On the vanishing of negative K-groups

On the vanishing of negativ e K -groups Thomas Geisser · Lars Hesselholt Abstract W e show that for a d -dimensional scheme X essentially of finite type over an infinite perfect field k of characteristic p > 0, the negati ve K -groups K q ( X ) vanish for q < − d provided that strong resolution of singularities holds ov er the field k . Keywords Negati ve K -group s · topolog ical cyclic hom ology · cdh-topo logy Mathematics Subject Classification (2000 ) 19 D35 · 14F20 · 19D55 Introduction A conjectu re of W eibel [ 22, Que stion 2 .9] pre dicts that for every noetherian scheme X , the negative K -gro ups K q ( X ) vanish for q < − dim ( X ) . It was proved recently by Corti ˜ nas, Hae semeyer , Schlichting, and W eibel [4, Theorem 6 .2] that th e co njecture holds if X is essentially of finite typ e over a field o f char acteristic 0. In th is p aper, we prove similarly that the conjecture holds if X is essentially of finite ty pe over an infinite per fect field k of ch aracteristic p > 0 provided that stro ng resolution of singularities holds ov er k . Th e proofs ar e by comparison with Con nes’ cyclic ho mol- ogy [1 8] and the topo logical cyclic ho mology o f B ¨ oksted t, Hsiang , and Madsen [ 3], respectively . W e say that stro ng resolutio n of singularities holds over k if for ev ery integral scheme X separated and of finite type over k , there exists a sequence of blo w-ups X r → X r − 1 → · · · → X 1 → X 0 = X The first aut hor was suppor ted in part by NSF Grant No. 090 1021 and by the JSPS. The second aut hor recei ved partial support from NSF Grant No. 0306519. Thomas Geisser Uni versi ty of Southern Californi a, Los Angeles, Californ ia E-mail: geisser@usc.e du Lars Hesselholt Nagoya Uni versity , Nagoya, Japan E-mail: larsh@math.na goya- u.ac.jp 2 such that the r educed scheme X red r is smooth ov er k ; the center Y i of the blo w-up X i + 1 → X i is connected an d smoo th over k ; the clo sed e mbeddin g of Y i in X i is no r- mally flat; an d Y i is no where d ense i n X i . Strong re solution of singu larities holds over fields of characteristic zero by Hironaka [15, T heorem 1*]. I n g eneral, the field k must necessarily b e p erfect. W e say that a sch eme is essentially of finite type over k if it can be covered by fin itely m any affine open subsets of the fo rm Spec S − 1 A with A a finitely genera ted k -algeb ra and S ⊂ A a multiplicative subset. The fo llowing r esult was conjectured by W eibel [22, Question 2.9]: Theorem A. Let k be an infinite perfect field of characteristic p > 0 such th at str o ng r e solution of sing ularities ho lds over k , a nd let X be a d -d imensional scheme essen- tially of finite type over k. Then K q ( X ) vanishes for q < − d . In genera l, the group K − d ( X ) is non-zero. F or instance, by closed Mayer-V ietor is, the grou p K − d ( ∂ ∆ d + 1 k ) is readily seen to be an infinite cyclic group. Th erefore , the vanishing re sult above is optimal. W e further show in The orem 5.3 belo w th at, under the assumptio n that strong r esolution of singu larities h olds over all infinite perfec t fields of chara cteristic p > 0 , the con clusion of Th eorem A is tr ue for any sche me X of finite type over a ny field of characteristic p . T o prove Theorem A, we consider the cyclotomic trace map tr : K ( X ) → { TC n ( X ; p ) } from the non-co nnective Bass comp lete K -theor y spectrum of X to the topolog ical cyclic h omolog y pro -spectrum of X an d define the pro-spectru m { F n ( X ) } to be the lev el-wise map ping fiber ; comp are [7, Section 1]. Then { F n ( − ) } defin es a presheaf of pro-spectra on the cate gory S ch / X of schemes s eparated and o f finite type over X . W e first show that, in the situation of Theore m A, this p resheaf satisfies descent with respect to the cdh-topolog y of V oev odsky [1 9]. Theorem B. Let k be a n infi nite perfect fie ld of po sitive characteristic p such that str ong resolution o f singularities holds over k , an d let X b e a scheme essentially o f finite type over k. Then for all integ ers q , the canonica l map { F n q ( X ) } → { H − q cdh ( X , F n ( − )) } is an isomorphism of pr o- abelian gr oup s. W e rem ark that if the dimension of X is zero then T heorem B reduce s to the statement that th e map { F n q ( X ) } → { F n q ( X red ) } in duced by the canonical inclu sion is an isomorph ism of pro -abelian gr oups. This statement, in turn, is a special case of the general fact that { F n q ( − ) } is inv ariant with respect to nilpoten t extension s of unital associativ e F p -algebras [7, T heorem B]. It would be very inter esting to similarly extend Theorem B to a statement v alid for all unital associati ve F p -algebras. T o prove Theorem B we gen eralize o f a theorem o f Corti ˜ n as, Haesemeyer , Schlicht- ing, and W eibel [4, Theo rem 3.12] to show that if strong resolutio n of singularities holds ov er the infinite perfect field k , then a p resheaf o f pro-spectra satisfies cdh- descent fo r schem es essentially of finite type over k , provide d that it takes infin ites- imal thinkenin gs to weak equ iv alen ces and finite abstract blow-up sq uares to homo- topy cartesian squares, and provided further that the individual preshea ves o f spectra 3 satisfy Nisnevich descent and take squar es defined by blo w-ups along r egular embed- dings to homotopy cartesian squares. The presheaf { F n ( − ) } satisfies all four proper- ties ac cording to theore ms of Th omason [20, The orem 2.1] and [21, Th eorem 10.8 ], Blumberg and Man dell [ 2, Theorem 1.4], an d the au thors [7, Theorem B and D]. Hence, Theor em B follows. In the situation o f T heorem A, every cdh-covering of X admits a refinemen t to a cdh-covering by sch emes essentially smooth over k . T o gether with a co homolo gical dimension result o f Su slin and V oevodsky [19, Theore m 12.5] this sho ws that the group H − q cdh ( X , K ( − )) vanishes for q < − d . T herefor e, in v iew of Theor em B, to prove Th eorem A, it suffi ces to prove the fo llowing r esult. Theorem C. Let k be a perfect fie ld of po sitive char acteristic p such tha t r esolutio n of sin gularities ho lds over k, and let X be a d -dimensiona l scheme es sentially of finite type over k. Then the canon ical map { TC n q ( X ; p ) } → { H − q cdh ( X , TC n ( − ; p )) } is an isomorphism of p r o-a belian gr oups for q < − d , and an ep imorphism o f pr o - abelian gr o ups for q = − d . W e note that Theorem C uses the weaker assumptio n that resolution o f singulari- ties ho lds over k : Ev ery integral k - scheme separated and of finite t ype a dmits a p roper bi-rationa l mo rphism p : X ′ → X from a smooth k -schem e. W e expect the map in the statement of Th eorem C to b e an isomor phism of pro-abelian group s for q 6 − d , an d an epimorp hism f or q = − d + 1. T o prove Th eorem C, we take advantage of th e fact th at topolo gical cyclic ho- mology , as opposed to K -th eory , satisfies ´ etale descent. This imp lies that, to prove Theorem C, we may replace the cdh-topolog y by the fi ner eh-topolo gy defined in [ 6, Definition 2.1 ] to be the smallest Grothend ieck top ology on Sch / X for which both ´ etale and cd h-coverings are coverings. The proof of Theorem C is then completed by a careful co homolo gical analysis of the eh-sheaves a eh TC n q ( − ; p ) associated with the presheaves of ho motopy g roups TC n q ( − ; p ) = π q TC n ( − ; p ) in co mbination with the following cohomological dimension r esult. Theorem D. Let X be a sc heme essentially of finite type over a field k of characteris- tic p > 0 . Th en the p -coho mological dimension of X with respect to the eh -to pology is less than or equal to dim ( X ) + 1 . W e remark th at Th eorems B and C both concern p re-sheaves of pro-spectr a. In- deed, we do not know whether the analog of Theorem C holds if the p re-sheaf of pro-spec tra { TC n ( − ; p ) } is replaced by th e pre -sheaf o f spectra TC ( − ; p ) given b y the homotopy limit. Our recen t p aper [ 7] was written primar ily with the purp ose of proving Theorem B above. Let X b e a noetherian sch eme. W e define Sch / X to be the category of schemes separated and of fin ite type over X and denote by a τ : ( Sch / X ) ∧ → ( Sch / X ) ∼ τ and i τ : ( Sch / X ) ∼ τ → ( Sch / X ) ∧ the sheafification fu nctor and the inclu sion functor, re- spectiv ely , between the categories of pre sheav es an d τ -sheaves of sets. W e further denote by α : ( Sch / X ) eh → ( Sch / X ) et the canonical morph ism of sites. 