An Artin-Rees Theorem and applications to zero cycles

AN AR TIN-REES THEOREM AND APPLICA TIONS TO ZER O CYCLES AMALENDU KRISHNA Abstract. F or the smooth normalization f : X → X of a sing ular v ariety X ov er a ﬁeld k of characteristic zero, we show that fo r an y conducting subsc heme Y fo r the no rmalization, and for any i ∈ Z , the natural map K i ( X, X , nY ) → K i ( X, X , Y ) is zer o for all suﬃciently la rge n . As a n application, we pr ov e a formula fo r the Chow group of zero cycles on a quasi-pro jectiv e v ar ie t y X over k with Cohen-Macaulay isolated singular ities, in terms of an inverse limit of relative Chow g roups of a desing ularization e X relative to the mu ltiples of the exceptional divisor. W e use this formula to verify a conjecture of Sriniv as a bo ut the Cho w group of zero cycles on the aﬃne cone over a smo o th pro jectiv e v ar ie t y which is arith- metically Cohen-Maca ulay . 1. Introduction It has b een sho wn o v er t he years that the relativ e and the double relative K - groups pla y a v ery imp o rtan t r ole in the study o f algebraic K -theory and algebraic cycles on singular v ar ieties. These are particularly ve ry useful to ols in comparing the K - groups (or Cho w groups) of a singular sc heme and its normalization. The double relativ e K -groups are in general v ery diﬃcult to compute. Our aim in this paper is t wo fo ld. First w e prov e an Artin-Rees t ype theorem for the do uble relativ e K -groups, and then g iv e some v ery imp ortan t applications of this result to con vince the reader wh y suc h Artin-Rees t yp e results are indeed desirable if one w an ts to study a lg ebraic cycles on singular v arieties. In this pap er, a v a r iety will mean a reduced, connected and separated sc heme of ﬁnite type ov er a ﬁeld k of c haracteristic zero. All the underlying ﬁelds in this pap er will b e of c haracteristic zero. Let X b e singular v ariet y of dimension d o v er a ﬁeld k and let f : X → X b e the nor malization morphism. F or an y closed subsc heme Y of X , t he relativ e K -gr oups K ∗ ( X , Y ) a re deﬁned as the stable homotop y groups of the homotopy ﬁb er K ( X , Y ) of the map K ( X ) → K ( Y ) of non-connectiv e spectra as deﬁned in [27]. If Y ֒ → X is a conducting subsc heme for the normalization, w e put Y = Y × X X . W e deﬁne the double r elat ive K -gro ups K ∗ ( X , X , Y ) as the stable homotop y groups of the homotop y ﬁb er K ( X , X , Y ) of the induced natural map K ( X , Y ) → K ( X , Y ) o f no n-connectiv e sp ectra. As sho wn in [27 ], the underlying sp ectra to deﬁne the relativ e a nd double relative K -groups are non-connectiv e, and henc e the gro ups K i ( X , X , Y ) ma y be non-zero 1991 Mathematics Subje ct Classiﬁc ation. P rimary 14C15, 14C25; Seconda ry 14C35. Key wor ds a nd p hr ases. Algebraic cycles, Singular v arieties, K-theory , Ho ch schild homolog y . 1 2 AMALENDU KR ISHNA ev en wh en i is negative . F or a conducting subsc heme Y , w e let I b e the sheaf of ideals on X deﬁning Y . W e denote b y nY (for n ≥ 1), the conducting subsc heme of X for the normalization that is deﬁned by the sheaf of ideals I n . The f unctorialit y of the double relative g roups deﬁnes the natural maps K ∗ ( X , X , ( n + 1) Y ) → K ∗ ( X , X , nY ) for all n ≥ 1. If X is a ﬃne, it is classically know n t ha t K i ( X , X , Y ) = 0 f or i ≤ 0. F ur- thermore, it follows from [9] (Theorem 3.6) and a result of Cortinas ([8], Coro l- lary 0 .2) that K 1 ( X , X , Y ) ∼ = I Y / I 2 Y ⊗ Ω Y /Y . This sho ws that for X aﬃne, the map K i ( X , X , 2 Y ) → K i ( X , X , Y ) is zero for i ≤ 1. This has already b een ve ry useful in the study of zero cycles on singular sc hemes. W e refer the reader to [16], [14], [3] among sev eral others for some applications of suc h results to zero cycles on surfaces and threefolds. One o f the main diﬃcultie s in adv ancing the study of K -theory and alg ebraic cycles to the higher dimensional singular v arieties has b een the need for the follo wing general Artin-Rees t yp e r esult. Theorem 1.1. L et X b e a quasi-pr oje ctive v a riety of dimens ion d over a ﬁeld k and let f : X → X b e the normaliza tion of X . Assume that X is smo oth. Then for any c on d ucting subscheme Y ֒ → X for the n o rm alization, a n d for any i ∈ Z , the n a tur al map K i ( X , X , nY ) → K i ( X , X , Y ) is zer o for a l l suﬃciently la r ge n . As a lready said b efore, one of the motiv ations for proving the ab ov e theorem is its applications in computing the K -groups and the Chow groups of algebraic cycles on singular v arieties. These Cho w groups are us ually v ery diﬃcult to compute, ev en for v arieties with isolated singularities. In this pap er, we use the ab ov e theorem t o pro v e the follow ing form ula for the Cho w group of zero cycles on a quasi-pro jectiv e v ariet y X with Cohen-Macaula y is olated singularities, in terms of an inv erse limit of the relativ e Chow groups of a desingularization e X relativ e to the m ultiples of the exceptional divisor. Suc h a formula for the normal surface singularities was conjectured b y Blo c h and Sriniv a s ([24]) and prov ed b y Sriniv as and the author in [16] (Theorem 1.1). This formula was later v eriﬁed for the thr eefolds with Cohen-Macaula y isolated singularities in [14] (Theorem 1.1 ) . Before we state our next theorem, w e v ery brieﬂy recall the deﬁnition of the Cho w group o f zero cycles on singular v arieties and some other related notions. Let X b e a quasi-pro jectiv e singular v ariety of dimension d ov er a ﬁeld k . The cohomological Cho w group of zero cycles C H d ( X ) w as deﬁned by Levine and W eib el in [19] as the free ab elian group on the s mo oth closed points of X , mo dulo the zero cycles giv en as the sum of divisors of suitable rational functions on certain Cartier curve s on X . These curv es can b e assumed to be irreducible and disjoin t from the singular locus o f X , if it is normal. W e refer the reader to [12] fo r the full deﬁnition of C H d ( X ) and s ome of its impor tan t prop erties. Let F d K 0 ( X ) denote the subgroup of the Grothendiec k group of v ector bundles K 0 ( X ), generated b y the classes of smo oth co dimension d closed p oin ts on X . There is a natural surjectiv e map C H d ( X ) → F d K 0 ( X ). This map is know n ( cf. [17], Corollary 2.6 and Theorem 3.2) to b e an isomorphism, if the underlying ﬁeld k is algebraically closed and X is either an aﬃne or a pro jectiv e v a riet y o v er k with only normal is olated singularities. This isomorphism will b e used thro ughout this pap er. F or any closed subsc heme Z of X , let F d K 0 ( X , Z ) b e the subgroup AN AR TIN -REES THEOREM AN D APPLICA TIONS TO ZER O CYCLES 3 of K 0 ( X , Z ) ( deﬁned ab ov e) g enerated by the classes of smo oth closed p oints of X − Z ( cf. [7]). W e shall also call F d K 0 ( X , Z ) the relativ e Cho w group of zero cycles of X relativ e to Z . No w let X be a n ir r educible quasi-pro jectiv e v a riet y of dimension d with isolated singularities o v er a ﬁeld k , and let p : e X → X b e a resolution of singularit ies of X . Let E denote t he reduced exce ptional divisor on e X and let nE denote the n th inﬁnitesimal thic k ening of E . F rom the ab ov e deﬁnitions a nd from Lemma 7.1 b elo w (see also [14 ], T heorem 1.1 ), one has the follo wing comm utativ e diag r am for eac h n > 1 with all arr ows surjectiv e. (1.1) F d K 0 ( X ) t t t t i i i i i i i i i i i i i i i i i i     p ∗ * * * * U U U U U U U U U U U U U U U U U F d K 0 ( e X , nE ) / / / / F d K 0 ( e X , ( n − 1) E ) / / / / F d K 0 ( e X ) W e a lso recall from [14] that a resolution of singularities p : e X → X o f X is called a go o d resolution of singularities if p is obtained as a blow up of X along a closed subsc heme Z whose supp ort is the set of singular p oin ts of X . Suc h a resolution of s ingularities alw ay s exis ts. In fact, it is kno wn ( cf. [14], Lemma 2.5 ) using Hironak a’s theory that ev ery resolution of singularities is dominated by a go o d resolution of singularities. Hence in the study of the Cho w group of zero cycles o n X , it suﬃc es to consider only the go o d res olutions of singularities since C H d ( e X ) is a birational in v aria nt of a smo oth pro jectiv e v a riet y e X of dimension d . W e no w state our next result. Theorem 1.2. L et X b e a quasi-pr oje ctive variety of dimension d ≥ 2 over a ﬁeld k . L et p : e X → X b e a go o d r esolution of singularities of X with the r e duc e d exc eptional divisor E on e X . Assume that X has onl y Coh en-Mac aulay isolate d singularities. Then for al l suﬃciently lar ge n , the maps ( i ) F d K 0 ( e X , nE ) → F d K 0 ( e X , ( n − 1) E ) and ( ii ) F d K 0 ( X ) → F d K 0 ( e X , nE ) ar e isomorphisms. In p articular, if k = k and X is either aﬃne or p r oje ctive, then C H d ( X ) ∼ = lim ← − n F d K 0 ( e X , nE ) . Since rationa l singularit ies are Cohen-Macaula y , we imme diately get Corollary 1.3. If X is a quasi - pr oje ctive variety with only r ational and isola te d singularities, then one has the a b ove formula for C H d ( X ) . It turns out that the r elat ive C ho w groups (as deﬁned ab o v e) of the resolution of singularities, relative to the m ultiples of the exceptional divisor a r e often com- putable. Suc h computations hav e b een carried o ut b efore to compute the Cho w groups o f zero cycles on nor ma l surfaces ( cf. [16]) and certain threefolds ( cf. [14]). W e now pro ceed to carry these computations further in all dimensions and then use Theorem 1.2 to compute the Cho w group of zero cycles on certain classes of sin- gular v a r ieties. These results in particular giv e man y new examples of the v alidity 4 AMALENDU KR ISHNA of the singular analogue of the we ll kno wn conjecture Blo c h and its generalization in higher dimensions. Our result in this direction is inspired b y the following conjecture of Sriniv as ([25], Section 3), whic h in itself was probably motiv ated by the aﬃne analogue of a conjecture of Blo c h ab out the Cho w groups of zero cycles on smo oth pro jectiv e v arieties. Conjecture 1.4 (Sriniv a s) . L et Y ֒ → P N C b e a s mo oth and pr oje ctively normal variety over C of dimension d and let X = C ( Y ) b e the aﬃne c o n e over X . Then C H d +1 ( X ) = 0 if and only if H d ( Y , O Y (1)) = 0 . The ‘only if ’ part of this conjecture w as prov ed by Sriniv as himself in [25] (Corol- lary 2). So the ‘if ’ part remains op en at presen t. This conjecture w as v eriﬁed in [16] (Coro lla ry 1.4 ) when Y is a curve. This w as also veriﬁe d b y Consani ([6], Prop osition 3.1) in a v ery sp ecial case when Y is a h yp ersurface in P 3 . The fol- lo wing result v eriﬁes this conjecture when the aﬃne cone X is Cohen-Macaula y . In part icular, Sriniv as’ conjecture is v eriﬁed wh en Y is a complete in tersection in P N . Theorem 1.5. L et k b e an algebr aic al ly close d ﬁeld of char acteristic zer o and let Y ֒ → P N k b e a smo oth pr oje ctive variety of dimen sion d . L et X = C ( Y ) b e the aﬃne c one over Y . Assume that X is Cohen-Mac aulay. Then C H d +1 ( X ) = 0 if H d ( Y , O Y (1)) = 0 . Mor e ov e r, if k is a universal d o m ain and if X is the pr oje ctive c one over Y , then t he fol lowing ar e e q uiva lent. ( i ) H d ( Y , O Y (1)) = 0 ( ii ) H d +1 ( X , O X ) = 0 ( iii ) H d +1 ( X , Ω i X /k ) ∼ = − → H d +1 ( W , Ω i W /k ) for al l i ≥ 0 , wher e W is a r esol ution of singularities of X . ( iv ) C H d +1 ( X ) = 0 ( v ) C H d +1 ( X ) ∼ = C H d ( Y ) . If Y ֒ → P N k is a hypersurface of degree d , then there is an isomorphism H N − 1 ( Y , O Y (1)) ∼ = H 0  P N k , O P N ( d − N − 2)  and w e obtain the f ollo wing imme- diate consequenc e of Theorem 1.5. Corollary 1.6. L et Y ֒ → P N k b e a smo oth hyp ersurfac e of d e gr e e d and le t X = C ( Y ) b e the aﬃne c one over Y . Then C H N ( X ) = 0 if d ≤ N + 1 . The c on verse also holds if k is a univers a l do main. The follo wing application of Theorem 1.5 to the pro jectiv e mo dules on singular aﬃne algebras fo llows at once from Murthy ’s result ([21], Corollary 3.9). Corollary 1.7. L et Y ֒ → P N k b e as in The or em 1.5 a n d let A b e its homo gene ous c o or d inate ring. If H d ( Y , O Y (1)) = 0 , then every p r oje c tive m o dule over A of r ank at le ast d