∼ → X is an isomor phism. Prop ositi o n 3 .5 If X is normal in the curve U , t hen H S i ( U, Z /p ) ∼ = H S i ( X, Z /p ) . Pr o of. This follows by applying the arg umen t of Pr op osition 3.3 to X ′ the normalizatio n of X , Z the c lo sed subset where X ′ → X is not an isomor - phism, Z ′ = X ′ × X Z and U ′ = X ′ × X U . Since X is nor mal in U , we hav e Z ⊆ U and Z ′ ⊆ U ′ . ✷ On Suslin’s singular homology and cohomology 9 4 Galois prop ert ies Suslin homolog y is cov a riant, i.e. a separ ated map f : X → Y of finite type induces a map f ∗ : Co r k ( T , X ) → Co r k ( T , Y ) b y sending a closed ir reducible subscheme Z o f T × X , finite ov er T , to the subs cheme [ k ( Z ) : k ( f ( Z ))] · f ( Z ) (note that f ( Z ) is closed in T × Y and finite ov er T ). On the o ther hand, Suslin homology is contrav ariant for finite flat maps f : X → Y , b ecaus e f induces a map f ∗ : Cor k ( T , Y ) → Cor k ( T , X ) by comp osition with the gr aph of f in Co r k ( Y , X ) (note that the graph is a universal rela tive cycle in the sense of [20]). W e examine the pr op erties of Suslin homolog y under change o f base-fields. Lemma 4. 1 L et L/k b e a finite extension of fields, X a scheme over k and Y a scheme over L . Then Cor L ( Y , X L ) = Cor k ( Y , X ) and if X is sm o oth, then Cor L ( X L , Y ) = Cor k ( X, Y ) . In p articular, Suslin homolo gy do es not dep end on t he b ase fi eld. Pr o of. The fir st s tatement follows b ecause Y × L X L ∼ = Y × k X . The second statement follows b ecause the map X L → X is finite and separated, hence a closed subscheme of X L × L Y ∼ = X × k Y is finite and surjective ov er X L if and o nly if it is finite and surjective over X . ✷ Given a scheme over k , the g raph o f the pr o jection X L → X in X L × X gives ele ments Γ X ∈ Cor k ( X, X L ) and Γ t X ∈ Cor k ( X L , X ). 4.1 Cov ariance Lemma 4. 2 a) If X and Y ar e sep ar ate d schemes of finite typ e over k , then the t wo maps Cor L ( X L , Y L ) → Co r k ( X, Y ) induc e d by c omp osition and pr e c omp osition, re sp e ctively, with Γ t Y and Γ X agr e e. Both maps send a gener ator Z ⊆ X L × k Y ∼ = X × k Y L to its image in X × Y with multiplicity [ k ( Z ) : k ( f ( Z ))] , a divisor of [ L : k ] . b) If F /k is an infinite algebr aic ex tension, then lim L/k Cor L ( X L , Y L ) = 0 . Pr o of. The firs t part is easy . If Z is of finite t yp e over k , then k ( Z ) is a finitely generated field extens io n of k . F or every c o mpo nent Z i of Z F , we obta in a map F → F ⊗ k k ( Z ) → k ( Z i ), and sinc e F is not finitely g e ne r ated ov er k , neither is k ( Z i ). Hence g o ing up the to wer of finite extensions L /k in F , the degree of [ k ( W L ) : k ( Z )], for W L the comp onent of Z L corres p o nding to Z i , go es to infinity . ✷ 10 Thomas Geisser 4.2 Contra v ariance Lemma 4. 3 a) If X and Y ar e schemes over k , then the two maps Cor k ( X, Y ) → Cor L ( X L , Y L ) induc e d by c omp osition and pr e c omp osition, r esp e ctively with Γ Y and Γ t X agr e e. Both m aps send a gener ator Z ⊆ X × Y to the cycle asso ciate d t o Z L ⊆ X × k Y L ∼ = X L × k Y . If L/ k is sep ar able, this is a sum of the inte gr al subschemes lying over Z with mult iplicity one. If L/ k is Galois with gr oup G , then t he maps induc e an isomorphism Cor k ( X, Y ) ∼ = Cor L ( X L , Y L ) G . b) V arying L , Co r L ( X L , Y L ) forms a et aleshe af on Sp ec k with stalk M = colim L Cor L ( X L , Y L ) ∼ = Cor ¯ k ( X ¯ k , Y ¯ k ) , wher e L runs t hr ough the fin ite extensions of k in an algebr aic closur e ¯ k of k . In p articular, Cor L ( X L , Y L ) ∼ = M Gal( ¯ k/L ) . Pr o of. Again, the firs t par t is ea sy . If L/ k is separable, Z L is finite and eta- leov er Z , hence Z L ∼ = P i Z i , a finite sum of the integral cycles lying over Z with mult iplicity one each. If L/k is moreover Ga lo is, then Cor k ( X, Y ) ∼ = Cor L ( X L , Y L ) G and Cor ¯ k ( X ¯ k , Y ¯ k ) ∼ = colim Cor L ( X L , Y L ) by E GA I V Thm. 8.10.5. ✷ The pro po sition suggests to work with the complex C X ∗ of etale sheav es on Sp ec k given by C X ∗ ( L ) := Cor L ( ∆ ∗ L , X L ) ∼ = Cor k ( ∆ ∗ L , X ) . Corollary 4. 