4 1 cdh -descent In this sectio n, we for mulate and prove a generalization of [4, Theo rem 3.12] to pre-sheaves of pro -spectra. W e apply this theor em to prove Theore m B of the intro- duction. W e first recall some definitio ns. Let X be a noeth erian scheme, and let F ( − ) be a p resheaf of fibrant symmet- ric spe ctra on Sch / X . If τ is a Grothend ieck topolo gy o n Sch / X that has eno ugh points, we define the h yperco homolo gy spectrum H · τ ( X , F ( − )) to be the Gode ment- Thomason constructio n [2 0, Definition 1.33] of the site ( Sch / X ) τ with coefficients in the pr esheaf F ( − ) ; see also [8, Section 3.1] . W e recall fro m [8, Proposition 3.1.2] that, in this situation, there is a condition ally c onv ergent spectral sequen ce E 2 s , t = H − s τ ( X , a τ F t ( − )) ⇒ H − s − t τ ( X , F ( − )) from the shea f coh omolog y g roups of X with coefficients in the τ -sheaf on Sch / X associated with the p resheaf F t ( − ) = π t ( F ( − )) and with abutment the homotopy group s H − q τ ( X , F ( − )) = π q H · τ ( X , F ( − )) . W e next r ecall th e cdh-topolo gy on Sch / X f rom [19]. W e say that the cartesian square of X -schemes Z ′ i ′ / / p ′   Y ′ p   Z i / / Y is an abstract blo w-up squ ar e if i is a closed immersion an d p is a proper map that in- duces an isomorphism of Y ′ r Z ′ onto Y r Z . W e s ay that the square is a finite abstract blow-up squa r e if it is an abstract blow-up square and the p roper map p is finite. W e say that the squ are is an elem entary Nisnevich squ are if i is an op en immersion and p is an ´ etale map that in duces an isomorphism of Y ′ r Z ′ onto Y r Z . W e say that i : Z → Y is an infinitesimal thick ening if i is a closed immer sion and the correspond - ing quasi-coherent ideal I ⊂ O Y is nilpotent. Th e cd h-topolog y on Sch / X is defined to be the smallest Grothe ndieck topolog y such that for e very abstract blo w-up squ are and every elementar y Nisne vich square, the family of morph isms { p : Y ′ → Y , i : Z → Y } is a cov ering of Y . In particular , the closed covering of the scheme Y by its irredu cible compon ents is a cdh-covering as is th e closed immersion Y red → Y . For the purpose o f this paper, we d efine a pr o- spectrum to be a functor fr om the partially ordered set of positive integers viewed as a category with a sing le m orphism from m to n , if n 6 m , to the categor y of fibrant symmetric spectra, a nd we define a strict map of pr o-spectra to be a na tural transfor mation. W e define the strict map of pr o-spectra f : { X n } → { Y n } to be a weak equiva lence if for every integer q , the induced map of homotopy groups f ∗ : { π q ( X n ) } → { π q ( Y n ) } is an isomorp hism of p ro-abe lian grou ps. This, we recall, means that for e very n , there exists m > n , such that th e maps induced by the structu re map s f rom the kernel a nd 5 cokernel of the m ap f m ∗ : π q ( X m ) → π q ( Y m ) to the kernel and cokern el, respectiv ely , of the map f n ∗ : π q ( X n ) → π q ( Y n ) are both zero . W e say that the squ are diagram of strict maps of pro-sp ectra { X n } / /   { Y n }   { Z n } / / { W n } is homotop y ca rtesian if the canonical map { X n } → { h olim ( Y n → W n ← Z n ) } is a weak equiv alence. The following r esult generalizes [11, Theorem 6.4] and [4, Theorem 3.12]. Theorem 1. 1. Let k b e an infinite p erfect fi eld such th at str o ng resolution of singu- larities holds over k , and let { F n ( − ) } be a pr esheaf of pr o-spec tra on the category of schemes e ssentially of fi nite type over k . Assume that { F n ( − ) } takes infi nitesi- mal thick enings to weak equivalences and finite abstract blow-u p squa r es to hom o- topy cartesian sq uar es. Assume fu rther tha t each F n ( − ) takes elemen tary Nisnevich squar es and squar es associated with blow-ups alo ng r e gular embeddin gs to ho mo- topy cartesian squar es. Then the canonical map defines a weak equivalence { F n ( X ) } ∼ − → { H · cdh ( X , F n ( − )) } of pr o-sp ectra fo r every sc heme X essentially of finite type over k. Pr oo f. The proof in outline i s analogous to the p roof of [1 1, Th eorem 6 .4]. But som e extra care is ne eded, sin ce for a map { A n ( − ) } → { B n ( − ) } o f pro-objects in the cate- gory of she av es of abelian gr oups on the categor y of schemes essentially o f finite type over k to be an isomorphism, it does not suffice to show that for every such scheme X and every po int x ∈ X , the map { A n ( − ) X , x } → { B n ( − ) X , x } o f the pro-abelian groups of stalks is a n isomo rphism. Instead, o ne must show that for every scheme X essen- tially of finite type over k an d e very p oint x ∈ X , there exists a Zariski open neigh- borho od x ∈ U ⊂ X such that the map { A n ( U ) } → { B n ( U ) } is an iso morph ism of pro-ab elian groups. W e p oint out the necessary changes in the proof of loc. cit. First, it follows from [4, Cor ollary 3.9 ] that fo r every scheme X smooth over k and positive integer n , the canonical map defines a weak equiv alence of spe ctra F n ( X ) ∼ − → H · cdh ( X , F n ( − )) . In particular, the canon ical map defines a weak equiv a lence of pro-spectra { F n ( X ) } ∼ − → { H · cdh ( X , F n ( − )) } . Now , it f ollows verbatim from the p roof of [11, Pro position 3.1 2] that the statement holds f or every schem e X wh ich is a normal cro ssing scheme over k in the sen se of loc. cit., Definition 3.10. Next, let X b e a Coh en-Macaulay scheme over k and let D ⊂ X b e an in tegral subscheme alon g wh ich X is nor mally flat. Let X ′ be the blow-up of X alo ng D an d 6 let D ′ be the exceptional fiber . W e indicate the change s necessary to the pro of of op. cit., Theorem 5.7, in order to show that the square of pro-spe ctra { F n ( D ′ ) } { F n ( X ′ ) } o o { F n ( D ) } O O { F n ( X ) } o o O O is homotopy cartesian . I t follows f rom lo c. cit., Proposition 5.4, that af ter replacing X by a Zariski o pen neighb orhoo d of a g iv en poin t x ∈ X , th ere exists a r eduction ˜ D of D in the sense of loc . cit., Definition 5.1, such th at ˜ D is regu larly embedde d in X . Let X ˜ D be the b low-up of X alo ng ˜ D and let ˜ D ′ be the exceptio nal fiber . W e co nsider the fo llowing diagram of schemes, wher e every square is cartesian, and the induced diagram of pro-sp ectra. D ′   / /   ˜ D ′′   / /   X ′   { F n ( D ′ ) } { F n ( ˜ D ′′ ) } o o { F n ( X ′ ) } o o ˜ D ′   / /   X ˜ D   { F n ( ˜ D ′ ) } O O { F n ( X ˜ D ) } o o O O D   / / ˜ D   / / X { F n ( D ) } O O { F n ( ˜ D ) } o o O O { F n ( X ) } o o O O The lower right-han d square in the left-ha nd d iagram is a blow-up along a r egular embedd ing, an d therefo re, the lower right-h and square in the r ight-han d diag ram is homoto py cartesian by assumption. Similar ly , the upper right-hand squa re in the left- hand diagr am is a fin ite abstra ct blow-up square, and th erefore, the up per right-h and square in the right-hand d iagram also is homotopy cartesian. Moreover , the left-hand horizon tal maps in the left-hand diagr am are infinitesimal thickenings, and therefore, the left-h and hor izontal maps in the right- hand diagram are weak equivalences. It fol- lows that th e o uter squ are in the right-han d diagram is homotopy cartesian as desired. Now , it fo llows verbatim from t he pr oof o f op . cit., Theorem 6.1, t hat the theorem holds, if X is a h ypersur face in a sch eme essentially smooth over k . Similarly , the proof of o p. cit., Corollary 6.2, sh ows that the theorem h olds, if X is a local complete intersection in a scheme essentially smooth over k . Finally , in the gen eral case, we pr oceed as in th e pro of of op. cit., Theor em 6 .4. W e may assume that X is integral, since th e presheaf { F n ( − ) } is in variant under infinitesimal thickenings and satisfies descen t fo r finite clo sed coverings. W e now argue by ind uction on th e d imension d o f X . The ca se d = 0 follows from wh at was p roved earlier since X is smooth over k , k being per fect. So we let d > 0 and assume the theorem has been prov ed fo r schemes o f smaller d imension. Replacing X by an affine o pen neighb orhoo d of a giv en point x ∈ X , we may embed X as a closed subscheme X = Z ( p ) ⊂ U of an af fine scheme U = Spec A essentially smooth o ver k . Since U is regular, there exists a r egular sequen ce in p of length ht ( p ) . Th is sequence defines a clo sed subscheme ˜ X ⊂ U which co ntains X as an ir reducible compo nent 7 and which, after possibly replacing U by a smaller open neig hborh ood of x , is a local complete intersectio n. Hence, the theo rem holds for ˜ X by what was pr oved ab ove. Let X c ⊂ ˜ X b e the union of th e compon ents othe r th an X . Then the theo rem also holds for the intersection X ∩ X c by induction. Therefore, it h olds for X (and for X c ) by the Mayer-V ietoris sequence associated with the closed co vering of ˜ X by X an d X c . Remark 1. 