has a unimo dular element. If X is a no r ma l pro jectiv e surface with a resolution of singularities p : e X → X , then it w as shown in [16] ( Theorem 1 .3) that C H 2 ( X ) ∼ = C H 2 ( e X ) iﬀ H 2 ( X , O X ) ∼ = AN AR TIN -REES THEOREM AN D APPLICA TIONS TO ZER O CYCLES 5 H 2 ( e X , O e X ). If X has higher dimension, it is not exp ected that the isomor phism o f the top cohomology of the structure shea v es is the suﬃcie n t condition to conclude the isomorphism of the Cho w g r oups of zero cycles. Ho w ev er, Theorem 1.5 suggests that this might still b e true in the case of isolated singularities. As can b e seen in the a b o v e results , our applications o f Theorem 1.1 has b een restricted to computing some pieces of the group K 0 of singular v arieties. On the other hand, the conclusion of this t heorem applies also to hig her K i ’s. W e hop e that this result will b e a signiﬁcan t too l to study K 1 of singular v arieties for whic h hardly any thing is known. W e conclude this section with a brief outline o f this pap er. Our strategy of pro ving Theorem 1.1 is to use the Brown-Gersten spectral seq uence of [27] and the Cortinas’ pro of o f KABI-conjecture in [8] to reduce the problem to pro ving similar results fo r the Ho c hsc hild and cyclic homology . Since these homology groups ha v e canonical decomp ositions in terms of Andr´ e-Quillen homolog y , we ﬁrst pro v e a n Artin-Rees theorem for these homolo g y gro ups. W e giv e an o v ervie w of the Andr´ e- Quillen homology and then g eneralize a result of Quillen ([23], Theorem 6.15) in the nex t section. Section 3 con tains a pro of of our Artin-Rees t yp e result for the Andr ´ e-Quillen homology . In Section 4, w e deal with proving some reﬁnemen ts of this result for the sp ecial case of conducting ideals for the smo oth normalization of the essen tially of ﬁnite t yp e k - algebras, where k is a ﬁeld. Section 5 generalizes these results f o r the Andr ´ e-Quillen homology o v er any base ﬁeld whic h is contained in k . W e then use these resu lts in Section 6 to pro v e the Artin-Rees theorem for the double relativ e Ho c hsc hild a nd cyclic homology , and then giv e the pro of of Theorem 1 .1 . Section 7 con tains the pro of of Theorem 1.2. In Section 8, we relate the Cho w group of zero cycles with ce rtain cohomology of Milnor K -shea v es and then pro v e some general results ab out t hese cohomology groups, whic h are then used in the ﬁna l section to compute the Chow group of zero cycles o n the aﬃne cones. 2. Andr ´ e-Quillen Homology All the rings in this section will b e assumed to b e comm utativ e k -a lgebras, where k is a given ﬁeld of c haracteristic zero. Our aim in this section is to give a n o v erview of Andr ´ e-Quillen homology and related conce pts. This homology theory of alg ebras will b e o ur ma in to ol to prov e Theorem 1.1. W e also pro v e here a generalization of an Artin-Rees type theorem of Quillen [23 ], Theorem 6.15) for the Andr´ e-Quillen homology o f algebras whic h are ﬁnite ov er the base r ing . As Quillen sho ws (see also [1]), suc h a result itself has many intere sting consequences for the ho mology o f comm utativ e algebras. Apart from the ab ov e cited w orks of Andr ´ e and Quillen, our o ther basic reference for this ma t eria l including t he Ho c hsc hild and cyclic homolog y , is [20]. Let A be a comm uta tiv e ring whic h is e ssen tia lly of ﬁnite type o v er the ﬁeld k , and let B be an A -algebra. A simplicial A -algebra will mean a simplicial ob ject in the category of A -alg ebras. Let P ∗ b e a simplicial A - algebra. W e sa y that P ∗ is B - augmen ted if the natural map A → B fa ctors thro ugh A → P ∗ → B , where any A -algebra B is naturally consid ered a simplicial A -algebra with all the face and degeneracy maps taken as iden tit y map of B . The homotopy groups of a simplicial A -algebra P ∗ is deﬁned as the homoto py groups of the simplicial set P ∗ , whic h is same as the homotop y groups of t he simplicial A -mo dule P ∗ . The Dold-Kan corresp ondence implies that these homotop y g r oups are same as the 6 AMALENDU KR ISHNA homology gro ups of t he correspo nding c hain complex (whic h we also denote b y P ∗ ) o f A - mo dules. Using this equiv alence b etw een simplicial A -mo dules and c hain complexes of A -mo dules, w e shall o ften write the homot o p y groups π i ( M ∗ ) of a simplicial A -mo dule M ∗ homologically as H i ( M ∗ ) without an y ado. W e say that P ∗ is a free A -algebra if eac h P i ( i ≥ 0) is a symm etric algebra o v er a free A -module. Deﬁnition 2.1. A free simplicial A -algebra P ∗ is called a simplicial resolution of an A -algebra B if P ∗ is B - augmen ted suc h that the natural map H i ( P ∗ ) → H i ( B ) is an isomorphism fo r all i . It is known ( cf. [20], Lemma 3.5.2) that an y A - algebra B admits a free simplicial resolution and a ny t w o suc h resolutions are homotopy equiv alent. Before w e deﬁne the cotangen t mo dules, w e recall ( cf. [20], 1.6 .8) that for t wo simplicial B mo dules M ∗ and N ∗ , their tensor and w edge pro ducts are deﬁned degree-wise, i.e., ( M ∗ ⊗ B N ∗ ) i = M i ⊗ B N i and ( ∧ r B M ∗ ) i = ∧ r B M i for r , i ≥ 0 . The face and degeneracy maps of the tensor (or w edge) pro duct are degree-wise tensor (or wed ge) pro duct of the corresp onding maps. Since w e are in c haracteristic zero, the f ollo wing lemma relating tensor and exterior p ow ers of a simplicial B - mo dule is elemen tary . Lemma 2.2. F o r an y simplicial B -mo dule M ∗ and for r ≥ 0 , ∧ r B M ∗ is c anonic al ly a r etr ac t of ⊗ r B M ∗ . In p articular, H i ( ∧ r B M ∗ ) is a c anonic al dir e ct summan d of H i ( ⊗ r B M ∗ ) for al l i ≥ 0 . Pr o of . This is w ell know n and we only giv e a very brief ske tc h. Since the tensor and exterior p o w ers are deﬁned degree-wise, it suﬃces to pro v e the lemma fo r a B -mo dule M . One deﬁn es a B -linear map ⊗ r M alt − → ⊗ r M as alt ( a 1 ⊗ · · · ⊗ a r ) = 1 /r ! Σ σ ∈ S r sg n ( σ ) a σ (1) ⊗ · · · ⊗ a σ ( r ) . It is easy to c hec k that ‘ alt ’ is a pro jector and is natural with resp ect to map of B -mo dules. Moreo v er, if e ∧ r M denotes the image of this map, then the composite e ∧ r M → ⊗ r M → ∧ r M is a canonical isomorphism.  