4 We have H S i ( ¯ X , A ) ∼ = colim L H S i ( X L , A ) , and ther e is a sp e c- tr al se quenc e E s,t 2 = lim s H t S ( X L , A ) ⇒ H s + t S ( ¯ X , A ) . The maps in the dir e ct and inverse system ar e induc e d by c ontr avariant func- toriality of Sus lin homolo gy for finite flat maps. Pr o of. This follows from the quasi-iso morphisms R Hom Ab ( C X ∗ ( ¯ k ) , Z ) ∼ = R Hom Ab (colim L C X ∗ ( L ) , Z ) ∼ = R lim L R Hom Ab ( C X ∗ ( L ) , Z ) . ✷ 4.3 Coi n v ariants If G k is the abso lute Ga lois gro up of k , then Cor ¯ k ( ¯ X , ¯ Y ) G k can b e identified with Cor k ( X, Y ) by as so ciating orbits of p oints of ¯ X × ¯ k ¯ Y with their image in On Suslin’s singular homology and cohomology 11 X × k Y . How ever, this identification is ne ither compatible with cov ariant no r with co ntrav aria nt functoriality , a nd in particula r not with the differentials in the complex C X ∗ ( k ). But the obstructio n is torsion, and we ca n r emedy this problem b y tensoring with Q : Define an isomorphism τ : (Cor ¯ k ( ¯ X , ¯ Y ) Q ) G k → Cor k ( X, Y ) Q . as follows. A genera to r 1 ¯ Z corres p o nding to the closed irreducible subscheme ¯ Z ⊆ ¯ X × ¯ Y is s ent to 1 g Z 1 Z , wher e Z is the image of ¯ Z in X × Y and g the nu mber of irreducible comp onents of Z × k ¯ k , i.e. g Z is the size o f the Galois orbit of ¯ Z . Lemma 4. 5 The isomorphism τ is fun ct orial in b oth variables, henc e it in- duc es an isomorphism of c omplexes ( C X ∗ ( ¯ k ) Q ) G k ∼ = C X ∗ ( k ) Q . Pr o of. This can b e pr ov ed by direct verification. W e give an alternate pro o f. Consider the comp osition Cor k ( X, Y ) → Cor ¯ k ( ¯ X , ¯ Y ) G k → Cor ¯ k ( ¯ X , ¯ Y ) G k τ − → Cor k ( X, Y ) Q . The middle map is induced by the identit y , and is mult iplication by g Z on the comp onent co rresp onding to Z . All maps are isomor phisms up on tensoring with Q . The firs t map, the s econd map, and the comp os ition are functorial, hence so is the τ . ✷ 5 Et ale theory Let ¯ k b e the algebraic closure of k with Galois group G k , and let A b e a con- tin uous G k -mo dule. Then C X ∗ ( ¯ k ) ⊗ A is a complex of co ntin uous G k -mo dules, and if k has finite cohomologic a l dimension we define Galo is-Suslin homo logy to b e H GS i ( X, A ) = H − i RΓ ( G k , C X ∗ ( ¯ k ) ⊗ A ) . By c o nstruction, there is a sp ectr a l se quence E 2 s,t = H − s ( G k , H S t ( ¯ X , A )) ⇒ H GS s + t ( X, A ) . The case X = Sp ec k shows that Suslin homology doe s not a gree with eta le- Suslin homology , i.e. Suslin homolo gy do es not hav e Galo is descent. W e define etaleSuslin cohomolo gy to be H i GS ( X, A ) = Ext i G k ( C X ∗ ( ¯ k ) , A ) . (8) This a grees with the old definition if k is algebraica lly closed. Let τ ∗ be the functor from G k -mo dules to contin uous G k -mo dules which sends M to colim L M G L , where L runs thro ug h the finite extens io ns of k . It is eas y to s e e that R i τ ∗ M = colim H H i ( H, M ). 12 Thomas Geisser Lemma 5. 1 We have H i GS ( X, A ) = H i RΓ G k Rτ ∗ Hom Ab ( C X ∗ ( ¯ k ) , A ) . In p ar- ticular, ther e is a sp e ctr al se quenc e E s,t 2 = H s ( G k , R t τ ∗ Hom Ab ( C X ∗ ( ¯ k ) , A )) ⇒ H s + t GS ( X, A ) . (9) Pr o of. This is [16, Ex. 0.8]. Since C X ∗ ( ¯ k ) is a complex of free Z -modules , Hom Ab ( C X ∗ ( ¯ k ) , − ) is exact and preser ves injectives. Hence the der ived func- tor of τ ∗ Hom Ab ( C X ∗ ( ¯ k ) , − ) is R t τ ∗ applied to Ho m Ab ( C X ∗ ( ¯ k ) , − ). ✷ Prop ositi o n 5 .2 If A is a Q -ve ctor sp ac e with trivial G k -action, then H GS i ( X, A ) ∼ = H S i ( X, A ) H i GS ( X, A ) ∼ = H i S ( X, A ) . Pr o of. Since − ⊗ A is exa ct, H S i ( X, A ) = H i ( C X ∗ ( ¯ k ) G k ⊗ A ) as well as H GS i ( X, A ) = H i (( C X ∗ ( ¯ k ) ⊗ A ) G k ) are isomor phic to the homology of the kernel of the ma p of co mplex es C X ∗ ( ¯ k ) ⊗ A ϕ − 1 → C X ∗ ( ¯ k ) ⊗ A. Since higher Galois cohomolog y is tors ion, we have R t τ ∗ Hom( C X i ( ¯ k ) , A ) = 0 for t > 0, and H s ( G k , τ ∗ Hom( C X ∗ ( ¯ k ) , A )) = 0 for s > 0. Hence H i GS ( X, A ) is isomorphic to the i th cohomolog y o f Hom G k ( C X ∗ ( ¯ k ) , A ) ∼ = Hom Ab ( C X ∗ ( ¯ k ) G k , A ) ∼ = Hom Ab ( C X ∗ ( k ) , A ) . The latter equality follows with Lemma 4 .5. ✷ Theorem 5.3 If m is invertible in k and A is a finitely gener ate d m - torsion G k -mo dule, t hen H i GS ( X, A ) ∼ = H i et ( X, A ) . Pr o of. This follows with the ar gument o f Suslin-V o ev o dsky [19]. Indeed, let f : ( S ch/k ) h → E t k be the cano nical ma p fr o m the larg e site with the h- top ology of k to the small eta le site of k . Clearly f ∗ f ∗ F ∼ = F , a nd the pro of of Thm.4.5 in lo c.cit. shows that the cokernel