2 . In the situation o f Th eorem B, suppo se that { F n ( − ) } takes elementary Nisnevich squares of schemes essentially of finite typ e over k to h omotopy cartesian squares. Then for every scheme X essentially of finite typ e ov er k , the canon ical ma p defines a weak equivalence { F n ( X ) } ∼ − → { H · Nis ( X , F n ( − )) } . Therefo re, if { F n ( − ) } takes infin itesimal thickenings (resp. finite blow-up squar es) of affine schemes essentially of finite type over k to weak e quiv alences (resp. homotopy cartesian squares) the n { F n ( − ) } takes infinitesima l thickenin gs (resp. finite blo w- up squ ares) o f all schem es essentially of finite ty pe over k to w eak equ iv a lences (resp. homotopy cartesian squares). Pr oo f of Theor em B. W e s how that the presheaf of p ro-spectra { F n ( − ) } , where F n ( X ) is the mapping fiber of the cyclotomic trace map tr : K ( X ) → T C n ( X ; p ) , satisfies the hypoth esis of Th eorem 1.1. The functors K ( − ) and TC n ( − ; p ) both take elementary Nisnevich square s to homoto py cartesian sq uares. I ndeed, this is proved for K ( − ) in [2 1, Th eorem 1 0.8] and for T C n ( − ; p ) in [8, Pro position 3 .2.1]. T here- fore, the fu nctor F n ( − ) takes elementar y Nisnevich squares to ho motopy cartesian squares. N ext, it f ollows from Remark 1.2 and f rom [7, Theo rem B] that { F n ( − ) } takes infinitesimal th ickenings to weak equ i valences. Similarly , Remarks 1.2 and [ 7, Theorem D] show that { F n ( − ) } takes finite abstrac t blow-up squares to homo topy cartesian squares. Finally , two theo rems of Th omason [20, The orem 2.1] and Blum - berg and Mandell [2, Theorem 1.4 ] show tha t the functor s K ( − ) and TC n ( − , ; p ) take squares associated with b low-ups along regular embeddings to homo topy c arte- sian squares. Hence, th e same holds for the functor F n ( − ) . Now , Theorem B f ollows from Theorem 1.1. 2 The eh -topolo gy Let X b e a noetherian scheme . W e recall from [6, Definition 2.1] that the eh-topology on the category Sch / X of schemes separated and of finite type over X is defined to b e the smallest Grothen dieck topo logy f or whic h both ´ etale coverings and cdh- coverings are coverings. In this section , we estimate the cohomolo gical dimension of the eh-topo logy . Our argum ents closely follow Suslin-V oe vodsky [19, Section 12]. Lemma 2.1. Let X b e a noetherian sc heme, and let F be an eh -sheaf of abelian gr ou ps on Sch / X . Then the cano nical closed immer sion induces an isomorphism H ∗ eh ( X , F ) ∼ − → H ∗ eh ( X red , F ) . 8 Pr oo f. W e choose an injective resolutio n F → I · in the cate gory of eh- sheav es of abelian g roups on Sch / X . Sin ce th e clo sed immersio n i : X red → X is a cdh-covering , we have the equ alizer diagram I · ( X ) i ∗ / / I · ( X red ) pr ∗ 1 / / pr ∗ 2 / / I · ( X red × X X red ) . But the closed immersion i : X red → X is also a univ ersal hom eomorp hism so the two projection pr 1 and pr 2 are eq ual. Henc e, the map i ∗ is an isomorphism of com plexes of abelian groups. The lemma follows. For e very morphism f : Y → X between noeth erian schemes, we have the induced functor f − 1 : Sch / X → Sch / Y defined by f − 1 ( X ′ / X ) = X ′ × X Y / Y . This functor , in turn, giv es rise to the adjoint pair of functors ( Sch / X ) ∧ f p / / ( Sch / Y ) ∧ f p o o where f p is the re striction along f − 1 and f p the left Kan extension along f − 1 . Since the functor f p preserves eh-shea ves, we obtain a morp hism of s ites f : ( Sch / Y ) eh → ( Sch / X ) eh with the direct image functo r f ∗ giv en by the re striction of the fu nctor f p and with the in verse im age functor giv en by f ∗ = a eh f p i eh . W e co nsider two special ca ses. First, if f : Y → X is separa ted and o f finite typ e, the in verse imag e f unctor is gi ven by f ∗ F ( Y ′ / Y ) = F ( Y ′ / X ) and has an exact left adjoint functor f ! . Hence, in this case, we see that f ∗ preserves injectiv es an d that the canonical map defines an isomorp hism H ∗ eh ( Y , F ) ∼ − → H ∗ eh ( Y , f ∗ F ) . Second, sup pose that f : Y → X is the limit of a cofiltered diag ram { X i } with affine transition maps o f schemes separ ated and of finite type over X . It then f ollows f rom [10, Theorem IV .8.8.2] that for e very scheme Y ′ separated and of finite type over Y , there exists an ind ex i and a scheme X ′ i separated and of finite type over X i such that Y ′ = X ′ i × X i Y . Moreover , in this situation, loc. cit. implies that f p F ( Y ′ / Y ) = colim j → i F ( X ′ i × X i X j / X ) . It is proved in [19, Section 12] that, in this case, th e functor f p preserves eh-sheav es, and hence, that the in verse im age functor f ∗ is giv en by the same formula. Proposition 2 .2. Let X be a noeth erian scheme, and let f : Y → X be the limit of a cofilter ed diagram { X i } with affine transition maps of sc hemes s eparated and of finite type over X . Th en for every eh -sheaf F of abelian gr oups on Sch / X , the cano nical map defines an isomorphism colim i H ∗ eh ( X i , F ) ∼ − → H ∗ eh ( Y , f ∗ F ) . 9 Pr oo f. Let F → I · be an injective resolution in the category of eh-sheaves of ab elian group s o n Sch / X . Then, since filtered colimits and finite limits of diagr ams o f sets commute, the explicit formula for the functor f ∗ giv es an isomorphism colim i H ∗ eh ( X i , F ) = colim i H ∗ Γ ( X i , I · ) ∼ − → H ∗ ( colim i Γ ( X i , I · )) = H ∗ Γ ( Y , f ∗ I · ) . W e claim that for every injectiv e eh-sheaf I of abelian gro ups on Sch / X , the eh-sheaf f ∗ I of abelian gro ups on Sch / Y is Γ ( Y , − ) -acyclic. T o show this, it suffices to sh ow that th e ˇ Cech co homolo gy groups of ˇ H ∗ ( Y , f ∗ I ) vanish [1, Propo sition V .4.3] . Since Y is noetherian , every eh-covering of Y adm its a refinem ent b y a finite eh-c overing. Therefo re, it suffices to sho w that th e ˇ Cech coh omolog y groups of f ∗ I with respect to ev ery finite eh-covering of Y vanish. But this follows immed iately fr om th e explicit formu la fo r the f unctor f ∗ , since fil tered colimits and finite limits of dia grams of sets commute. This proves the claim. Since f ∗ is exact, we conc lude tha t f ∗ F → f ∗ I · is a resolution by Γ ( Y , − ) -acyclic o bjects in the category of eh-sheaves of abelian g roups on Sch / Y . Therefore , th e canonical map H ∗ eh ( Y , f ∗ F ) → H ∗ Γ ( Y , f ∗ I · ) is an isomorph ism. Th is shows that the map of the statement is an isomorp hism. Theorem 2.3. Let X be a sc heme separated an d of fi nite type o ver a separ ably closed field k of positive characteristic p. Then the p-co homologica l dime nsion o f X with r e spect to the eh -topology is less than or equal to dim ( X ) . Pr oo f. W e must show that f or every eh-sheaf F of p - primary tor sion abelian grou ps on Sch / X , the coh omolog y gro up H q eh ( X , F ) vanishes for q > d = d im ( X ) . W e follow the proof of [19, Theo rem 12. 5] an d pro ceed b y ind uction on d . Su ppose fir st that d = 0. By Lem ma 2 .1 we may fu rther assume that X is reduced. I t follows f rom [6, Proposition 2.3] that e very eh-covering o f X ad mits a refinemen t by an ´ etale cov ering of X . The refore, we con clude from [1, T heorem III .4.1] that the chang e-of-to pology map defines an isomorph ism H q et ( X , α ∗ F ) ∼ − → H q eh ( X , F ) . The statement now f ollows f rom [1, X Corollary 5.2]. W e n ext let d > 0 an d assume that the statement h as been p roved for schemes of smaller dimension . By Lemma 2.1 we may assume that X is reduced , a nd by a de- scending indu ction on the num ber of con nected comp onents we may furth er assume that X is integral [6, Prop 2.3]. W e consider the Lera y spectral sequence E s , t 2 = H s et ( X , R t α ∗ F ) ⇒ H s + t eh ( X , F ) . Let x ∈ X be a p oint of codim ension c < d , let ¯ x be a geo metric point lyin g above x , and let f : Spe c O sh X , ¯ x → X be th e canonical map. It follows from Proposition 2.2 that the stalk of R t α ∗ F a t ¯ x is given by ( R t α ∗ F ) ¯ x = H t eh ( Spec O sh X , ¯ x , f ∗ F ) . 