F or an A -alg ebra B , we deﬁne the cotangen t mo dule of B to b e the simplicial B -mo dule L B / A giv en b y (2.1)  L B / A  i = Ω 1 P i / A ⊗ P i B , where P ∗ is any free simplicial resolution o f B and the face and degeneracy maps of L B / A are induced b y those of P ∗ . The homotop y eq uiv alence of diﬀeren t simplicial resolutions of B implies that L B / A is a we ll-deﬁned o b ject in the homot op y categor y of simplicial B -mo dules. The And r ´ e - Quillen homology of B with coeﬃcien ts in a B -mo dule M is deﬁned as D q ( B / A, M ) = H q  L B / A ⊗ B M  , q ≥ 0 . AN AR TIN -REES THEOREM AN D APPLICA TIONS TO ZER O CYCLES 7 One deﬁnes the higher Andr´ e - Quillen ho mology o f B with co eﬃcien ts in a B - mo dule M as D ( r ) q ( B / A, M ) = H q  L ( r ) B / A ⊗ B M  for r , q ≥ 0 , where L ( r ) B / A = ∧ r B  L B / A  for r ≥ 0 . When M is same as B , w e write D ( r ) q ( B / A, M ) simply as D ( r ) q ( B / A ). It is easy to see from these deﬁnitions that D (0) 0 ( B / A, M ) = M and D (0) ≥ 1 ( B / A, M ) = 0. It is also known ( cf. [20], Theorem 3.5.8, Theorem 4.5.12) that for an y r ≥ 0 , there is a canonical isomorphism (2.2) D ( r ) 0 ( B / A, M ) ∼ = − → Ω r B / A ⊗ B M . Let A b e a k -a lgebra a s ab ov e and let { B n } n ≥ 0 b e an in v erse system of A - algebras. W e denote this in v erse s ystem b y B • . A B • -mo dule is a n in ve rse system { M n } n ≥ 0 of A -modules suc h tha t f o r each n ≥ 0, M n is in fact a B n -mo dule and the map M n f n − → M n − 1 is B n -linear suc h that these maps are compatible with maps in the inv erse system B • . A t ypical example in whic h w e shall be mostly in tereste d in is when I is a n ideal of A and M n = B n = A/I n +1 . Prop osition 2.3. L et B • b e an inverse system of A -algebr as. L et M • ∗ and N • ∗ b e the ﬂat simpl i cial B • -mo dules such that for e ach q ≥ 0 and for e ac h n 0 ≥ 0 , the map H q ( M n ∗ ) → H q ( M n 0 ∗ ) is zer o for al l n ≫ n 0 . Then for e ach q ≥ 0 and for e ach n 0 ≥ 0 , the ma p H q ( M n ∗ ⊗ B n N n ∗ ) → H q  M n 0 ∗ ⊗ B n 0 N n 0 ∗  is zer o for al l n ≫ n 0 . Pr o of . F or any A alg ebra B , let C : S impM od ( B ) → C h ≥ 0 ( B ) b e the D old- Kan functor fro m the cat ego ry of simplicial B - mo dules to the category o f c hain complexes of B -mo dules whic h are b ounded b elo w at zero. This functor tak es a simplicial mo dule M ∗ to itself and the diﬀeren t ial at eac h lev el is the alternating sum of the face maps at that lev el. Then the Eilen berg- Zilb er theorem ( cf. [2 0], 1.6.12) implies that there is a natural Alexander-Whitney map C ( M ∗ ⊗ B N ∗ ) → C ( M ∗ ) ⊗ B C ( N ∗ ) whic h is a quasi-isomorphism. Here the term o n the righ t is the tensor pro duct in the category of c hain complexes, whic h is give n as the total complex of t he double complex ( M ∗ , N ∗ ) i,j = M i ⊗ B N j . Hence it suﬃces to pro v e the lemma for the tensor pro duct of c hain complexes. Since the double complex { M n i ⊗ B n N n j } i,j ≥ 0 lies o nly in the ﬁrst quadra nt, there is a con v ergen t sp ectral sequence ([28], 5.6.1) n E 2 p,q = H p  · · · → H q ( M n ∗ ⊗ B n N n j ) → · · · → H q ( M n ∗ ⊗ B n N n 1 ) → H q ( M n ∗ ⊗ B n N n 0 )  ⇒ H p + q ( M n ∗ ⊗ B n N n ∗ ) . 8 AMALENDU KR ISHNA This spectral sequence is compatible with the maps in the in v erse systems { B n } , { M n } and { N n } and w e g et an in vers e sys tem of spectral se quences n E 2 p,q   ⇒ H p + q ( M n ∗ ⊗ B n N n ∗ )   n − 1 E 2 p,q ⇒ H p + q  M n − 1 ∗ ⊗ B n − 1 N n − 1 ∗  . No w as N n j is a ﬂat B n -mo dule fo r eac h n and j , w e see that n E 2 p,q is same as H p ( H q ( M n ∗ ) ⊗ B n N n ∗ ). In particular, we see that for eac h p, q ≥ 0 and for each n 0 ≥ 0, the natural map n E 2 p,q → n 0 E 2 p,q is ze ro fo r all n ≫ n 0 and hence the map n E i p,q → n 0 E i p,q is ze ro for all n ≫ n 0 and f o r a ll i ≥ 2. F ro m this w e conclude that for a ﬁxed q ≥ 0, there is a ﬁltration 0 = F n − 1 ⊂ F n 0 ⊂ · · · F n q − 1 ⊂ F n q ( H q ( M n ∗ ⊗ B n N n ∗ )) = H q ( M n ∗ ⊗ B n N n ∗ ) and a map of ﬁltered B n -mo dules H q ( M n ∗ ⊗ B n N n ∗ ) → H q  M n − 1 ∗ ⊗ B n − 1 N n − 1 ∗  suc h that for eac h j, n 0 ≥ 0, the map F n j /F n j − 1 = n E ∞ j,q − j → n 0 E ∞ j,q − j = F n 0 j /F n 0 j − 1 is zero for all n ≫ n 0 . No w w e sho w by induction that f or an y 0 ≤ j ≤ q and any n 0 ≥ 0, the map F n j → F n 0 j is zero f o r all n ≫ n 0 . This will ﬁnis h t he pro of of the prop osition. The follo wing tric k (whic h w e call the doubling tric k ) to do this will b e used rep eatedly in this pap er. W e ﬁx j with 0 ≤ j ≤ q and b y induction, assume that there e xist n 1 ≫ n 0 and n 2 ≫ n 1 suc h that in the commutativ e diagram (2.3) 0 / / F n j − 1   / / F n j / /   n E ∞ j,q − j   / / 0 0 / / F n 1 j − 1   / / F n 1 j / /   n 1 E ∞ j,q − j   / / 0 0 / / F n 0 j − 1 / / F n 0 j / / n 0 E ∞ j,q − j / / 0 , the bott o m left and the b ottom righ t v ertical arrows are zero for all n ≥ n 1 , and the top left and the top right vertical arrows are ze ro f o r all n ≥ n 2 . A diagram c hase sho ws that the comp osite middle vertic al arro w is zero for all n ≥ n 2 .  Corollary 2.4. L et B • b e an inverse s ystem o f A -algebr as and let M • ∗ b e a ﬂat simplicial B • -mo dule. Assume that fo r e ach q ≥ 0 and for e ach n 0 ≥ 0 , the m ap H q ( M n ∗ ) → H q ( M n 0 ∗ ) is ze r o for al l n ≫ n 0 . Then for e ach r ≥ 1 and q , n 0 ≥ 0 , the m ap H q  ∧ r B n M n ∗  → H q  ∧ r B n 0 M n 0 ∗  is zer o for al l n ≫ n 0 . Pr o of . By Lemma 2.2, it suﬃces to pro v e the corollary when the exterior p ow ers are replaced b y the corresp onding tensor p o w ers, when it fo llo ws directly from Prop osition 2.3 and induc tion on r .  