of the injection f ∗ f ∗ F → F is uniquely m - divisible, fo r any homo topy inv aria nt pr esheaf with transfers (like, for example, C X i : U 7→ Cor k ( U × ∆ i , X )). Hence Ext i h ( F ∼ h , f ∗ A ) ∼ = Ext i h ( f ∗ f ∗ F ∼ h , f ∗ A ) ∼ = Ext i E t k ( f ∗ F ∼ h , A ) ∼ = Ext i G k ( F ( ¯ k ) , A ) . Then the a rgument of section 7 in lo c.cit. together with Theorem 6.7 can b e descended from the algebraic closure o f k to k . ✷ On Suslin’s singular homology and cohomology 13 5.1 Duality results Dualit y results fo r the Galois cohomo logy of a field k lea d via theo rem 5.3 to duality r esults betw een Galois -Suslin ho mology and co ho mology over k . Theorem 5.4 L et k b e a finite field, A a fi nite G k -mo dule, and A ∗ = Hom( A, Q / Z ) . Then ther e is a p erfe ct p airing of finit e gr oups H GS i − 1 ( X, A ) × H i GS ( X, A ∗ ) → Q / Z . Pr o of. According to [16, Example 1.10] we hav e Ext r G k ( M , Q / Z ) ∼ = Ext r +1 G k ( M , Z ) ∼ = H 1 − r ( G k , M ) ∗ . Hence Ext r G k ( C X ∗ ( ¯ k ) , Hom( A, Q / Z )) ∼ = Ext r G k ( C X ∗ ( ¯ k ) ⊗ A, Q / Z ) ∼ = H 1 − r ( G k , C X ∗ ( ¯ k ) ⊗ A ) ∗ ∼ = H GS r − 1 ( X, A ) ∗ . ✷ The ca se o f non-torsion sheaves is disc ussed b elow. Theorem 5.5 L et k b e a lo c al field with finite r esidue field and sep ar able closur e k s . F or a finite G k -mo dule A let A D = Hom( A, ( k s ) × ) . Then we have isomorphi sms H i GS ( X, A D ) ∼ = Hom( H GS i − 2 ( X, A ) , Q / Z ) . Pr o of. According to [16, Thm.2.1] w e hav e Ext r G k ( M , ( k s ) × ) ∼ = H 2 − r ( G k , M ) ∗ for every finite G k -mo dule M . The rest o f the pr o of is the same as ab ov e. ✷ 6 Finite base fields F r om now on we fix a finite field F q with algebr aic closure ¯ F q . T o obtain the following r e sults, we assume r esolution of s ingularities. This is nee de d to use the sequenc e s (2) and (3) to reduce to the smo oth and pro jectiv e case on the one hand, and the pro of o f Jannsen-Saito [11] of the Kato conjecture on the other hand (how ever, K erz and Saito announced a pro of of the prime to p -pa rt of the Kato conjectur e which doe s no t requir e reso lution of singularities). The critical r eader is invited to view the following results as co njectur es which are theorems in dimension a t most 3 . W e firs t pres ent r e s ults on finite genera tion in the spirit of [11] and [7]. 14 Thomas Geisser Theorem 6.1 F or any X / F q and any inte ger m , the gr oups H S i ( X, Z /m ) and H i S ( X, Z /m ) ar e finitely gener ate d. Pr o of. I t suffices to consider the case of homolog y . If X is s mo oth and pro p e r of dimension d , then H S i ( X, Z /m ) ∼ = C H 0 ( X, i, Z /m ) ∼ = H 2 d − i c ( X, Z /m ( d )), and the result follows fro m work of Jannsen-Sa ito [11]. The usual devisag e then shows that H j c ( X, Z /m ( d )) is finite for all X and d ≥ dim X , hence H S i ( X, Z /m ) is finite for smo oth X . Finally , one pro ceeds by inductio n on the dimension o f X with the blow-up long-exact sequence to r educe to the case X smo o th. ✷ 6.1 Ratio nal Suslin- homology W e have the following unconditiona l r esult: Theorem 6.2 F or every c onne cte d X , the map H S 0 ( X, Q ) → H S 0 ( F q , Q ) ∼ = Q is an isomorphism. Pr o of. B y inductio n on the n umber o f ir reducible comp onents and (2) we ca n first assume that X is irr educible and then reduce to the situation where X is smo o th. In this ca se, we use (3) and the following P rop osition to reduce to the smo o th and prop er case , where H S 0 ( X, Q ) = C H 0 ( X ) Q ∼ = C H 0 ( F q ) Q . ✷ Prop ositi o n 6 .3 If n > dim X , then H i c ( X, Q ( n )) = 0 for i ≥ n + dim X . Pr o of. By induction o n the dimension and the lo ca lization sequence for motivic cohomolog y with compact supp or t one sees tha t the statement for X and a dense o pen subscheme of X a r e equiv alent. Hence we can assume tha t X is smo oth a nd prop er o f dimensio n d . Compa ring to hig he r Chow gr oups, one sees tha t this v anishes for i > d + n for dimension (o f cy c le s) rea sons. F or i = d + n , we obtain from the niveau sp ectr al s equence a surjection M X (0) H n − d M ( k ( x ) , Q ( n − d )) ։ H d + n M ( X, Q ( n )) . But the summands v anish for n > d b ecause higher Milnor K -theory o f finite fields is to rsion. ✷ By definition, the g roups H i ( X, Q ( n )) v anish for i < n . W e will consider the following conjecture P n of [5 ]: Conjecture P n : F or al l smo