10 By [1, X Lem ma 3.3 (i)] there exists a morp hism g : X ′ → X from a c -dimension al scheme separ ated and of finite type over a separ ably closed field k ′ and a point x ′ ∈ X ′ lying under ¯ x such that g ( x ′ ) = x and such that the induced map g ′ : Spec O sh X ′ , ¯ x → Spec O sh X , ¯ x is an isomorph ism. Le t f ′ : Spec O sh X ′ , ¯ x → X ′ be the canonical map. Then ( R t α ∗ F ) ¯ x = H t eh ( Spec O sh X ′ , ¯ x , g ′ ∗ f ∗ F ) = H t eh ( Spec O sh X ′ , ¯ x , f ′ ∗ g ∗ F ) . Hence, we conclude from Proposition 2.2 that ( R t α ∗ F ) ¯ x = c olim H t eh ( U ′ , g ∗ F ) where the colimit ran ges over a ll ´ etale neig hborh oods U ′ → X ′ of ¯ x . Each U ′ is a c -dimension al scheme separated an d of finite typ e over the separab ly closed field k ′ . Therefo re, the in ductive hypoth esis sho ws that ( R t α ∗ F ) ¯ x is ze ro for t > c . It fo llows that the sheaf R t α ∗ F is supported in dimen sion max { 0 , d − t } . W e recall from [1, X Corollary 5.2] that if Z is a scheme o f finite ty pe over a separab ly c losed field of ch aracteristic p > 0, th en the ´ etale p -coh omolog ical dim ension of Z is less than or equal to dim ( Z ) . This shows that in th e Leray spectral seq uence, E s , t 2 is zero fo r s > max { 0 , d − t } , and hence, the edge-ho momor phism H q eh ( X , F ) → H 0 et ( X , R q α ∗ F ) is an isomorph ism for q > d . W e now fix a cohomo logy class h ∈ H q eh ( X , F ) with q > d an d proceed to show that h is zero. T here exists an eh -covering Y → X such that the restriction of h to Y is zero. Moreover , by [6, Proposition 2. 3], th e covering Y → X adm its a refine ment of the fo rm U ′ → X ′ → X , where U ′ → X ′ is an ´ etale covering and X ′ → X a proper bi-rationa l cdh-covering. W e let X ′′ ⊂ X ′ be the closure of the inv erse im age of the generic poin t of X , an d let U ′′ = U × X ′ X ′′ → X ′′ . It fo llows from [19, Lemm a 12.4 ] that X ′′ is a schem e separated and o f finite typ e over k of dim ension at most d and that the morp hism X ′′ → X is p roper and bi-ratio nal. W e claim that the restriction h ′′ of the class h to X ′′ is zero. Ind eed, the restriction o f h ′′ to U ′′ is zero, and theref ore, the image of h ′′ by the edge homomo rphism o f the Leray spectral sequence H q eh ( X ′′ , F ) → H 0 et ( X ′′ , R q α ∗ F ) is zero. But we proved above that the edge homomorp hism is an i somorp hism, s o we find that h ′′ is zero as claimed . T o con clude that h is zero, we cho ose a proper closed subscheme Z ⊂ X such that th e morph ism X ′′ → X is an isomo rphism outside Z an d define Z ′′ = X ′′ × X Z . Then by [6, Proposition 3 .2], we ha ve a long exact cohomolo gy sequence · · · → H q − 1 eh ( Z ′′ , F ) → H q eh ( X , F ) → H q eh ( Z , F ) ⊕ H q eh ( X ′′ , F ) → · · · The sche mes Z and Z ′′ are of finite ty pe over k and their dimension s are strictly smaller than d . Therefore, by the inductive hypoth esis, the restriction map H q eh ( X , F ) → H q eh ( X ′′ , F ) 11 is an is omorp hism for q > d . Sinc e the ima ge h ′′ of h by this map is zero, we co nclude that h is zero as stated. This completes the proof. Pr oo f of Theor em D. W e consider the Leray spectral sequence E s , t 2 = H s et ( X , R t α ∗ F ) ⇒ H s + t eh ( X , F ) . Let x ∈ X be a point of codimen sion c 6 d , and let ¯ x be a geom etric point lying above x . W e claim that stalk of R t α ∗ F at ¯ x vanishes for t > c . T o p rove the claim, we may assume that X is affine. W e write X as a localizatio n j : X → X 1 of a schem e X 1 separated and of finite type over k . The n x 1 = j ( x ) ⊂ X 1 is a poin t of codimen sion c . Hence, we find as in the proo f of T heorem 2.3, that the stalk may be rewritten as a filtered colimit ( R t α ∗ F ) ¯ x = c olim H t eh ( U ′ , g ∗ F ) where the U ′ are c -dimen sional sch emes separated and of fin ite type over a separa - bly clo sed field k ′ . T he claim now f ollows from Theore m 2. 3. W e conclude th at th e sheaf R t α ∗ F is zer o for t > d , and is suppo rted in dimen sion d − t for t 6 d . Finally , we re call fr om [1, X Theorem 5. 1] that the ´ etale p -co homolo gical dimension o f a noetherian F p -scheme Z is less than or eq ual to dim ( Z ) + 1. The refore, E s , t 2 is zero for s + t > d + 1. This completes the proof. 3 The de Rham-Witt shea ves W e say that the sch eme X is essentially smoo th over the field k if it can be cov- ered by finitely many affine open subsets of the for m Sp ec S − 1 A with A a smo oth k -alge bra and S ⊂ A a multiplica ti ve set. If X is a scheme essentially smoo th over a field k , we let Sm / X be th e full subcategory of Sch / X whose o bjects are the schemes smooth over X . W e define the Zariski, ´ etale, cdh, and eh-topo logy on S m / X to be th e Grothend ieck topo logy induced by th e Zariski, ´ etale, cdh, and eh-topology o n Sch / X , respectively , in the sense of [1, III.3 .1]. W e denote by φ p the restriction fu nctor from the category of preshea ves on Sch / X to the cate gory of presheaves o n Sm / X . Lemma 3.1 . Let k b e a perfect field of positive characteristic p such that resolution of singu larities hold s over k, an d let X be a scheme essentially smooth over k. Then for every pr eshea f F of abelia n gr oups on Sch / X , the canonical map H ∗ τ ( X , a τ φ p F ) → H ∗ τ ( X , a τ F ) is an isomorphism for τ the Zariski, ´ etale, cdh , and eh -topology . Pr oo f. Let φ l and φ r be the left an d right ad joint functor s o f φ p giv en by the two Kan exten sions. The f unctor φ p preserves τ -sheaves by the d efinition of the ind uced Grothend ieck topo logy [1, III.3. 1]. W e c laim that also φ r preserves τ -sheaves. In deed, for the Zarisk i topolo gy an d ´ etale topolog y this follows fr om [1, Coro llary III.3.3 ], 12 and for the c dh-topolog y and eh-topolo gy it follows fr om [ 1, Theorem III.4.1]. Hence , we get the diagrams of adjunction s ( Sch / X ) ∧ a τ / / φ p   ( Sch / X ) ∼ τ i τ o o φ ∗   ( Sm / X ) ∧ a τ / / φ r   ( Sm / X ) ∼ τ i τ o o φ !   ( Sm / X ) ∧ a τ / / φ l O O ( Sm / X ) ∼ τ i τ o o φ ∗ O O ( Sch / X ) ∧ a τ / / φ p O O ( Sch / X ) ∼ τ i τ o o φ ∗ O O where φ ∗ and φ ! are th e restrictions o f φ p and φ r , resp ectiv ely , to the subca tegory of τ -sheaves, and whe re φ ∗ = a τ φ l i τ . It f ollows that th e fun ctor φ ∗ preserves all limits and co limits. Mor eover , since th e two diagrams of rig ht adjo int functo rs commute, the two diagra ms of left adjoint f unctors commu te up to natural isomor phism. In particular, we have a natur al is omorp hism a τ φ p F ∼ − → φ ∗ a τ F . Now , the map of the statement is equal to the compo sition H ∗ τ ( X , a τ φ p F ) → H ∗ τ ( X , φ ∗ a τ F ) → H ∗ τ ( X , a τ F ) of the indu ced isomor phism of coho mology gro ups and the ed ge-ho momor phism of the Leray spectral sequence E s , t 2 = H s τ ( X , R t φ ∗ a τ F ) ⇒ H s + t τ ( X , a τ F ) . Since φ ∗ is exact the spectral sequen ce collap ses and the ed ge-ho momorp hism is an isomorph ism. T his completes the proof. W e let X b e a noetherian F p -scheme and let O X be the p resheaf o n Sch / X that to the X -sche me X ′ assigns th e F p -algebra Γ ( X ′ , O X ′ ) . I t is a sheaf for the ´ etale topolog y , but not for the cdh-topolo gy . W e r ecall the presheaf W n Ω q X = W n Ω q O X of de Rh am-W itt fo rms of Bloch-Deligne-Illusie [16, Definition I.1.4]. It f ollows from Proposition I.1.14 of op. cit. that the associated sheav es a Zar W n Ω q X and a et W n Ω q X agree and are quasi-coh erent she av es of W n ( O X ) -modules on Sch / X . Now , supp ose that X is a sch eme essentially smo oth over a perf ect field k of characteristic p > 0. W e recall the structure of the sheaf a et W n Ω q X from [16]. W e will abuse n otation and write W n Ω q X for the ´ etale sheaf a et W n Ω q X on Sch / X . There is a short exact sequence of shea ves o f abelian grou ps on Sm / X f or the ´ etale topology 0 → gr n − 1 W n Ω q X → W n Ω q X R − → W n − 1 Ω q X → 0 (3.2) where gr n − 1 W n Ω q X is the subsheaf gener ated by the imag es of V n − 1 : Ω q X → W n Ω q X and dV n − 1 : Ω q − 1 X → W n Ω q X . Let Z Ω q X and B Ω q + 1 X be the kern el an d imag e she av es of the differential d : Ω q X → Ω q + 1 X . The inv erse Cartier operator C − 1 : Ω q X → Z Ω q X / B Ω q X 13 is defin ed as follows. Let F : W 2 Ω q X → Ω q X be the Froben ius map. It satisfies the relations d F = pF d , F V = p , and F d V = d . The first r elation shows th at F factors throug h the inc lusion of the subsheaf Z Ω q X in Ω q X , a nd the two r emaining relation s show th at the composition W 2 Ω q X → Z Ω q X → Z Ω q X / B Ω q X of the Froben ius map and the canonical projection annihilates the subsheaf gr 1 W 2 Ω q X . The in verse Cartier op erator is now defined t o be the comp osition Ω q X ∼ ← − W 2 Ω q X / gr 1 W 2 Ω q X ¯ F − → Z Ω q X / B Ω q X . It is an isomorp hism of sheaves of abelian group s on Sm / X for th e ´ etale