The following result w as prov ed by Quillen ([23], Theorem 6.15) for r = 1 . AN AR TIN -REES THEOREM AN D APPLICA TIONS TO ZER O CYCLES 9 Corollary 2.5. L et A b e an essential ly of ﬁnite typ e k -algebr a. L e t I b e an ide al of A and put B n = A/I n +1 for n ≥ 0 . Then for e ach r ≥ 1 a n d q , n 0 ≥ 0 , the natur al map D ( r ) q ( B n / A ) → D ( r ) q ( B n 0 / A ) is zer o for al l n ≫ n 0 . Pr o of . W e see fr om 2.1 that L B n / A is a free simplicial B n -mo dule. Since A is no etherian, we can apply [23] (Theorem 6.15) to conclude tha t for eac h q , n 0 ≥ 0, there exists an N such that the natural map D q ( B nn 0 / A, B n 0 ) → D q ( B n 0 / A, B n 0 ) = D q ( B n 0 / A ) is zero for all n ≥ N . Since the map D q ( B nn 0 / A ) → D q ( B n 0 / A ) is the comp osite of the map D q ( B nn 0 / A ) = D q ( B nn 0 / A, B nn 0 ) → D q ( B nn 0 / A, B n 0 ) → D q ( B n 0 / A, B n 0 ) , w e see that the map D q ( B nn 0 / A ) → D q ( B n 0 / A ) is zero f o r all n ≥ N . This in turn implies tha t the natural map D q ( B n / A ) → D q ( B n 0 / A ) is zero for all n ≫ n 0 (in f a ct for a ll n ≥ N n 0 ). Now w e a pply Corollary 2 .4 to the inv erse system B • = { B n } and t he free simplicial module M • ∗ = { L B n / A } to conclude the pro of of the corollary .  3. Ar tin-R ees theore m for Andr ´ e-Quillen Homology Let k b e a ﬁeld and let A b e a k -alg ebra whic h is essen tially of ﬁnite t yp e ov er k . Let I b e a n ideal of A and put B n = A/I n +1 for n ≥ 0. Then B • = { B n } is an inv erse sy stem of ﬁnite A -algebras. Moreov er, n D ( r ) q ( B n /k ) D ( r ) q ( A/k ) o n ≥ 0 is a B • -mo dule. Similarly , n Ker  D ( r ) q ( A/k , B n ) → D ( r ) q ( B n /k ) o n ≥ 0 is also a B • -mo dule. Our aim in this section is to pro v e an Artin-Rees t yp e theorem for these t wo mo dules. W e b egin with the follo wing elemen tary result. Lemma 3.1. L et A b e any k -alge b r a and let 0 → M ′ ∗ → M ∗ → M ′′ ∗ → 0 b e a short exact se quenc e of fr e e simplicial A -mo dules . Then ther e exi s ts a c onver- gent sp e ctr al se quenc e E 1 p,q = H q − p  ∧ p M ′ ∗ ⊗ A ∧ r − p M ′′ ∗  ⇒ H q − p ( ∧ r M ∗ ) . This sp e ctr al se quenc e is natur a l for morphisms of k -algebr as and morphisms of short e xact se quenc es of fr e e simplicial mo d ules . Pr o of . Exactne ss of simplicial mo dules means that it is exact at eac h lev el a nd the exactness is compatible with the face and the degeneracy maps. F or each i ≥ 0, w e can deﬁne a decreasing ﬁnite ﬁltration on ∧ r M i b y deﬁning F j ∧ r M i to b e the A -submo dule generated b y the forms of the t yp e  a 1 ∧ · · · ∧ a r | a i 1 , · · · , a i j ∈ M ′ i for some 1 ≤ i 1 ≤ · · · ≤ i j ≤ r  . Then w e ha v e ∧ r M i = F 0 ∧ r M i ⊇ · · · ⊇ F r ∧ r M i ⊇ F r +1 ∧ r M i = 0 and it is easy to c hec k that for 0 ≤ j ≤ r , the map β j i : ∧ j M ′ i ⊗ A ∧ r − j M i → F j ∧ r M i , 10 AMALENDU KR ISHNA β j i (( a 1 ∧ · · · ∧ a j ) ⊗ ( b 1 ∧ · · · ∧ b r − j )) = a 1 ∧ · · · ∧ a r ∧ b 1 ∧ · · · ∧ b r − j descends to an isomorphism of quotien ts (3.1) β j i : ∧ j M ′ i ⊗ A ∧ r − j M ′′ i ∼ = − → F j ∧ r M i F j +1 ∧ r M i . W e also see from the ab ov e deﬁnition of the ﬁltration and the maps β j i that this ﬁltration and the isomorphisms in 3.1 are compatible with the morphisms of short exact seque nces. In particular, they are compatible w ith the face and the degen- eracy maps. Th us w e g et a decreasing ﬁltr a tion { F j ∧ r M ∗ } 0 ≤ j ≤ r of the simplicial mo dule ∧ r M ∗ suc h that for each 0 ≤ j ≤ r , there is a nat ura l isomorphism (3.2) ∧ j M ′ ∗ ⊗ A ∧ r − j M ′′ ∗ ∼ = − → F j ∧ r M ∗ F j +1 ∧ r M ∗ . This ﬁltration on the simplicial mo dule ∧ r M ∗ giv es ([28], 5.5) a conv ergen t sp ectral sequence E 1 p,q = H q − p  F p ∧ r M ∗ F p +1 ∧ r M ∗  ⇒ H q − p ( ∧ r M ∗ ) with diﬀeren tial E 1 p,q → E 1 p +1 ,q . The isomorphism of 3.2 now complete s the pro of of the existence of the sp ectral sequence. The f unctoria lit y with the morphisms of k -algebras and morphisms of exact sequences of simplicial mo dules is clear from the deﬁnition of the ﬁltration a b o v e, whic h is preserv ed under a morphism of exact sequence s.  Corollary 3.2. L et A b e an essential ly of ﬁnite typ e algebr a over a ﬁ eld k and let l ⊂ k b e any subﬁeld. Then ther e is a c on v e r gent sp e ctr al se quenc e E 1 p,q = Ω p k /l ⊗ k D ( r − p ) q − p ( A/k ) ⇒ D ( r ) q − p ( A/l ) . Pr o of . W e hav e ([2 3 ], pro of o f Theorem 5.1) a short exact sequence o f free simplicial A -mo dules 0 → L k /l ⊗ k A → L A/l → L A/k → 0 . Put K A/l = L k /l ⊗ k A . Then Lemma 3.1 giv es us a con v ergen t spectral sequence (3.3) E 1 p,q = H q − p  ∧ p K A/l ⊗ A ∧ r − p L A/k  ⇒ H q − p  ∧ r L A/l  . T o iden tify the E 1 -terms, w e see from the pro of o f Prop osition 2.3 tha t fo r eac h p, q ≥ 0, there is a con v ergent sp ectral sequence ′ E 2 i,j = H i  H j  ∧ p K A/l  ⊗ A ∧ r − p L A/k  ⇒ H i + j  ∧ p K A/l ⊗ A ∧ r − p L A/k  . Since A is k -ﬂat, w e hav e H j  ∧ p K A/l  = H j  ∧ p L k /l ⊗ k A  = H j  ∧ p L k /l  ⊗ k A = D ( p ) j ( k /l ) ⊗ k A. Since this last gro up is a free A - mo dule, w e obta in ′ E 2 i,j = D ( p ) j ( k /l ) ⊗ k D ( r − p ) i ( A/k ) . No w as k is a direct limit of its subﬁe lds whic h are ﬁnitely generated o v er l , and since the Andr ´ e-Quillen homology comm utes with direct limits ([23], 4.11), w e see AN AR TIN -REES THEOREM AN D APPLICA TIONS TO ZER O CYCLES 11 that D ( p ) j ( k /l ) is a direct limit of the Andr ´ e-Quillen homology of the subﬁelds o f k whic h are ﬁnitely generated ov er l . In particular, w e ha ve ([20], Theorem 3.5.6 ) D ( p ) j ( k /l ) =  Ω p k /l if p ≥ 0 , j = 0 0 otherwise . Th us w e get ′ E 2 i,j =  Ω p k /l ⊗ k D ( r − p ) i ( A/k ) if i ≥ 0 , j = 0 0 if j > 0 . In particular, this spectral sequence degene rates at ′ E 2 and w e get fo r p, i ≥ 0, H i  ∧ p K A/l ⊗ A ∧ r − p L A/k  = Ω p k /l ⊗ k D ( r − p ) i ( A/k ) . Putting this in our spectral sequence of 3.3, w e get the pro of of the coro llary .  