oth and pr oje ctive schemes X over the finite field F q , the gr oups H i ( X, Q ( n )) vanish for i 6 = 2 n . This is a specia l case of Parshin’s conjecture: If X is smo o th and pro jective of dimension d , then On Suslin’s singular homology and cohomology 15 H i ( X, Q ( n )) ∼ = H 2 d − i M ( X, Q ( d − n )) ∼ = K i − 2 n ( X ) ( d − n ) and, accor ding to Parshin’s co njecture, the latter K -gr o up v anishes for i 6 = 2 n . By the pr o jective bundle for mula, P n implies P n − 1 . Prop ositi o n 6 .4 a) L et U b e a curve. Then H S i ( U, Q ) ∼ = H S i ( X, Q ) for any X normal in U . b) Assu m e c onje ctur e P − 1 . Then H i ( X, Q ( n )) = 0 for al l X and n < 0 , and if X has a desingularization p : X ′ → X which is an isomorphism outside of t he dense op en subset U , then H S i ( U, Q ) ∼ = H S i ( X, Q ) . In p articular, Su slin homolo gy and higher Chow gr oups of weight 0 ar e bir ational invariant. c) Under c onje ctu r e P 0 , t he gr oups H S i ( X, Q ) ar e fin ite dimensional and vanish u nless 0 ≤ i ≤ d . d) Conje ctu r e P 0 is e quivalent t o the vanishing of H S i ( X, Q ) for al l i 6 = 0 and al l smo oth X . Pr o of. The argument is the sa me as in Theo rem 3.2. T o prov e b), we hav e to show that H i c ( X, Q ( n )) = 0 for n > d = dim X under P − 1 , a nd for c) we hav e to show that H i c ( X, Q ( d )) is finite dimensional and v anishes unless d ≤ i ≤ 2 d under P 0 . By induction on the dimension and the lo ca lization sequence we can assume that X is smo oth and pro jective. In this ca se, the statement is Con- jecture P − 1 and P 0 , resp ectively , plus the fact tha t H S 0 ( X, Q ) ∼ = C H 0 ( X ) Q is a finite dimens ional vector spa ce. The final statement follows from the ex a ct sequence (3) and the v anishing o f H i c ( X, Q ( n )) = 0 for n > d = dim X under P − 1 . ✷ Prop ositi o n 6 .5 Conje ctur e P 0 holds if and only if the map H i M ( X, Q ) → H i S ( X, Q ) of (7) is an isomorphism for al l X/ F q and i . Pr o of. The second statement implies the first, b ecause if the map is an is o- morphism, then H i S ( X, Q ) = 0 fo r i 6 = 0 and X smo oth a nd prop er, and hence so is the dual H S i ( X, Q ). T o show that P 0 implies the seco nd sta tement, first note that b ecause the map is c o mpatible with long exact blow-up seq uences, we can b y inductio n on the dimension a ssume that X is smo oth of dimensio n d . In this case, motivic cohomolo g y v anishes ab ov e degree 0, and the same is true for Suslin coho mology in view of Prop os itio n 6.4d). T o show that for connected X the map (7) is an isomorphism o f Q in degree zero, we cons ider the commut ative diag ram induced by the structure map H 0 M ( F q , Q ) − − − − → H 0 S ( F q , Q )   y   y H 0 M ( X, Q ) − − − − → H 0 S ( X, Q ) This reduces the pro ble m to the c a se X = Spec F q , where it can b e dire ctly verified. ✷ 16 Thomas Geisser 6.2 Integral co efficients Combining the torsion results [11] with the r a tional r esults, we o btain the following Prop ositi o n 6 .6 Conje ctur e P 0 is e quivalent to the finite gener ation of H S i ( X, Z ) for al l X / F q . Pr o of. If X is smo oth and prop er, then according to the main theor em o f Jannsen-Saito [11], the gro ups H S i ( X, Q / Z ) = C H 0 ( X, i, Q / Z ) ar e is o morphic to etale ho mology , and hence finite for i > 0 by the W eil-conjectures . Hence finite generation of H S i ( X, Z ) implies that H S i ( X, Q ) = 0. Conv ersely , we ca n by induction on the dimensio n assume that X is smo oth a nd has a s mo oth and pr o p er model. Expressing Suslin homolog y of s mo oth schemes in terms of cohomology with compact supp ort a nd a gain using induction, it s uffices to show that H i M ( X, Z ( n )) is finitely g enerated for smo oth and prop er X and n ≥ dim X . Using the pro jective bundle for- m ula we can assume that n = dim X , a nd then the statement follows b ecause H i M ( X, Z ( n )) ∼ = C H 0 ( X, 2 n − i ) is finitely ge ne r ated a ccording to [7, Thm 1.1]. ✷ Recall the pair ings of Lemma 2.1. W e call them p erfect if they ident ify one group with the dual of the other group. In the tor sion case, this implies that the g roups ar e finite, but in the free c a se this is not true: F o r ex a mple, ⊕ I Z and Q I Z are in p erfect duality . Prop ositi o n 6 .7 L et X b e a s ep ar ate d scheme of fi nite typ e over a finite field. Then the fol lowing statement s ar e e quivalent: a) The