topol- ogy; see [17, Theorem 7.1]. The in verse is omorp hism C : Z Ω q X / B Ω q X ∼ − → Ω q X is the Cartier operato r . It gi ves rise to a ch ain of subsheav es of abelian groups 0 = B 0 Ω q X ⊂ B 1 Ω 1 X ⊂ · · · ⊂ B s Ω q X ⊂ · · · ⊂ Z s Ω q X ⊂ · · · Z 1 Ω q X ⊂ Z 0 Ω q X = Ω q X where B 0 Ω q X = 0, Z 0 Ω q X = Ω q X , B 1 Ω q X = B Ω q X , Z 1 Ω q X = Z Ω q X , and where for s > 2, B s Ω q X and Z s Ω q X are defined to be the sub sheav es of abelian gro ups of Ω q X character- ized by the short exact sequences 0 → B 1 Ω q X → B s Ω q X C − → B s − 1 Ω q X → 0 0 → B 1 Ω q X → Z s Ω q X C − → Z s − 1 Ω q X → 0 (3.3) It then follows from [16, Corollary I.3.9] that there is a short exact sequence 0 → Ω q X / B n − 1 Ω q X → gr n − 1 W n Ω q X → Ω q − 1 X / Z n − 1 Ω q − 1 X → 0 , (3.4) of sheaves of ab elian groups on Sm / X fo r the ´ e tale topolog y . Proposition 3.5. Let k be a perfect field of positive characteristic p and assume that r e solution of singula rities holds over k . Then fo r all sc hemes X essentially smoo th over k and all inte gers n > 1 and q > 0 , the change-of-top ology maps H ∗ Zar ( X , a Zar W n Ω q X ) / /   H ∗ et ( X , a et W n Ω q X )   H ∗ cdh ( X , a cdh W n Ω q X ) / / H ∗ eh ( X , a eh W n Ω q X ) ar e isomorphisms. Pr oo f. Since a Zar W n Ω q X and a et W n Ω q X are quasi-coh erent W n ( O X ) -modules, th e to p horizon tal map is an isomorp hism. W e sho w th at th e r ight-han d vertical map is an isomorph ism; the proo f f or the left-ha nd vertica l map is an alogou s. By Lemma 3 .1, it suffices to show that the change-of-to polog y map H ∗ et ( X , a et φ p W n Ω q X ) → H ∗ eh ( X , a eh φ p W n Ω q X ) 14 is an isomor phism. W e a gain abuse notatio n and wr ite W n Ω q X for the ´ etale sheaf a et φ p W n Ω q X on Sm / X . W e pro ceed b y inductio n on n > 1 as in [ 4, Proposition 6.3]. In the case n = 1, we let R et Γ ( − , Ω q X ) be th e pr esheaf of ch ain co mplexes gi ven by a fu nctorial m odel for the total righ t derived fu nctor for the ´ etale to pology of th e fun ctor Γ ( − , Ω q X ) from Sm / X to the categor y of abelian gro ups. W e mu st show that th e presheaf R et Γ ( − , Ω q X ) satisfi es descent for the eh-topolo gy . T o prove this, it suf fices by [4, Corollary 3.9] to show th at for e very blo w-up square of smooth X -schemes Z ′ / /   Y ′   Z / / Y , the induced square of complexes of abe lian groups R et Γ ( Z ′ , Ω q X ) R et Γ ( Y ′ , Ω q X ) o o R et Γ ( Z , Ω q X ) O O R et Γ ( Y , Ω q X ) o o O O is homoto py car tesian. But this is proved in [9, Chapter IV , The orem 1.2.1]. W e next assume the statem ent for n − 1 and prove it fo r n . By a five-lemma argu- ment b ased on th e sho rt exact sequence of shea ves ( 3.2), it suffices to show that the change- of-top ology map H ∗ et ( X , gr n − 1 W n Ω q X ) → H ∗ eh ( X , α ∗ gr n − 1 W n Ω q X ) is an isomorph ism. Furtherm ore, by a fiv e-lemma argument based on the exact se- quence (3.4), it suffices to show that for all s > 0 , the change -of-top ology maps H ∗ et ( X , B s Ω q X ) → H ∗ eh ( X , α ∗ B s Ω q X ) H ∗ et ( X , Z s Ω q X ) → H ∗ eh ( X , α ∗ Z s Ω q X ) (3.6) are isomo rphisms. T he case s = 0 was p roved above. T o prove the case s = 1, we argue by in duction o n q > 0 : Th e basic case q = 0 h as alre ady b een proved, since B 1 Ω 0 X = 0 and C : Z 1 Ω 0 X = Z 1 Ω 0 X / B 1 Ω 0 X ∼ − → Ω 0 X , and th e inductio n step follows by a fi ve-lemma argument based o n the exact sequences of shea ves 0 → Z 1 Ω q − 1 X → Ω q − 1 X d − → B 1 Ω q X → 0 0 → B 1 Ω q X → Z 1 Ω q X C − → Ω q X → 0 . Finally , we let s > 2 an d assume, in ductively , th at th e maps (3.6) h av e been proved to be isomorphisms fo r s − 1. Th en a fi ve-lemma argumen t based o n th e short exact sequence of sheav es (3.3) show that the maps (3 .6) are isom orphisms for s . This completes the proo f. 15 Lemma 3 .7. Let X be a noetherian F p -scheme and le t x ∈ X be a point. Then the canon ical ma p f : Sp ec O X , x → X indu ces an isomorphism f ∗ a Zar W n Ω q X → a Zar W n Ω q Spec O X , x of sheaves of abelian gr oup s on Sch / Spec O X , x for the Zariski topology . Pr oo f. Let X be a topos. W e recall from [16, Definition I.1.4 ] that, by definition , the de Rham-W itt complex is the left adjoin t of the fo rgetful functor g from the category of V -p ro-com plexes in X to the category of F p -algebras in X . W e let x ∈ U ⊂ X be an o pen neighb orhoo d, an d let i U be the cor respondin g point of the topo s ( Sch / X ) ∧ giv en by i ∗ U ( F ) = Γ ( U , F ) and i U ∗ ( E )( X ′ ) = E Hom X ( U , X ′ ) . Since g ◦ i U ∗ = i U ∗ ◦ g , we conclud e tha t there is a natural isomorp hism Γ ( U , W n Ω q X ) = Γ ( U , W n Ω q O X ) ∼ − → W n Ω q Γ ( U , O X ) . Now , let Y = Spec O X , x . T o pr ove the lemma, it suffices to show that for every scheme Y ′ affine and of finite type ov er Y , the cano nical map Γ ( Y ′ , f p W n Ω q X ) → Γ ( Y ′ , W n Ω q Y ) is an isomorphism. The discussion p receeding Pr oposition 2.2 shows that there exists an affine open neighborho od x ∈ U ⊂ X and a s cheme U ′ affine and of finite type over U with Y ′ = Y × U U ′ such that the map in question is the canonical map colim x ∈ V ⊂ U Γ ( V × U U ′ , W n Ω q X ) → Γ ( Y × U U ′ , W n Ω q Y ) . Here t he colimit ranges over the af fine ope n neigh borho ods x ∈ V ⊂ U . No w , by what was proved above, we may identify this map with the canonical map colim x ∈ V ⊂ U W n Ω q Γ ( V × U U ′ , O X ) → W n Ω q Γ ( Y × U U ′ , O Y ) . Here t he left-hand side is the colimit in the categor y of s ets. Ho we ver , since th e in dex category f or t he colimit is filtered, th e left-hand side is a lso eq ual to the co limit in the category o f V -pr o-comp lexes in the category of sets. Therefore, the cano nical map in question is an isomo rphism. Ind eed, being a left adjo int, the de Rham-Witt complex preserves colimits, and the canonical map of F p -algebras colim x ∈ V ⊂ U Γ ( V × U U ′ , O X ) → Γ ( Y × U U ′ , O Y ) is an isomorph ism. Theorem 3.8. Let k be a p erfect field of characteristic p > 0 such that resolution of singularities ho lds over k, and let X be a d -dimensiona l scheme essentia lly o f fi nite type over k. Then the change-of-top ology map H i et ( X , a et W n Ω q X ) → H i eh ( X , a eh W n Ω q X ) is a surjection if i = d , and both gr oup s vanish if i > d . 16 Pr oo f. W e fo llow the pr oof of [4, Th eorem 6.1]. Since the sheaves a Zar W n Ω q X and a et W n Ω q X are qu asi-coheren t W n ( O X ) -modules, the change-o f-topo logy map de fines an isomorph ism H i Zar ( X , a Zar W n Ω q X ) ∼ − → H i et ( X , a et W n Ω q X ) . In particu lar , the common group vanishes for i > d . W e con sider the following com- mutative diagram, where the hor izontal map s ar e the change-of -topolo gy maps, and where the vertical maps are induced from the canonical closed immersion. H i Zar ( X , a Zar W n Ω q X ) / /   H i eh ( X , a eh W n Ω q X )   H i Zar ( X red , a Zar W n Ω q X red ) / / H i eh ( X red , a eh W n Ω q X red ) . By Lemma 2 .1, the right- hand vertical map is an isomorp hism for all i > 0 . Moreover, the co homolo gy groups on the left-h and side m ay be ca lculated on the small Zariski sites and X an d X red , respectively , and th e left-h and vertical ma p may be iden tified with the map of sheaf co homo logy grou ps of th e sm all Zariski site of X induced b y the surjectiv e map of shea ves a Zar W n Ω q X → i ∗ a Zar W n Ω q X red induced by the closed imm ersion i : X red → X . It fo llows that the left-hand vertical map is a surjection for i > d . Therefore, we may assume that X is reduced . W e proceed by indu ction on d . The case d = 0 follows from Proposition 3.5, since every reduced 0-dimen sional scheme X of finite ty pe over the perfect field k is smooth over k . So we let d > 0 and assume that t he statement for schemes o f smaller dimension. W e must sh ow that the statemen t hold s for every reduced d -dimension al scheme X essentially of finite ty pe over k . Su ppose fir st that X is affine, an d h ence, separated. By resolution of singularities, there exists an abstract blo w-up square Y ′ i ′ / / p ′   X ′ p   Y i / / X where X ′ is essentially smooth over k an d where the dimensions of Y and Y ′ are smaller than d . The group H i eh ( X ′ , a eh W n Ω q X ′ ) vanishes fo r i > d by Propo sition 3.5 and the group s H i eh ( Y , a eh W n Ω q Y ) and H i eh ( Y ′ , a eh W n Ω q Y ′ ) vanish fo r i > d − 1 by the in- duction. Therefor e, th e Mayer-V ie toris lon g exact sequence of e h-cohomo logy grou ps associated with the abstract blow-up s