Let A b e an essen tially of ﬁnite t yp e a lg ebra ov er a ﬁeld k and let B • = { B n = A/I n +1 } be the in v erse system of ﬁnite A -algebras as deﬁned in the b eginning of this section. Lemma 3.3. F or a n y given r, q , n 0 ≥ 0 , the na tur al m ap D ( r ) q ( A/k , B n ) D ( r ) q ( A/k ) → D ( r ) q ( A/k , B n 0 ) D ( r ) q ( A/k ) is zer o for al l n ≫ n 0 . Pr o of . F or r = 0, b oth sides are zero, so w e can a ssume r ≥ 1. W e ﬁrst observ e that for any n ≥ 0, one has D ( r ) q ( A/k , B n ) D ( r ) q ( A/k ) ∼ = − → D ( r ) q ( A/k , B n ) D ( r ) q ( A/k ) ⊗ A B n . By [23] (4.7), there is a con v ergent sp ectral se quence n E 2 p,q = T or A p  D ( r ) q ( A/k ) , B n  ⇒ D ( r ) p + q ( A/k , B n ) . This sp ectral sequence is compatible with the maps B n ։ B n − 1 and giv es a ﬁnite ﬁltration of D ( r ) q ( A/k , B n ) 0 = F n − 1 ⊆ F n 0 ⊆ · · · ⊆ F n q − 1 ⊆ F n q = D ( r ) q ( A/k , B n ) suc h that n E ∞ j,q − j = F n j /F n j − 1 for 0 ≤ j ≤ q and the edge map giv es D ( r ) q ( A/k ) ⊗ A B n ։ F n 0 . Hence it suﬃces to show that the natural map (3.4) D ( r ) q ( A/k , B n ) F n 0 D ( r ) q ( A/k , B n ) → D ( r ) q ( A/k , B n 0 ) F n 0 0 D ( r ) q ( A/k , B n 0 ) is zero f or all n ≫ n 0 . Using the ab ov e ﬁltration, an induction on j and the doubling tric k of 2.3, this is reduced to show ing t hat for 1 ≤ j ≤ q , the map n E 2 j,q − j → n 0 E 2 j,q − j is zero for all n ≫ n 0 . But f o r j ≥ 1, w e hav e n E 2 j,q − j = 12 AMALENDU KR ISHNA T or A j  D ( r ) q − j ( A/k ) , B n  . F urthermore, A is a lo calization of a ﬁnite t yp e k -alg ebra and so b y [23] (Prop osition 4.12, Theorem 5.4(i)), D ( r ) q ( A/k ) is a ﬁnite A -mo dule for all r , q ≥ 0. Hence by [1] (Prop osition 1 0, Lemma 11), the map T or A j  D ( r ) q − j ( A/k ) , B n  → T or A j  D ( r ) q − j ( A/k ) , B n 0  is zero for all n ≫ n 0 .  Prop osition 3.4. L et A and B • b e as ab o ve. Then for any given r , q , n 0 ≥ 0 , the natur al map D ( r ) q ( B n /k ) D ( r ) q ( A/k ) → D ( r ) q ( B n 0 /k ) D ( r ) q ( A/k ) is zer o for al l n ≫ n 0 . Pr o of . F or r = 0, b oth sides are zero, so w e assume r ≥ 1. Since the map D ( r ) q ( A/k ) → D ( r ) q ( B n /k ) factors through the map D ( r ) q ( A/k ) → D ( r ) q ( A/k , B n ), one has for all n , the natural exact sequence (3.5) D ( r ) q ( A/k , B n ) D ( r ) q ( A/k ) → D ( r ) q ( B n /k ) D ( r ) q ( A/k ) → D ( r ) q ( B n /k ) D ( r ) q ( A/k , B n ) → 0 . Using the doubling t r ic k of 2.3 and Lemma 3.3, w e only need to sho w that for an y giv en r ≥ 1 and q , n 0 ≥ 0, the natural map (3.6) D ( r ) q ( B n /k ) D ( r ) q ( A/k , B n ) → D ( r ) q ( B n 0 /k ) D ( r ) q ( A/k , B n 0 ) is zero for all n ≫ n 0 . F or n ≥ 0 , we put K B n /k = L A/k ⊗ A B n . The n w e o bserv e t hat for r , q ≥ 0, D ( r ) q ( A/k , B n ) is same as H q  ∧ r A L A/k  ⊗ A B n  = H q  ∧ r B n K B n /k  . F or all n ≥ 0, w e ha v e an exact sequence ([23 ], Theorem 5.1) of free simplicial B n -mo dules 0 → L A/k ⊗ A B n → L B n /k → L B n / A → 0 . Hence b y Lemma 3.1, there is a con v ergent sp ectral se quence n E 1 p,q = H q − p  ∧ p K B n /k ⊗ B n ∧ r − p L B n / A  ⇒ H q − p  ∧ r L B n /k  . This sp ectral sequence is compatible with the maps B n ։ B n − 1 and giv es a ﬁnite ﬁltration of H q  ∧ r L B n /k  (3.7) H q  ∧ r L B n /k  = F n 0 ⊇ F n 1 ⊇ · · · ⊇ F n r ⊇ F n r +1 = 0 with F n j /F n j +1 ∼ = n E ∞ j,q + j for 0 ≤ j ≤ r and a morphism of ﬁltered mo dules H q  ∧ r L B n /k  → H q  ∧ r L B n − 1 /k  . F urthermore, the edge map give s a surjec- tion H q  ∧ r K B n /k  ։ F n r H q  ∧ r L B n /k  . In particular, w e ha v e H q ( ∧ r L B n /k ) H q ( ∧ r K B n /k ) ∼ = − → AN AR TIN -REES THEOREM AN D APPLICA TIONS TO ZER O CYCLES 13 H q ( ∧ r L B n /k ) F n r H q ( ∧ r L B n /k ) . Hence using the ﬁltra t ion in 3.7, a n induction on j and the dou- bling tric k of 2.3, it suﬃces to sho w that for 0 ≤ j ≤ r − 1, the natural map n E 1 j,q + j → n 0 E 1 j,q + j is zero for all n ≫ n 0 . But this follows fro m Prop osition 2.3 and Corollary 2.5. This prov es 3.6 and hence the prop osition.  Lemma 3.5. L et the k -algebr as A and B • b e as ab ove. T hen for any given r , q , n 0 ≥ 0 , the na tur al m ap Ker  D ( r ) q ( A/k , B n ) → D ( r ) q ( B n /k )  → Ker  D ( r ) q ( A/k , B n 0 ) → D ( r ) q ( B n 0 /k )  is zer o for al l n ≫ n 0 . Pr o of . F or r = 0, b oth sides are zero, so w e assume r ≥ 1. W e ha v e seen in the pro of of Prop osition 3.4 that fo r all n ≥ 0 , D ( r ) q ( B n /k ) = H q  ∧ r L B n /k  has a ﬁnite ﬁltration { F n j } 0 ≤ j ≤ r suc h that H q  ∧ r K B n /k  ։ F n r H q  ∧ r L B n /k  . Th us w e can replace D ( r ) q ( B n /k ) b y F n r H q  ∧ r L B n /k  = n E ∞ r,q + r = n E r +1 r,q + r in the statemen t of the lemma. No w for 0 ≤ j ≤ r , there is an exact sequence n E j +1 r − j − 1 ,q + r + j → n E j +1 r,q + r → n E j +2 r,q + r → 0 , whic h giv es a ﬁnite increasing ﬁltrat io n of Ker  D ( r ) q ( A/k , B n ) = n E 1 r,q + r → n E r +1 r,q + r  0 = F n − 1 ⊆ F n 0 ⊆ · · · ⊆ F n r − 1 ⊆ F n r = Ker  n E 1 r,q + r → n E r +1 r,q + r  suc h that for all 0 ≤ j ≤ r , n E j +1 r − j − 1 ,q + r + j ։ F n j / F n j − 1 . Th us b y using an induction on j and the doubling tric k as b efore, it suﬃces to sho w that for 0 ≤ j ≤ r and for giv en n 0 ≥ 0, the natural map n E 1 r − j − 1 ,q + r + j → n 0 E 1 r − j − 1 ,q + r + j is ze ro for all n ≫ n 0 . But this ag ain f o llo ws from Prop o sition 2.3 and Corollary 2.5.  4. Andr ´ e-Quillen Homology and Normaliza tion I Let A b e an inte gral do main whic h is essen tially of ﬁnite type algebra o v er a ﬁeld k . Assume that A is singular, and let f : A → B b e t he normalization morphis m of A . W e assume that B is smooth o ver k . Our aim in this section is to estimate the Andr ´ e-Quillen homology of the conducting ideals for this normalization. W e b egin with estimating the k ernels of the maps b et w een diﬀeren tial forms. F or an y conducting ideal I ⊂ A for the normalizatio n and for r ≥ 0, let Ω r ( A,I ) /l (resp Ω r ( B ,I ) /l ) denote the k ernel of the map Ω r A/l ։ Ω r ( A/I ) /l (resp Ω r B /l ։ Ω r ( B /I ) /l ) for an y subﬁeld l ⊂ k . Lemma 4.1. L et A and B b e as ab ove. Then for any given c onducting ide al I ⊂ A for t he normalization and fo r any r ≥ 0 , the natur al map Ω r ( A,I n ) /k → Ω r ( B ,I n ) /k is inje c tive fo r al l suﬃciently lar ge n . 14 AMALENDU KR ISHNA Pr o of . W e see from 2.2 and Lemma 3.5 that for an y g iv en n 0 ≥ 0 the natural map (4.1) Ker  Ω r A/k ⊗ A A/I n u A n − → Ω r ( A/I n ) /k  → Ker  Ω r A/k ⊗ A A/I n 0 u A n 0 − − → Ω r ( A/I n 0 ) /k  is zero for all n ≫ n 0 . In the same wa y , t he map Ker( u B n ) → Ker( u B n 0 ) is zero for all n ≫ n 0 . On the other hand, we hav e a comm utativ e diag r a m of ex act sequence s for an y n ≥ 0. 