gr oups H S i ( X, Z ) ar e finitely gener ate d for al l i . b) The gr oups H i S ( X, Z ) ar e finitely gener ate d for al l i . c) The gr oups H i S ( X, Z ) ar e c ou n table for al l i . d) The p airings of L emma 2.1 ar e p erfe ct for al l i . Pr o of. a) ⇒ b) ⇒ c) are clear , and c) ⇒ a) follows from [9, Pr op.3F.12], which states that if A is not finitely generated, then either Hom( A, Z ) or Ext( A, Z ) is uncountable. Going throug h the pro of of Lemma 2.1 it is easy to se e that a) im- plies d). Conversely , if the pairing is p er fect, then tor H S i ( X, Z ) is finite. Let A = H i S ( X, Z ) / tor and fix a prime l . Then A/l is a quotient of H i S ( X, Z ) /l ⊆ H i S ( X, Z /l ), and which is finite by Theorem 6 .1. Cho os e lifts b i ∈ A o f a bas is of A/l and let B b e the finitely genera ted free a b elia n subgr oup of A gen- erated by the b i . B y co nstruction, A/B is l -divisible, hence H S i ( X, Z ) / tor = Hom( A, Z ) ⊆ Hom( B , Z ) is finitely g e ne r ated. ✷ On Suslin’s singular homology and cohomology 17 6.3 Alg ebraically closure of a finite fiel d Suslin homolo gy has prop erties simila r to a W eil-co homology theory . Let X 1 be separated and of finite type over F q , X n = X × F q F q n and X = X 1 × F q ¯ F q . F r om Co rollar y 4 .4, we o btain a s hort exact s equence 0 → lim 1 H t +1 S ( X n , Z ) → H t S ( X, Z ) → lim H t S ( X n , Z ) → 0 . The outer terms can b e calculated with the 6-term lim-lim 1 -sequence asso ci- ated to (6 ). The theor em of Suslin and V o evodsky implies that lim H i S ( X, Z /l r ) ∼ = H i et ( X, Z l ) for l 6 = p . F or X is prop er and l = p , w e get the same r esult from [6] H i S ( X, Z /p r ) ∼ = Hom( C H 0 ( X, i, Z/ p r ) , Z /p r ) ∼ = H i et ( X, Z /p r ) . W e show that this is true int egr ally: Prop ositi o n 6 .8 L et X b e a smo oth and pr op er cu rve over the algebr aic clo- sur e of a finite field k of char acteristic p . Then the non- vanishing c ohomolo gy gr oups ar e H i S ( X, Z ) ∼ =      Z i = 0 lim r Hom GS ( µ p r , Pic X ) × Q l 6 = p T l Pic X ( − 1) i = 1 Q l 6 = p Z l ( − 1) i = 2 . The homomorphisms ar e m aps of gr oup schemes. Pr o of. By prop er ne s s and smo othness we hav e H S i ( X, Z ) ∼ = H 2 − i M ( X, Z (1)) ∼ =      Pic X i = 0; k × i = 1; 0 i 6 = 0 , 1 . Now Ext 1 ( k × , Z ) = Hom(colim p 6| m µ m , Q / Z ) ∼ = Y l 6 = p Z l ( − 1) and s inc e P ic X is finitely gener ated by torsio n, Ext 1 (Pic X, Z ) ∼ = Hom(colim m m Pic X , Q / Z ) ∼ = lim Hom GS ( m Pic X , Z /m ) ∼ = lim m Hom GS ( µ m , m Pic X ) by the W eil-pair ing. ✷ 18 Thomas Geisser Prop ositi o n 6 .9 L et X b e smo oth, pr oje ctive and c onne cte d over the alge- br aic closur e of a finite field. Assuming c onje ctur e P 0 , we have H i S ( X, Z ) ∼ = ( Z i = 0 Q l H i et ( X, Z l ) i ≥ 1 . In p articular, the l -adic c ompletion of H i S ( X, Z ) is l -adic c ohomolo gy H i et ( X, Z l ) for al l l . Pr o of. Let d = dim X . By pr op erness a nd smo o thness we have H S i ( X, Z ) ∼ = H 2 d − i M ( X, Z ( d )) . Under hypothesis P 0 , the gro ups H S i ( X, Z ) ar e tor sion for i > 0, and H S 0 ( X, Z ) = C H 0 ( X ) is the pro duct of a finitely gener ated group and a torsio n group. Hence for i ≥ 1 we get by (6 ) that H i S ( X, Z ) ∼ = Ext 1 ( H S i − 1 ( X, Z ) , Z ) ∼ = Hom( H S i − 1 ( X, Z ) tor , Q / Z ) ∼ = Hom( H 2 d − i +1 M ( X, Z ( d )) tor , Q / Z ) ∼ = Hom( H 2 d − i et ( X, Q / Z ( d )) , Q / Z ) ∼ = Hom(colim m H 2 d − i et ( X, Z /m ( d )) , Q / Z ) ∼ = lim m Hom( H 2 d − i et ( X, Z /m ( d )) , Z /m ) . By Poincare-dua lity , the latter agree s w ith lim H i et ( X, Z /m ) ∼ = Q l H i et ( X, Z l ). ✷ 7 Ar ithmetic homology and cohomology W e r e call some definitions a nd results from [3]. Let X b e sepa rated and of finite type ov er a finite field F q , ¯ X = X × F q ¯ F q and G b e the W eil-gro up of F q . Let γ : T G → T ˆ G be the functor from the categor y o f G -mo dules to the ca tegory o f contin uous ˆ G = Gal( F q )-mo dules which as s o ciated to M the mo dule γ ∗ = c o lim m M mG , where the index set is or dered by divisibility . It is easy to see that the forg etful functor is a left adjoint of γ ∗ , hence γ ∗ is left exact and pres erves limits. The der ived functors γ i ∗ v a nish for i > 1, and γ 1 ∗ M = R 1 γ ∗ M = colim M mG , where the tr ansition maps a re g iven by M mG → M mnG , x 7→ P s ∈ mG/mnG sx . Consequently , a complex M · of G - mo