quare above · · · → H i − 1 eh ( Y ′ , a eh W n Ω q Y ′ ) → H i eh ( X , a eh W n Ω q X ) → H i eh ( X ′ , a eh W n Ω q X ′ ) ⊕ H i eh ( Y , a eh W n Ω q Y ) → · · · 17 shows that the grou p H i eh ( X , a eh W n Ω q X ) vanishes for i > d as stated. W e must also show t hat H d eh ( X , a eh W n Ω q X ) is zero. W e first show that the common group H d Zar ( X ′ , a Zar W n Ω q X ′ ) ∼ − → H d eh ( X ′ , a eh W n Ω q X ′ ) is zero. Th e lef t-hand group m ay be evaluated on the small Zariski site of X ′ . Now , the th eorem o f fo rmal func tions [10, Corollar y III.4 .2.2] shows that fo r every quasi- coheren t W n ( O ′ X ) -module F on the small Zariski site of X ′ , the higher direct image sheaf R d p ∗ F on the small Z ariski site of X is zero . Since X is affine, we con clude from the Leray spectral sequence tha t the H d Zar ( X ′ , a Zar W n Ω q X ′ ) is zero as desired. W e next sho w that the lo wer horizonta l map in the following diagram is surjectiv e. H d − 1 Zar ( X ′ , a Zar W n Ω q X ′ ) i ′ ∗ / /   H d − 1 Zar ( Y ′ , a Zar W n Ω q Y ′ )   H d − 1 eh ( X ′ , a eh W n Ω q X ′ ) i ′ ∗ / / H d − 1 eh ( Y ′ , a eh W n Ω q Y ′ ) Here the vertical maps are the c hange-o f-topo logy maps. By induc tion, the right- hand vertical m ap is surjective, so we m ay instead show that the up per h orizon- tal map is sur jectiv e. Th e co homo logy gro ups in the uppe r row may be ev aluated on small Zariski sites of X ′ and Y ′ , re spectiv ely . Moreover, the theorem of formal function s shows th at the R d − 1 p ∗ is a right-exact fun ctor f rom th e category of quasi- coheren t W n ( O X ′ ) -modules to the category of quasi-coher ent W n ( O X ) -modules. Since the closed immersion i ′ giv es rise to a surjection W n Ω q X ′ → i ′ ∗ W n Ω q Y ′ of quasi-coh erent W n ( O X ′ ) -modules on the small Zariski site of X ′ , we conclude that the induced map R d − 1 p ∗ W n Ω q X ′ → R d − 1 p ∗ i ′ ∗ W n Ω q Y ′ is a surjection of W n ( O X ) -modules on the s mall Zariski site o f X . As X is assumed to be affine, the Leray spectral sequence shows th at i ′ ∗ : H d − 1 Zar ( X ′ , a Zar W n Ω q X ′ ) → H d − 1 Zar ( Y ′ , a Zar W n Ω q Y ′ ) is surjecti ve as desired. W e conclu de from the Mayer-V ieto ris exact sequen ce that the group H d eh ( X , a eh W n Ω q X ) is zer o. This proves the statement of the th eorem for X a d -dimensional reduced affi ne scheme essentially of finite type over k . It rem ains to prove the stateme nt fo r X a gen eral d -dimensio nal reduced schem e essentially of finite ty pe over k . T o this en d, we let ε : ( Sch / X ) eh → ( Sch / X ) Zar be the canonical morph ism of si tes and consider the Leray spectral sequence E s , t 2 = H s Zar ( X , R t ε ∗ a eh W n Ω q X ) ⇒ H s + t eh ( X , a eh W n Ω q X ) . Let x ∈ X b e a point of codim ension c . T hen Proposition 2.2 and Lemma 3 .7 show that the stalk of R t ε ∗ a eh W n Ω q X at x is given b y ( R t ε ∗ a eh W n Ω q X ) x = H t eh ( Spec O X , x , a eh W n Ω q Spec O X , x ) . 18 W e h av e proved that this gro up v anishes if eithe r c > 0 an d t > c or c = 0 an d t > 0, or equivalently , if t > 0 and c 6 t . It follows that for t > 0, the high er direct im age sheaf R t ε ∗ a eh W n Ω q X is sup ported in dimension < d − t . Hence, E s , t 2 vanishes if t > 0 and s + t > d . Th is shows that H i eh ( X , a eh W n Ω q X ) is zero f or i > d and that the edge homom orphism defines a su rjection H d Zar ( X , ε ∗ a eh W n Ω q X ) ։ H d eh ( X , a eh W n Ω q X ) . Finally , it follows from Proposition 3.5 that the cokernel of the unit map a Zar W n Ω q X → ε ∗ a eh W n Ω q X is supp orted on the singular set of X wh ich has dimension strictly less that d . Since the functor H d Zar ( X , − ) is right-exact, we conc lude that the induced map H d Zar ( X , a Zar W n Ω q X ) → H d Zar ( X , ε ∗ a eh W n Ω q X ) is surjective. This proves the induction step and the theorem. 4 The sheav es a eh TR n q ( − ; p ) and a eh TC n q ( − ; p ) Let X b e a noetherian F p -scheme. W e b riefly re call the preshea ves of fibr ant symm et- ric spectra TR n ( − ; p ) and TC n ( − ; p ) on Sch / X and refer to [8 ] and [2] for a detailed discussion. T o pological Hochschild hom ology gives a presheaf THH ( − ) of fibrant symmetric spectra with an ac tion by the mu ltiplicativ e group T of complex number s of modulu s 1. W e let C p n − 1 ⊂ T be the subgro up of order p n − 1 and define TR n ( − ; p ) = THH ( − ) C p n − 1 to be th e presheaf of fibrant symmetric spectr a g i ven b y the C p n − 1 -fixed poin ts. There are two maps of presheaves of fibrant symmetric spectra R , F : T R n ( − ; p ) → TR n − 1 ( − ; p ) called the restrictio n map and the Fro benius map , respectively , and the pre sheaf of fibrant symme tric spectra TC n ( − ; p ) is defin ed to be their h omotopy equalizer . It follows immediately from the definition that the associated p resheaves of hom otopy group s are related by a long exact sequence · · · / / TC n q ( X ; p ) / / TR n q ( X ; p ) R − F / / TR n − 1 q ( X ; p ) / / · · · Moreover , by [1 4, Proposition 6 .2.4], the sheaves a Zar TR n q ( − ; p ) and a et TR n q ( − ; p ) agree and are quasi-cohe rent W n ( O X ) -modules on Sch / X . 19 Proposition 4 .1. Let k be a perfect field of po sitive cha racteristic p such tha t reso- lution of singu larities holds over k, and let X be a scheme essentia lly smooth over k. Then for all inte gers q an d n > 1 , the change-of-topology maps H ∗ Zar ( X , a Zar TR n q ( − ; p )) / /   H ∗ et ( X , a et TR n q ( − ; p ))   H ∗ cdh ( X , a cdh TR n q ( − ; p )) / / H ∗ eh ( X , a eh TR n q ( − ; p )) ar e isomorphisms. Pr oo f. The sheav es a Zar TR n q ( − ; p ) an d a et TR n q ( − ; p ) ar e sheav es o f quasi-coheren t W n ( O X ) -modules. Therefore, the top horizontal map is an isomorph ism. W e show that the right-ha nd vertical map is an isomorph ism; the proof for the left-hand vertical map is an alogou s. I t suffices by Lemma 3.1, to sho w that the ch ange-of -topolo gy map H ∗ et ( X , a et φ p TR n q ( − ; p )) → H ∗ eh ( X , a eh φ p TR n q ( − ; p )) is an isomorph ism. W e recall from [1 2, Theorem B] that there is a canonical isomor- phism of sheaves o f abelian groups on Sm / X for the Zariski top ology M m > 0 a Zar φ p W n Ω q − 2 m X ∼ − → a Zar φ p TR n q ( − ; p ) . It in duces an iso morph ism of associated sheaves for the ´ etale top ology and fo r the eh-topolog y . Th e proposition now follows from P roposition 3.5 above. Corollary 4 .2. Let k be a perfect field of positive characteristic p and assume that r e solution of singularities holds over k. Let X b e a d -dimensiona l scheme essentially of finite type over k. Then for all inte gers q and n > 1 , H d + 1 eh ( X , a eh TR n q ( − ; p )) = 0 . Pr oo f. The pr oof is by ind uction on d and is ana logous to the first part of the proof of Theorem 3.8 above. Let k be a per fect field of characteristic p > 0 a nd let X be a scheme essentially of finite type over k . W e c onsider the long exact sequence · · · / / { a et TC n q ( − ; p ) } / / { a et TR n q ( − ; p ) } id − F / / { a et TR n q ( − ; p ) } / / · · · of ´ etale shea ves of pr o-abelian groups on Sch / X . Here the structure ma ps in t he pro- abelian gro ups are the restrictio n maps R . W e recall from [13] th at there is a canonical map compatib le with restriction and Froben ius operators a et W n Ω q X → a et TR n q ( − ; p ) (4.3) and that this map is an isom orphism, f or q 6 1 . W e remar k that the assumption in op. cit. that p be odd is unnecessary . Ind eed, [5, Theorem 4 .3] shows that the result is valid als o for p = 2 . W e examine the map id − F in degrees q 6 1. 