dules gives rise to an ex act tr iangle of co n tinuous G k -mo dules γ ∗ M · → Rγ ∗ M · → γ 1 ∗ M · [ − 1] . (10) If M = γ ∗ N is the restriction of a contin uous ˆ G -mo dule, then γ ∗ M = N and γ 1 ∗ M = N ⊗ Q . In particular , W eil-eta le c ohomolog y and etalecoho mology of torsion sheaves a gree. Note that the de r ived functors γ ∗ restricted to the ca t- egory of ˆ G -mo dules do es no t agre e with the derived functors of τ ∗ considered in Lemma 5 .1. Indeed, R i τ ∗ M = colim L H i ( G L , M ) is the colimit of Galois cohomolog y gro ups, whereas R i γ ∗ M = co lim m H i ( mG, M ) is the colimit of cohomolog y g roups of the disc r ete g roup Z . On Suslin’s singular homology and cohomology 19 7.1 Ho m ology W e define ar ithmetic homo logy with co efficients in the G -mo dule A to b e H ar i ( X, A ) := T o r G i ( C X ∗ ( ¯ k ) , A ) . A concrete representativ e is the double complex C X ∗ ( ¯ k ) ⊗ A 1 − ϕ − → C X ∗ ( ¯ k ) ⊗ A, with the left and right term in homolog ical degrees one and zero , r e sp e ctively , and with the F rob enius endomorphism ϕ ac ting diago nally . W e obtain sho rt exact sequences 0 → H S i ( ¯ X , A ) G → H ar i ( X, A ) → H S i − 1 ( ¯ X , A ) G → 0 . (11) Lemma 7. 1 The gr oups H ar i ( X, Z /m ) ar e finite. In p articular, H ar i ( X, Z ) /m and m H ar i ( X, Z ) ar e finite. Pr o of. The first statement follows from the short exa ct sequence (11 ). Indeed, if m is prime to the characteristic, then we apply (1) to gether with finite generation o f etale cohomology , and if m is a p ower of the characteristic, we apply Theore m 3 .2 to obtain finiteness of the outer terms o f (11). The final statements follows from the long exact s e q uence · · · → H ar i ( X, Z ) × m − → H ar i ( X, Z ) → H ar i ( X, Z /m ) → · · · ✷ If A is the r estriction of a ˆ G -mo dule, then (10) applied to the complex of contin uous ˆ G -mo dules C X ∗ ( ¯ k ) ⊗ A , gives a long exact s equence · · · → H GS i ( X, A ) → H ar i +1 ( X, A ) → H GS i +1 ( X, A Q ) → H GS i − 1 ( X, A ) → · · · With rational co efficients this sequence breaks up into iso morphisms H ar i ( X, Q ) ∼ = H S i ( X, Q ) ⊕ H S i − 1 ( X, Q ) . (12) 7.2 Coho mology In a nalogy to (8), we define arithmetic cohomolo gy with co efficients in the G -mo dule A to b e H i ar ( X, A ) = Ext i G ( C X ∗ ( ¯ k ) , A ) . (13) Note the difference to the definition in [14], which do e s not give well-behav ed (i.e. finitely g enerated) groups for s chemes which are not smo oth and pro p e r. A concrete representativ e is the double complex 20 Thomas Geisser Hom( C X ∗ ( ¯ k ) , A ) 1 − ϕ − → Hom( C X ∗ ( ¯ k ) , A ) , where the left and rig ht hand term ar e in cohomolog ical degr ees ze ro and one, resp ectively . There are short exa ct sequences 0 → H i − 1 S ( ¯ X , A ) G → H i ar ( X, A ) → H i S ( ¯ X , A ) G → 0 . (14) The pro of of Lemma 7.1 also s hows Lemma 7. 2 The gr oups H i ar ( X, Z /m ) ar e finite. In p articular, m H i ar ( X, Z ) and H i ar ( X, Z ) /m ar e finite. Lemma 7. 3 F or every G -mo dule A , we have an isomorphism H i ar ( X, A ) ∼ = H i GS ( X, Rγ ∗ γ ∗ A ) . Pr o of. Since M G = ( γ ∗ M ) ˆ G , W eil-Suslin coho mology is the Galois co homology of the der ived functor of γ ∗ Hom Ab ( C X ∗ ( ¯ k ) , − ) on the catego ry o f G -mo dules. By Lemma 5.1, it suffices to show that this derived functor agrees with the derived functor of τ ∗ Hom Ab ( C X ∗ ( ¯ k ) , γ ∗ − ) on the c ategory of G -mo dules. But given a contin uous ˆ G -mo dules M and a G -mo dule N , the inclusion τ ∗ Hom Ab ( M , γ ∗ N ) ⊆ γ ∗ Hom Ab ( γ ∗ M , N ) induced by the inclusion γ ∗ N ⊆ N is a n is omorphism. Indeed, if f : M → N is H -inv aria nt and m ∈ M is fixed by H ′ , then f ( m ) is fixed by H ∩ H ′ , hence f factor s thr ough γ ∗ N . ✷ Corollary 7. 4 If A is a c ontinuous ˆ G -mo dule, then ther e is a long exact se quenc e · · · → H i GS ( X, A ) → H i ar ( X, A ) → H i − 1 GS ( X, A Q ) → H i +1 GS ( X, A ) → · · · . Pr o of. This fo llows from the Lemma by a pplying the lo ng exact Ext ∗ ˆ G ( C X ∗ ( ¯ k ) , − )- sequence to (10). ✷ 7.3 Fini te generation and duality Lemma 7. 