20 Lemma 4.4. Let X be a noetherian scheme over F p . Then th er e is an e xa ct s equenc e of sheaves of pr o -abelian gr ou ps on Sch / X for the ´ etale topology: 0 / / { Z / p n Z } / / { a et W n ( O X ) } id − F / / { a et W n ( O X ) } / / 0 . Pr oo f. Since X is a sch eme over F p , the Fro benius m ap F agrees with the m ap { W n ( ϕ ) } indu ced by the Frob enius endomor phism of X . W e prove that fo r ev ery strictly henselian noether ian local F p -algebra ( A , m , κ ) , the sequence W n ( A ) W n ( A ) / / Z / p n Z / / id − W n ( ϕ ) 0 / / 0 / / is exact. Since A is strictly henselian, the map id − ϕ : A → A is surjective. An ind uc- tion argument based on the diagram with e xact rows 0 / / A V n − 1 / / id − ϕ   W n ( A ) R / / id − W n ( ϕ )   W n − 1 ( A ) / / id − W n − 1 ( ϕ )   0 0 / / A V n − 1 / / W n ( A ) R / / W n − 1 ( A ) / / 0 shows that the map id − F is surjective as stated. T o ide ntify the kerne l of id − F , we consider the following diagram 0 / / W n ( m ) / / id − W n ( ϕ )   W n ( A ) / / id − W n ( ϕ )   W n ( κ ) / / id − W n ( ϕ )   0 0 / / W n ( m ) / / W n ( A ) / / W n ( κ ) / / 0 W e wish to show th at the u nit map η : W n ( F p ) → W n ( A ) is an isomor phism onto th e kernel of the middle vertical m ap. W e have o per at orn ameim ( η ) ⊂ ker ( id − W n ( ϕ )) . Mo reover , since κ is a domain, the compo- sition W n ( F p ) → W n ( A ) → W n ( κ ) of η and the can onical p rojection is an isom orphism of W n ( F p ) onto the kernel of the right- hand vertical map in the d iagram above. Hence, it will suffice to show that, in the diagr am above, the left-ha nd vertical map is in jectiv e. Moreover , an induction argument based on the diagram 0 / / W n − 1 ( m ) V / / id − W n − 1 ( ϕ )   W n ( m ) R n − 1 / / id − W n ( ϕ )   m / / id − ϕ   0 0 / / W n − 1 ( m ) V / / W n ( m ) R n − 1 / / m / / 0 shows that it suffices to co nsider the case n = 1. In this case, we recall th at by a theorem of Krull [10, Co rollary 0.7.3. 6], the m -ad ic topo logy on A is separated . It follows that the map id − ϕ : m → m is injecti ve as desired. 21 Lemma 4.5. Let X be a noethe rian scheme o ver F p . Then the map of sheaves of pr o- abelian gr oup s on Sch / X for the ´ etale topology id − F : { a et W n Ω 1 X } → { a et W n Ω 1 X } is an epimorphism. Pr oo f. It will suffice to show that for ev ery strictly henselian n oetherian local F p - algebra ( A , m , κ ) , the map R − F : W n Ω 1 A → W n − 1 Ω 1 A is surjec ti ve. Since A is local, the abelian gr oup W n Ω 1 A is gen erated by e lements of the form ad log [ x ] n with a ∈ W n ( A ) and x ∈ 1 + m . Moreover , we ha ve ( R − F )( ad log [ x ] n ) = ( R − F )( a ) d log [ x ] n − 1 . Hence, th e proof of Lemma 4.4 sho ws that R − F is an ep imorph ism. The lemma follows. Proposition 4.6. Let X be a noetherian scheme o ver F p . Then: (i) The sheaf of pr o-a belian gr oups { a et TC n q ( − ; p ) } is zer o for q < 0 . (ii) The sheaf of pr o- abelian g r oup s { a et TC n 0 ( − ; p ) } is canonically isomorph ic to the sheaf of pr o-a belian gr oups { Z / p n Z } . (iii) There is a lon g exact s equence of sheaves of pr o-abelian gr oup s · · · / / { a et TC n 1 ( − ; p ) } / / { a et TR n 1 ( − ; p ) } id − F / / { a et TR n 1 ( − ; p ) } / / 0 Pr oo f. This follows immed iately from the f act that the map (4.3) is an isomorphism, for q 6 1, and from Lemmas 4.4 and 4.5. Question 4.7 . W e do no t know whe ther or not the sequence of shea ves 0 → { a et TC n q ( − ; p ) } → { a et TR n q ( − ; p ) } 1 − F − − → { a et TR n q ( − ; p ) } → 0 on Sch / X f or the ´ etale topolo gy is e xact for q > 1. Theorem 4 .8. Let k be a field of positive characteristic p and assume that r esolution of singularities ho lds over k. Let X b e a d -dimension al scheme essentially o f fi nite type over k. Then the map of pr o -abelian gr oup s { H i et ( X , a et TC n q ( − ; p )) } → { H i eh ( X , a eh TC n q ( − ; p )) } induced by the change-of-to pology maps is an epimo rphism if q = 1 and i = d + 1 , an isomorphism if q = 0 , an d the two pr o- abelian gr oup s ar e zer o if q 6 − 1 . 22 Pr oo f. The statement f or q < 0 f ollows fro m Propo sition 4 .6 (i), and the statement for q = 0 follows from Proposition 4 .6 (ii) and from [6, Theorem 3 .6] wh ich shows that for the constant sheaf Z / p n Z , the change- of-topo logy map H ∗ et ( X , Z / p n Z ) → H ∗ eh ( X , Z / p n Z ) is an isomor phism. T o prove the s tatement fo r q = 1 , we fix a d -dimension al sch eme X e ssentially of fin ite type over k . The p -cohom ological dim ension of X with respect to both the ´ etale topolo gy an d the eh-topolo gy is less than or equal to d + 1; see Theorem D. Let us write · · · → { F n − 2 } → { F n − 1 } → { F n 0 } → 0 for th e long exact sequenc e of ´ etale sh eav es of pro-ab elian groups on Sch / X from Proposition 4.6 (iii). Then we have h ypercoh omolog y spe ctral sequences E s , t 1 = { H t et ( X , F n − s ) } ⇒ { H s + t et ( X , F n · ) } E s , t 1 = { H t eh ( X , α ∗ F n − s ) } ⇒ { H s + t eh ( X , α ∗ F n · ) } with the d 1 -differentials induc ed b y the d ifferential in the cochain co mplex { F n · } . Since the co mplex { F n · } is exact and since the coh omolog ical dimen sion of X is bound ed, the spectral seque nces co n verge and th eir abutment is zero. Mo reover , the change- of-top ology maps ind uce a map of spectral sequ ences from the top sp ectral sequence to the botto m spectr al sequ ence. W e proved in Cor ollary 4.2 th at the co- homolo gy g roups H d + 1 eh ( X , α ∗ F n − 3 ) and H d + 1 eh ( X , α ∗ F n − 1 ) vanish. Similarly , the coho- mology g roup s H d + 1 et ( X , F n − 3 ) an d H d + 1 et ( X , F n − 1 ) vanish, since the sheaves F n − 3 and F n − 1 are quasi-coh erent W n ( O X ) -modules. Hen ce, the map of hyperco homolo gy spec- tral sequences gives rise to a comm utative diagram { H d et ( X , F n − 1 ) } id − F / /   { H d et ( X , F n 0 ) } / /   { H d + 1 et ( X , F n − 2 ) } / /   0 { H d eh ( X , α ∗ F n − 1 ) } id − F / / { H d eh ( X , α ∗ F n 0 ) } / / { H d + 1 eh ( X , α ∗ F n − 2 ) } / / 0 with exact r ows. H ere the middle and left-h and vertical maps are bo th equal to the change- of-top ology map { H d et ( X , a et TR n 1 ( − ; p )) } → { H d eh ( X , a eh TR n 1 ( − ; p )) } which by [13] coincide s wi th the change-o f-topo logy map { H d et ( X , a et W n Ω 1 X ) } → { H d eh ( X , a eh W n Ω 1 X ) } . W e proved in Theorem 3.8 that this map is an ep imorphism . Therefore, also th e r ight- hand vertical map, { H d + 1 et ( X , a et TC n 1 ( − ; p )) } → { H d + 1 eh ( X , a eh TC n 1 ( − ; p )) } , is an epimorph ism. This completes the proof . 23 5 Proof of Theorems A a nd C W e first show th at, in the statement of Theorem C, we may replace the Zariski topol- ogy and cdh-topolo gy b y the ´ etale topolog y and eh-topolo gy , r espectively . Theorem 5.1. Let k be a p erfect fi eld of positive characteristic p and assume th at r e solution of singula rities holds over k. The n for every scheme X essentially of finite type over k, the horizon tal maps in the diagr am of change-of-topology maps H · Zar ( X , TC n ( − ; p )) / /   H · et ( X , TC n ( − ; p ))   H · cdh ( X , TC n ( − ; p )) / / H · eh ( X , TC n ( − ; p )) . ar e weak equivalences. Pr oo f. The diag ram in the statement ag rees with the diagram of hom otopy equ aliz- ers of th e m aps indu ced b y the restriction an d Frob enius ma ps from the fo llowing diagram of change -of-top ology maps to itself. H · Zar ( X , TR n ( − ; p )) / /   H · et ( X , TR n ( − ; p ))   H · cdh ( X , TR n ( − ; p )) / / H · eh ( X , TR n ( − ; p )) . Hence, it suffices to pr ove that the horiz ontal maps in this diagr am are weak eq uiv a- lences. The top horizo ntal m ap induces a map from the spectral sequence E 2 s , t = H − s Zar ( X , a Zar TR n t ( − ; p )) ⇒ H − s − t Zar ( X , TR n ( − ; p )) to the spectral sequence E 2 s , t = H − s et ( X , a et TR n t ( − ; p )) ⇒ H − s − t et ( X , TR n ( − ; p )) . The map of E 2 -terms is giv en by the change-of -topolo gy map H − s Zar ( X , a Zar TR n t ( − ; p )) → H − s et ( X , a et TR n t ( − ; p )) and is an isomorphism, since a Zar TR n t ( − ; p ) and a et TR n t ( − ; p ) are q uasi-coher ent W n ( O X ) -modules. It follows t hat the map H · Zar ( X , TR n ( − ; p )) → H · et ( X , TR n ( − ; p )) is a weak equ i valence as stated. It remains to prove th at also the lower horizon tal map is a weak equiv alence. Suppose first th at X is essentially smooth over k . T he left-hand vertical m ap in- duces a map from the spectral sequence E 2 s , t = H − s et ( X , a et TR n t ( − ; p )) ⇒ H − s − t et ( X , TR n ( − ; p )) . 