5 Ther e ar e natur al p airings H i ar ( X, Z ) / tor × H ar i ( X, Z ) / tor → Z and H i ar ( X, Z ) tor × H ar i − 1 ( X, Z ) tor → Q / Z . On Suslin’s singular homology and cohomology 21 Pr o of. F rom the adjunction Hom G ( M , Z ) ∼ = Hom Ab ( M G , Z ) and the fact that L ( − ) G = R ( − ) G [ − 1], we o btain by deriving a q ua si-isomor phism R Hom G ( C X ∗ ( ¯ k ) , Z ) ∼ = R Hom Ab ( C X ∗ ( ¯ k ) ⊗ L G Z , Z ) . Now we obtain the pair ing as in Lemma 2 .1 using the r esulting sp ectra l se- quence Ext s Ab ( H ar t ( X, Z ) , Z ) ⇒ H s + t ar ( X, Z ) . ✷ Prop ositi o n 7 .6 F or a given sep ar ate d scheme X of fi nite typ e over F q , the fol lowing statement s ar e e quivalent: a) The gr oups H ar i ( X, Z ) ar e finitely gener ate d. b) The gr oups H i ar ( X, Z ) ar e finitely gener ate d. c) The gr oups H i ar ( X, Z ) ar e c ou n table. d) The p airings of L emma 7.5 ar e p erfe ct. Pr o of. This is proved exactly as Prop os itio n 6.7, with Theor em 6.1 replaced by Le mma 7.1. ✷ W e need a W e il-version of motivic cohomolog y with compac t supp or t. W e define H i c ( X W , Z ( n )) to b e the i th cohomo logy o f RΓ ( G, R Γ c ( ¯ X , Z ( n ))), where the inner ter m is a complex defining motivic cohomolog y with com- pact supp ort of ¯ X . W e use this notation to distinguish it from arithmetic homology with compact supp ort co nsidered in [4], which is the cohomolo gy of RΓ ( G, RΓ c ( ¯ X et , Z ( n ))). Ho wev er, if n ≥ dim X , which is the case of mos t impo rtance for us, both theor ies agree. Similar to (3) w e obta in for a clo sed subscheme Z of a smo o th scheme X of pure dimension d with op en complement U a long exact sequence · · · → H ar i ( U, Z ) → H ar i ( X, Z ) → H 2 d +1 − i c ( Z W , Z ( d )) → · · · . (15) The shift b y 1 in deg rees o ccurs b eca use ar ithmetic homology is defined using homology of G , wher e a s cohomo logy with co mpact supp or t is defined using cohomolog y o f G . Prop ositi o n 7 .7 The fol lowing st atements ar e e quivalent: a) Conje ctur e P 0 . b) The gr oups H ar i ( X, Z ) ar e finitely gener ate d for al l X . Pr o of. a ) ⇒ b): By induction on the dimension of X and the blow-up square, we ca n a ssume tha t X is smo oth of dimension d , where H ar i ( X, Z ) ∼ = H 2 d +1 − i c ( X W , Z ( d )) . 22 Thomas Geisser By lo calization for H ∗ c ( X W , Z ( d )) and induction on the dimension we can reduce the questio n to X smo oth and pr o jective. In this case Z ( d ) has etale hyp e rcohomolo gical descent ov er an a lgebraica lly closed field by [6], hence H j c ( X W , Z ( d )) ag rees with the W eil- e tale cohomolo gy H j W ( X, Z ( d )) considered in [3]. Thes e groups are the finitely genera ted for i > 2 d by [3, Thm.7.3,7.5]. By co njecture P 0 , a nd the iso mo rphism H i W ( X, Z ( d )) Q ∼ = C H 0 ( X, 2 d − i ) Q ⊕ C H 0 ( X, 2 d − i + 1) Q of Thm.7.1c) lo c.cit., these g roups ar e torsion for i < 2 d , so that the finite gro up H i − 1 ( X et , Q / Z ( d )) surjects onto H i W ( X, Z ( d )). Finally , H 2 d W ( X, Z ( d )) is an extension o f the finitely g enerated group C H 0 ( ¯ X ) G by the finite gr o up H 2 d − 1 ( ¯ X et , Z ( d )) G ∼ = H 2 d − 2 ( ¯ X et , Q / Z ( d )) G . b) ⇒ a) Co nsider the sp ecial case that X is smo oth and pro jective. Then as ab ove, H ar i ( X, Z ) ∼ = H 2 d +1 − i W ( X, Z ( d )). If this gr oup is finitely g enerated, then we obtain from the co efficient sequence that H 2 d +1 − i W ( X, Z ( d )) ⊗ Z l ∼ = lim H 2 d +1 − i ( X et , Z /l r ( d )), and the latter group is tor sion for i > 1 by the W eil-c onjectures. Now use (12). ✷ Theorem 7.8 F or c onne ct e d X , the m ap H ar 0 ( X, Z ) → H ar 0 ( F q , Z ) ∼ = Z is an isomorphi sm. In p articular, we have H ar 0 ( X, Z ) ∼ = Z π 0 ( X ) . Pr o of. The pro of is s imilar to the pro of of Theor em 6 .2. Aga in we use induc- tion on the dimensio n and the blow-up sequence to reduce to the situation where X is ir reducible and smo o th. In this case, we can use (15) a nd the fol- lowing Prop ositio n to re duce to the smo oth and pr op er ca se, wher e we hav e H ar 0 ( X, Z ) = C H 0 ( ¯ X ) G ∼ = Z . ✷ Prop ositi o n 7 .9 If n > dim X , then H i c ( X W , Z ( n )) = 0 for i > n + dim X . Pr o of. By induction o n the dimension and the lo ca lization sequence for motivic cohomolog y with compact supp or t one sees tha t the statement for X and a dense o pen subscheme of X a r e equiv alent. Hence we can assume tha t X is smo oth a nd prop er of dimension d . In this case , H i c ( X W , Z ( n )) is a n extensio n of H i M ( ¯ X , Z ( n )) G by H i − 1 M ( ¯ X , Z ( n )) G . These g roups v anish for i − 1 > d + n for dimensio n (of cycles) reasons. F or i = d + n + 1 , we hav e to show that H