24 to the spectral sequence E 2 s , t = H − s eh ( X , a eh TR n t ( − ; p )) ⇒ H − s − t eh ( X , TR n ( − ; p )) . The map of E 2 -terms is the change-o f-topo logy map H − s et ( X , a et TR n t ( − ; p )) → H − s eh ( X , a eh TR n t ( − ; p )) which is an isomo rphism by Prop osition 4.1. Th erefore, the change-of-to pology map H · et ( X , TR n ( − ; p )) → H · eh ( X , TR n ( − ; p )) is a weak equiv alence. One sho ws similarly that the change-of-top ology map H · Zar ( X , TR n ( − ; p )) → H · cdh ( X , TR n ( − ; p )) is a weak equiv alence. Hence , for X essentially sm ooth over k , th e lower hor izontal map is a weak equiv alence. Finally , we show that f or a general schem e X essentially of finite type over k , the lower ho rizontal map in the d iagram at the beginn ing of the proo f is a weak equiv alence. Lemma 2.1 and th e ap propr iate d escent spectra l sequ ences show that the vertical maps in the diagram H · cdh ( X , TR n ( − ; p )) / /   H · eh ( X , TR n ( − ; p ))   H · cdh ( X red , TR n ( − ; p )) / / H · eh ( X red , TR n ( − ; p )) are weak eq uiv alences. H ence, we may assume that X is reduced . W e proceed by induction o n the dimension d = d im ( X ) . The case d = 0 has already been proved since every reduced 0-d imensional schem e o f finite ty pe over the per fect field k is smooth over k . So let X be a reduce d d -dimensional scheme separated and ess entially of finite type over k and assume th at the statement has b een proved fo r a ll schemes separated and essentially of finite type over k of dimension at m ost d − 1. W e m ay further assume that X is affine, a nd hence, separated . Therefore, by the assumptio n of resolution of singularities, there exists an abstract blo w-up square Y ′ i ′ / / p ′   X ′ p   Y i / / X where is X ′ essentially smooth over k and where the dim ensions of Y and Y ′ are strictly smaller than d . The change-of -topolo gy map gi ves rise to a m ap from the square of symmetric spectra H · cdh ( Y ′ , TR n ( − ; p )) H · cdh ( X ′ , TR n ( − ; p )) i ′ ∗ o o H · cdh ( Y , TR n ( − ; p )) p ′ ∗ O O H · cdh ( X , TR n ( − ; p )) i ∗ o o p ∗ O O 25 to the square of symmetric spectra H · eh ( Y ′ , TR n ( − ; p )) H · eh ( X ′ , TR n ( − ; p )) i ′ ∗ o o H · eh ( Y , TR n ( − ; p )) p ′ ∗ O O H · eh ( X , TR n ( − ; p )) i ∗ o o p ∗ O O both of wh ich are h omotopy cartesian. Th e map of u pper right- hand terms is a weak equiv alence, sinc e X ′ is essentially smooth over k . Moreover , since Y and Y ′ have dimension at most d − 1, the ma ps of the two left-ha nd terms are wea k equivalences by the ind uctive h ypothe sis. It follows th at th e map of lo wer r ight-han d terms is a weak equiv alence. This completes the proof. Pr oo f of Theor em C. Le t X be a d -dim ensional scheme essentially of finite ty pe o ver the field k . By Th eorem 5.1, it suffi ces to sho w that the change- of-topo logy map { H − q et ( X , TC n ( − ; p )) } → { H − q eh ( X , TC n ( − ; p )) } is an isomorphism o f pro- abelian gro ups for q < − d , an d an epimo rphism of pr o- abelian group s f or q = − d . W e recall from [1, X The orem 5.1] an d Theorem D that the p -coho mologica l dimension of X for the both the ´ etale topo logy and the eh-topology is at most d + 1. In particular, the spectra l sequences E 2 s , t = H − s et ( X , a et TC n t ( − ; p )) ⇒ H − s − t et ( X , TC n ( − ; p )) E 2 s , t = H − s eh ( X , a et TC n t ( − ; p )) ⇒ H − s − t eh ( X , TC n ( − ; p )) conv erge stro ngly and th e induce d filtration of the abutment is o f finite le ngth less than o r equal to d + 1. Therefo re, as n varies, these spectral sequences g iv e rise to strongly conv ergent spectral sequences of pro -abelian group s E 2 s , t = { H − s et ( X , a et TC n t ( − ; p )) } ⇒ { H − s − t et ( X , TC n ( − ; p )) } E 2 s , t = { H − s eh ( X , a eh TC n t ( − ; p )) } ⇒ { H − s − t eh ( X , TC n ( − ; p )) } . The map in question in duces a ma p of spectral sequences from the top spectra l se- quence to the botto m spectral sequ ence which , on E 2 -terms, is g i ven by th e cha nge- of-top ology map in sheaf coh omolog y . Th e two E 2 -terms are concentrated in the region wh ere − d − 1 6 s 6 0 and t > 0. Mo reover , T heorem 4 .8 shows that the change- of-top ology map is an isomorp hism of pro-abelian gr oups if t = 0, an d an epimorp hism of p ro-abe lian grou ps if t = 1 an d s = − d − 1. The theorem follows. Pr oo f of Theor em A. Let X be a d -dim ensional sch eme essentially o f finite ty pe ov er the field k . Then Th eorem B sho ws that the diagram of pro-spectr a K ( X ) / /   H · cdh ( X , K ( − ))   { TC n ( X ; p ) } / / { H · cdh ( X , TC n ( − ; p )) } 26 is homoto py car tesian. W e conclud e f rom Theorem C that the canonical map K q ( X ) → H − q cdh ( X , K ( − )) is an isomorphism, for q < − d , and an e pimorph ism for q = − d . T he g roups o n th e right-ha nd s ide are the abutment of t he spectral seque nce E 2 s , t = H s cdh ( X , a cdh K t ( − )) ⇒ H − s − t cdh ( X , K ( − )) . The assumption that resolution of sing ularities holds over k imp lies that ev ery cdh- covering of an objec t in Sch / X ad mits a refinemen t to a cdh-covering by sch emes essentially smo oth over k . This, in turn, implies that the sheaf a cdh K t ( − ) vanishes for t < 0 , and is canon ically isomor phic to the constant sheaf Z fo r t = 0. W e r ecall from [19, Theo rem 12.5] that the cdh-cohomo logical dimension of X is less than or equ al to d . Therefore, the g roups E 2 s , t are zero unless − d 6 s 6 0 and t > 0 . It follows th at H − q cdh ( X , K ( − )) is zero f or q < − d , and that H d cdh ( X , K ( − )) is cano nically isomorph ic to H d cdh ( X , Z ) . Hence K q ( X ) v anishes for q < − d as stated. Remark 5.2 . T he proof above also shows that K − d ( X ) surjects onto H d cdh ( X , Z ) . Theorem 5.3. Sup pose that str ong r eso lution of singularities holds over ever y infinite perfect fi eld of positive char acteristic p and let X b e a d - dimensiona l scheme o f fin ite type over some field of characteristi c p. Then K q ( X ) van ishes for q < − d . Pr oo f. W e first let F be a finite field of char acteristic p , and let X be a d -d imensional scheme essentially o f finite ty pe ov er F . W e let ℓ be a prime number, let F ′ be a Galois extension of F with Galois gr oup isomorph ic to the additive gr oup Z ℓ of ℓ - adic integers, an d let X ′ be the base-change o f X along Spec F ′ → Spec F . Then F ′ is an infinite perfect field. By assumption, strong resolution of singu larities hold s over F ′ , so T heorem A shows that K q ( X ′ ) vanishes for q < d . W e claim that the kern el of the pull-b ack map K ∗ ( X ) → K ∗ ( X ′ ) is an ℓ -pr imary torsion gr oup. T o see this, let F ′ i be the unique subfield of F ′ such that [ F ′ i : F ] = ℓ i , and let X ′ i be the base-ch ange of X along Spec F ′ i → Spec F . The composition of the pull-ba ck and push-f orwards map s K ∗ ( X ) → K ∗ ( X ′ i ) → K ∗ ( X ) is equa l to multiplication by ℓ i , an d hence, the kernel of the p ull-back map is an nihi- lated ℓ i . Moreover , it follows from [10, Section IV .8.5] that the canonical map colim i K ∗ ( X ′ i ) → K ∗ ( X ′ ) is an isomor phism. Since filtered colimits and finite limits of diagrams o f sets com- mute, we co nclude that the kernel o f the pull-back map K ∗ ( X ) → K ∗ ( X ′ ) is an ℓ - primary torsion g roup as claimed . It fo llows that for q < − d , K q ( X ) is an ℓ -primary torsion g roup . Since this is tr ue for every prim e num ber ℓ , we fin d that fo r q < − d , the group K q ( X ) is zero. W e let X b e a d - dimension al sche me of finite typ e over an arbitrary field k of characteristic p , and let k 0 ⊂ k be a perf ect su bfield. B y [10, Theo rem I V .8.8.2] , there exists an interm ediate field k 0 ⊂ k 1 ⊂ k such that k 1 is finitely g enerated over k 0 27 together with a sch eme X 1 of finite type over k 1 such that X is is omorp hic to the base- change of X 1 along Spec k → Spec k 1 . Let k 1 ⊂ k α ⊂ k be a finite g enerated extension of k 1 contained in k , and let X α be the base-chang e of X 1 along Spec k α → Spec k 1 . Then X α is of finite typ e over k α . Bu t the n X α is essentially of finite typ e over k 0 . Indeed , the field k α is the quotient field of a ring A α of finite type o ver k 0 , an d by [10, Theorem I V .8.8.2 ], we can find a sche me X α of finite type over A α such that X α is the g eneric fiber of X α over A α . Ther efore, the gro up K q ( X α ) vanishes for q < − d . Finally , it follows from [10, Proposition IV .8.5.5 ] that the canon ical ma p colim α K q ( X α ) → K q ( X ) from the filtered colimit indexed b y all intermedia te fields k 1 ⊂ k α ⊂ k fin itely g en- erated over k 1 is an isomo rphism. Hence, the g roup K q ( X ) vanishes for q < − d as stated. Acknowle dgements It is a pleasure to thank Christia n Haeseme yer f or a number of helpful con versat ions. W e are particular inde bted to Chuck W eibe l for pointing out a mistake in an earlier version to this paper . Finally , we are very grateful to an a nonymous referee for carefull y readi ng the paper and su ggesting sev eral improv ements. The work reported in this paper was done in part while the first author was visiting the Uni versi ty of T okyo. He wishes to thank the uni versity and T akeshi Saito in pa rticul ar for their hospi talit y . References 1. M. Arti n, A. Grothendieck, and J. L. 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