d + n M ( ¯ X , Z ( n )) G v a nishes. F rom the niveau sp ectr al sequence for motivic cohomolog y we obtain a sur jection M ¯ X (0) H n − d M ( k ( x ) , Z ( n − d )) ։ H d + n M ( ¯ X , Z ( n )) . The s ummands ar e isomo rphic to K M n − d ( ¯ F q ). If n > d + 1, then they v a n- ish beca use higher Milnor K -theor y of a lg ebraically closed fields v anishes. If n = d + 1, then the s ummands are is omorphic to ( ¯ F q ) × , whos e coinv aria nts v a nish. ✷ On Suslin’s singular homology and cohomology 23 8 A Kato type homology W e construc t a homolo gy theory mea suring the difference b etw een Suslin homology and arithmetic homo logy . The co homologic a l theory can b e de- fined analog ously . Kato-Suslin-homo logy with co efficients in the G -mo dule A , H K S i ( X, A ) is defined as the i th homolog y of the complex of coinv ariants ( C X ∗ ( ¯ k ) ⊗ A ) G . If A is triv ial as a G - mo dule, then since ( C X ∗ ( ¯ k ) ⊗ A ) G ∼ = C X ∗ ( k ) ⊗ A , we get the short exact sequence of co mplexes 0 → C X ∗ ( k ) ⊗ A → C X ∗ ( ¯ k ) ⊗ A 1 − ϕ − → C X ∗ ( ¯ k ) ⊗ A → ( C X ∗ ( ¯ k ) ⊗ A ) ϕ → 0 and hence a long exac t sequence · · · → H S i ( X, A ) → H ar i +1 ( X, A ) → H K S i +1 ( X, A ) → H S i − 1 ( X, A ) → · · · . By Theor e m 7.8 we hav e H K S 0 ( X, Z ) ∼ = H ar 0 ( X, Z ) ∼ = Z π 0 ( X ) . The following is a g eneralizatio n of the integral version [7] of Kato’s co njecture [1 2]. Conjecture 8.1 (Gener alize d int e gr al Kato-c onje ctu r e) If X is smo oth, then H K S i ( X, Z ) = 0 for i > 0 . Equiv alent ly , the ca nonical map H S i ( X, Z ) ∼ = H ar i +1 ( X, Z ) is a n isomor - phism for all smo o th X and all i ≥ 0, i.e . there a re short exact s equences 0 → H S i +1 ( ¯ X , Z ) G → H S i ( X, Z ) → H S i ( ¯ X , Z ) G → 0 . Theorem 8.2 Conje ctu r e 8.1 is e qu ivalent to c onje ctu r e P 0 . Pr o of. If Conjecture 8.1 holds, then H S i ( X, Q ) ∼ = H ar i +1 ( X, Q ) ∼ = H S i +1 ( X, Q ) ⊕ H S i ( X, Q ) implies the v a nishing o f H S i ( X, Q ) for i > 0 . Conv ersely , we first cla im that fo r smo oth and prop er Z , the canonica l map H i c ( Z, Z ( n )) → H i c ( Z W , Z ( n )) is an isomorphism for all i if n < dim Z , and for i ≤ 2 n if n = dim Z . Indee d, if n ≥ dim Z then the cohomology o f Z ( n ) agrees with the etale hyper cohomolog y of Z ( n ), see [6], hence satisfies Galois descent. But accor ding to (the pr o of of ) Pro p o sition 6 .4 b), these groups are torsion g roups, so that Galois desce n t RΓ G k agrees with RΓ G . Using lo calization for cohomolo gy with compact supp ort and induction on the dimension, w e ge t next that H i c ( Z, Z ( n )) ∼ = H i c ( Z W , Z ( n )) for all i and all Z with n < dim Z . Now choose a smo oth and pro per compactifica- tion C of X . Comparing the exact sequence s (3) and (1 5), we see with the 5-Lemma that the isomorphism H S i ( C, Z ) ∼ = H 2 d − i c ( C, Z ( d )) → H ar i +1 ( C, Z ) ∼ = H 2 d − i c ( C W , Z ( d )) for C implies the same isomorphism for X and i ≥ 0. ✷ 24 Thomas Geisser 9 T amely ramified class field theory W e pr o p o se the following conjecture relating W eil-Suslin homolog y to class field theory: Conjecture 9.1 (T ame r e cipr o city c onje ctur e) F or any X sep ar ate d and of finite typ e over a finite fi eld, ther e is a c anonic al inje ction t o the tame ab elian- ize d fundamental gr oup with dense image H ar 1 ( X, Z ) → π t 1 ( X ) ab . Note that the gr oup H ar 1 ( X, Z ) is conjecturally finitely generated. At this po int , we do no t have an explicit construction (asso cia ting elements in the Galois groups to algebr aic cycles) of the map. One might even hop e that H ar 1 ( X, Z ) ◦ := ker( H ar 1 ( X, Z ) → Z π 0 ( X ) is finitely genera ted a nd iso mo rphic to the a belia nized geometr ic pa rt of the tame fundamental gr oup defined in SGA 3 X § 6. Under Co njecture 8.1, H S 0 ( X, Z ) ∼ = H ar 1 ( X, Z ) for smo oth X , and Conjec- ture 9.1 is a theor em of Schmidt-Spiess [17]. Prop ositi o n 9 .2 a) We have H ar 1 ( X, Z ) ∧ l ∼ = π t 1 ( X ) ab ( l ) . In p articular, the prime to p -p art of Conje ctur e 9.1 holds if H ar 1 ( X, Z ) is finitely gener ate d. b) The analo g statement holds for t he p - p art if X has a c omp actific ation T which has a desingularization which is an isomorphism outside of X . Pr o of. a) By Theo r em 7.8, H ar 0 ( X, Z ) contains no divisible subgroup. 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