0 / / I n Ω r A/k / /   Ω r ( A,I n ) /k   / / Ker( u A n ) / /   0 0 / / I n Ω r B /k / / Ω r ( B ,I n ) /k / / Ker( u B n ) / / 0 W e claim that the map I n Ω r A/k → I n Ω r B /k is injectiv e for all n ≫ 0. T o see this, note that Ω r A/k is a ﬁnite A -mo dule and w e can apply the Artin-Rees t heorem to ﬁnd a c > 0 suc h that all n > c , one has  I n Ω r A/k ∩ Ω r ( A,B ) /k  ⊆ I n − c  I c Ω r A/k ∩ Ω r ( A,B ) /k  ⊆ I n − c Ω r ( A,B ) /k , where Ω r ( A,B ) /k = Ker  Ω r A/k → Ω r B /k  . On the o ther hand, the ﬁnite A -mo dule Ω r ( A,B ) /k is support ed on the supp ort of I and hence I n − c Ω r ( A,B ) /k = 0 for all n ≫ 0. This prov es the claim. Using the claim in the ab ov e diagram, w e see that there exists an n 0 suc h tha t Ker  Ω r ( A,I n ) /k → Ω r ( B ,I n ) /k  ֒ → Ker( u A n ) f o r all n ≥ n 0 . No w w e use 4.1 to conclude that the map Ker  Ω r ( A,I n ) /k → Ω r ( B ,I n ) /k  → Ker  Ω r ( A,I n 0 ) /k → Ω r ( B ,I n 0 ) /k  is ze ro for all n ≫ n 0 . Ho wev er, this map is clearly injectiv e. Hence w e m ust ha v e Ker  Ω r ( A,I n ) /k → Ω r ( B ,I n ) /k  = 0 for all n ≫ 0.  Lemma 4.2. L et f : A → B b e as ab ove. Th e n for any c on ducting ide al I for the normalization and for any r , q ≥ 1 , the natur al map D ( r ) q ( A/k ) u A n − → D ( r ) q ( A/I n /k ) is inje c tive fo r al l n ≫ 0 . Pr o of . W e ﬁrst obse rv e that D ( r ) q ( B /k ) = 0 for q ≥ 1 a s B is smoo t h ([20], The o- rem 3.5.6). Since the homology groups D ( r ) q ( A/k ) are ﬁnite A -mo dules and since they comm ute with t he lo calization ([23], Th eorem 5.4 (i)), w e see as b efore that I n D ( r ) q ( A/k ) = 0 for all n ≫ 0 whenev er q ≥ 1. In particular, for any r , q ≥ 1 there exits N ≫ 0 suc h that f o r all n ≥ N , (4.2) D ( r ) q ( A/k ) ∼ = − → D ( r ) q ( A/k ) ⊗ A A/I n . AN AR TIN -REES THEOREM AN D APPLICA TIONS TO ZER O CYCLES 15 F or r, q ≥ 1 and n ≥ 0, w e hav e a con ve rgen t spectral s equence n E 2 p,q = T or A p  D ( r ) q ( A/k ) , A/I n  ⇒ D ( r ) p + q ( A/k , A/I n ) with diﬀeren tial n E 2 p,q → n E 2 p − 2 ,q +1 . This giv es a ﬁltration 0 = F n − 1 ⊆ F n 0 ⊆ · · · ⊆ F n q − 1 ⊆ F n q = D ( r ) q ( A/k , A/I n ) and map of ﬁltered mo dules D ( r ) q ( A/k , A/I n ) → D ( r ) q ( A/k , A/I n − 1 ). The edge map further giv es a surjection D ( r ) q ( A/k ) ⊗ A A/I n = n E 2 0 ,q ։ F n 0 = n E ∞ 0 ,q = n E q +2 0 ,q . W e no w sho w that f o r r, q ≥ 1 and n 0 ≥ 0, the natural map (4.3) Ker  D ( r ) q ( A/k ) ⊗ A A/I n → F n 0 D ( r ) q ( A/k , A/I n )  → Ker  D ( r ) q ( A/k ) ⊗ A A/I n 0 → F n 0 0 D ( r ) q ( A/k , A/I n 0 )  is zero for all n ≫ n 0 . F or 0 ≤ j ≤ q , there is an exact sequence n E j +2 j +2 ,q − j − 1 → n E j +2 0 ,q → n E j +3 0 ,q → 0 . Th us by letting Γ n j = Ker  n E 2 0 ,q ։ n E j +2 0 ,q  for 0 ≤ j ≤ q , w e get a ﬁltration 0 = Γ n 0 ⊆ Γ n 1 ⊆ · · · ⊆ Γ n q of Γ n q suc h that n E j +2 j +2 ,q − j − 1 ։ Γ n j +1 / Γ n j . Th us to prov e 4.3, it suﬃces to s ho w by an induction on j and the doubling trick that for giv en r, q , n 0 ≥ 1 and for 0 ≤ j ≤ q , the natural map n E 2 j +2 ,q − j − 1 → n 0 E 2 j +2 ,q − j − 1 is zero for n ≫ n 0 . But for n, j ≥ 0 w e hav e n E 2 j +2 ,q − j − 1 = T or A j +2  D ( r ) q − j − 1 ( A/k ) , A/I n  and the map T or A j +2  D ( r ) q − j − 1 ( A/k ) , A/I n  → T or A j +2  D ( r ) q − j − 1 ( A/k ) , A/I n 0  is ze ro b y for n ≫ n 0 b y [1 ]( Pro p osition 10, Lemma 11) as D ( r ) q ( A/k ) are all ﬁnite A -mo dules. This prov es 4 .3 . If D ( r ) q ( A/k ) θ A n − → D ( r ) q ( A/k , A/I n ) and D ( r ) q ( A/k , A/I n ) v A n − → D ( r ) q ( A/I n /k ) denote the natural maps, then for an y r , q , n ≥ 1, we get a natural exact sequence 0 → Ker  θ A n  → Ker  u A n  → Ker  v A n  . No w for an y n 0 ≥ N , w e can use 4.2 and 4.3 to conclude that t he map Ker  θ A n  → Ker  θ A n 0  is zero for all n ≫ n 0 . The map Ker  v A n  → Ker  v A n 0  is zero for all n ≫ n 0 b y Lemma 3.5. The doubling tric k ag ain shows tha t for an y r , q ≥ 1 and n 0 ≥ N , the natural map K er  u A n  → Ker  u A n 0  is zero for all n ≫ n 0 . But this last map is clearly injectiv e. Hence we m ust hav e Ker  u A n  = 0 for all n ≫ 0.  16 AMALENDU KR ISHNA 5. Andr ´ e-Quillen Homology and Normaliza tion I I Most of the pro ofs in the previous s ection relied on the fact that the algebra A is es sen tially of ﬁnite t yp e o v er the ﬁeld k . Our aim in this section is to generalize the results of the previous section to the case when the base ring for the Andr ´ e- Quillen homology o f k -algebras is a n y subﬁeld of k . Our ev entual application will need these results when the base ring is the ﬁeld of ra tional n um b ers. So let A be an in tegral do ma in whic h is essen tially of ﬁnite ty p e alg ebra ov er a ﬁeld k . Let f : A → B b e the normalization of A suc h that B is smo oth. Let l ⊂ k b e any subﬁeld. The basic extra ingredien t to deal with the general case will b e our sp ectral sequence of Corollary 3.2 : (5.1) E 1 p,q ( A ) = Ω p k /l ⊗ k D ( r − p ) q − p ( A/k ) ⇒ D ( r ) q − p ( A/l ) . As sho wn in Lemma 3.1, this sp ectral sequenc e is clearly compatible with the mor- phisms of k -algebras. W e denote the corresponding sp ectral sequence for A/I n b y n E i p,q as b efor e and that for B b y E i p,q ( B ). Put E i p,q ( A, B ) = Ker  E i p,q ( A ) → E i p,q ( B )  . Lemma 5.1. F or a n y r , i ≥ 1 and p, q , n 0 ≥ 0 , the natur al map (5.2) n E i p,q E i p,q ( A ) → n 0 E i p,q E i p,q ( A ) is zer o for al l n ≫ n 0 . F urthermor e, the natur al map (5.3) E i p,q ( A, B ) → n E i p,q is inje c tive fo r al l n ≫ n 0 . Pr o of . W e pro v e b oth statemen ts b y induction o n i ≥ 1 . F o r i = 1, 5.2 follo ws directly from Prop osition 3.4 and 5.3 follows directly from Lemmas 4.1 and 4 .2 . So a ssume that 5.2 and 5.3 hold for a ll 1 ≤ j ≤ i . W e ﬁrst show that 5.3 holds for i + 1. Consider the following comm utativ e diagram of exact seq uences. E i p − i,q + i − 1 ( A ) / /   Ker  ∂ i p,q ( A )    / / E i +1 p,q ( A )   / / 0 E i p − i,q + i − 1 ( B ) / / Ker  ∂ i p,q ( B )  / / E i +1 p,q ( B ) / / 0 , where E i p,q ∂ i p,q − − → E i p + i,q − i +1 is t he diﬀeren tia l of the sp ectral sequence. If p 6 = q , then E 1 p,q ( B ) = Ω p k /l ⊗ k D ( r − p ) q − p ( B /k ) = 0 (as B is s mo oth o ve r k ) and so is E j p,q ( B ) for j ≥ 1. This giv es exact se quence (5.4) E i p − i,q + i − 1 ( A ) → Ker  Ker  ∂ i p,q ( A )  → Ker  ∂ i p,q ( B )  → E i +1 p,q ( A, B ) → 0 . If p = q , then q + i − 1 − p + i = 2 i − 1 ≥ 1 and w e get E i p − i,q + i − 1 ( B ) = 0. This again giv es the exact seq uence as ab o v e. Th us 5.4 holds f o r a ll p, q ≥ 0. Now w e AN AR TIN -REES THEOREM AN D APPLICA TIONS TO ZER O CYCLES 17 consider the fo llowing comm utative diagram with exact ro ws. E i p − i,q + i − 1 ( A ) / /   Ker  Ker  ∂ i p,q ( A )  → Ker  ∂ i p,q ( B )  / /   E i +1 p,q ( A, B ) / /   0 n E i p − i,q + i − 1 / / Ker  n ∂ i p,q  / / n E i +1 p,q / / 0 The middle ve rtical arrow is injectiv e for n ≫ n 0 b y induction. Let N b e the smallest in teger such that this arrow is injectiv e for n ≥ N . Then a diagram c hase shows that Ker  E i +1 p,q ( A, B ) → n E i +1 p,q  is an A -submo dule of a quotien t of n E i p − i,q + i − 1 E i p − i,q + i − 1 ( A ) for all n ≥ N . Th us to prov e 5 .3 for i + 1, it suﬃces to s ho w that t he natural map n E i p − i,q + i − 1 E i p − i,q + i − 1 ( A ) → N E i p − i,q + i − 1 E i p − i,q + i − 1 ( A ) is zero for all n ≫ N . But this is true as 5.2 holds for i b y induction. No w w e sho w that 5.2 holds for i + 1. W e hav e a comm utative diagram E i p,q ( A ) ∂ i p,q ( A ) / /   E i p + i,q − i +1 ( A )   E i p,q ( B ) ∂ i p,q ( B ) / / E i p + i,q − i +1 ( B ) . If p 6 = q , then E 1 p,q ( B ) = 0 = E i p,q ( B ) and if p = q , t hen q − i + 1 − p − i = 1 − 2 i < 0 as i ≥ 1 and hence E i p + i,q − i +1 ( B ) = 0. Now the a b o v e diagram shows that for p, q ≥ 0, one has a fa cto r ization (5.5) E i p,q ( A ) ∂ i p,q ( A ) − − − − → E i p + i,q − i +1 ( A, B ) ֒ → E i p + i,q − i +1 ( A ) . Next w e consider a nother comm utativ e diagram for n ≥ 0. 0 / / Ker  ∂ i p,q ( A )  / /   E i p,q ( A ) / /   Image  ∂ i p,q ( A )  / /   0 0 / / Ker  n ∂ i p,q  / / n E i p,q / / Image  n ∂ i p,q  / / 0 Since 5.3 holds for i , we see fro m 5.5 that the right vertical arro w is injective for all n ≫ n 0 . In particular, w e get an inclus ion Ker  n ∂ i p,q  Ker  ∂ i p,q ( A )  ֒ → n E i p,q E i p,q ( A ) 18 AMALENDU KR ISHNA for all n ≥ N ≫ n 0 . W e no w a pply induction on i in 5 .2 and this inclusion to conclude that the map Ker  n ∂ i p,q  Ker  ∂ i p,q ( A )  → Ker  N ∂ i p,q  Ker  ∂ i p,q ( A )  is zero for all n ≫ N . Since n E i +1 p,q E i +1 p,q ( A ) is a quotien t of the mo dule on the left for n ≥ 0, w e get that the map n E i +1 p,q E i +1 p,q ( A ) → N E i +1 p,q E i +1 p,q ( A ) is zero for all n ≫ N . Since N ≥ n 0 , we see tha t 5 .2 holds for i ≥ 1. This pro v es the lemma.  The follo wing is the generalization of Prop osition 3.4 and Lemmas 4.1 and 4.2 to the case when the base ring of the Andr ´ e-Quillen homo lo gy is any subﬁeld of the giv en ﬁeld k . Prop osition 5.2. L et f : A → B b e the smo o th normalization of an essential l y of ﬁnite typ e k -algebr a A . L et l ⊂ k b e a subﬁeld. L et I ⊂ A b e any given c onducting ide al for the normaliza tion . Then ( i ) F or any r , q , n 0 ≥ 0 , the na tur al m ap D ( r ) q ( A/I n /l ) D ( r ) q ( A/l ) → D ( r ) q ( A/I n 0 /l ) D ( r ) q ( A/l ) is zer o for al l n ≫ n 0 . ( ii ) F or any r, q , n 0 ≥ 0 , the na tur al m ap D ( r ) q ( B /I n /l ) D ( r ) q ( B /l ) → D ( r ) q ( B /I n 0 /l ) D ( r ) q ( B /l ) is zer o for al l n ≫ n 0 . ( iii ) F or any r, q ≥ 1 , the natur al maps D ( r ) q ( A/l ) → D ( r ) q ( A/I n /l ) D ( r ) 0 (( A, B ) / l ) → D ( r ) q ( A/I n /l ) ar e inje ctive for a l l n ≫ n 0 . Pr o of . F or r = 0, t he part ( i ) is ob vious as the g roups on the b oth s ides are zero. So w e assume r ≥ 1. The sp ectral sequenc e 5.1 ( cf. Corolla r y 3.2) gives for any r ≥ 1 and q ≥ 0, a ﬁnite ﬁltration of D ( r ) q ( A/l ) D ( r ) q ( A/l ) = F 0 ( A ) ⊇ F 1 ( A ) ⊇ · · · ⊇ F r ( A ) ⊇ F r +1 ( A ) = 0 AN AR TIN -REES THEOREM AN D APPLICA TIONS TO ZER O CYCLES 19 with F j ( A ) /F j +1 ( A ) ∼ = E ∞ j,q + j ( A ) = E r +1 j,q + j ( A ) for 0 ≤ j ≤ r . One has similar ﬁltrations for D ( r ) q ( B /l ) and D ( r ) q ( A/I n /l ) together with morphisms of ﬁltered A - mo dules. Th us for 0 ≤ j ≤ r and n ≥ 0, w e ha v e the exact sequence F j +1 ( A/I n ) F j +1 ( A ) → F j ( A/I n ) F j ( A ) → n E r +1 j,q + j E r +1 j,q + j ( A ) → 0 . No w b y comparing this exact sequence for n 0 and n ≥ n 0 , using the doubling tric k in 2.3 as befo r e and us ing the descending induction on j , w e se e that it suﬃces to sho w that for r ≥ 1, q , n 0 ≥ 0 and 0 ≤ j ≤ r , the natural map n E r +1 j,q + j E r +1 j,q + j ( A ) → n 0 E r +1 j,q + j E r +1 j,q + j ( A ) is zero for a ll n ≫ n 0 to pro v e part ( i ) of the prop o sition. But this follo ws from Lemma 5.1. T o pro v e ( ii ), w e ﬁrst observ e from the smoo thness of B and the ab o v e spectral sequence that D ( r ) q ( B /l ) =  Ω r B /l if q = 0 0 otherwise . F urthermore, as D ( r ) 0 ( B /I n /l ) = Ω r ( B /I n ) /l ( cf. 2.2), w e immediately get ( ii ) for q = 0 . W e also get from this that f o r q ≥ 1 and n ≥ 0, (5.6) D ( r ) q ( B /I n /l ) D ( r ) q ( B /l ) ∼ = − → D ( r ) q ( B /I n /l ) . Th us w e ne ed to sho w that the map D ( r ) q ( B /I n /l ) → D ( r ) q ( B /I n 0 /l ) is ze ro for all n ≫ n 0 . F or r , q ≥ 1, the natural map (5.7) D ( r ) q ( B /I n /k ) → D ( r ) q ( B /I n 0 /k ) is zero for all n ≫ n 0 b y 5.6 and Prop osition 3.4. No w the pro of of ( ii ) follows b y using 5.7 and the sp ectral sequ ence 5.1 and then b y following exactly the same ar- gumen t as in the pro of o f ( i ). Here the analog ue of Lemma 5.1 follows immediately from 5.1 and 5 .7. F or prov ing part ( iii ), w e consider the f ollo wing diagram of exact sequences f or 0 ≤ j ≤ r . 0 / / F j +1 ( A, B )   / / F j ( A, B ) / /   E r +1 j,q + j ( A, B )   0 / / n F j +1 / / n F j / / n E r +1 j,q + j / / 0 , where F j ( A, B ) = Ker ( F j ( A ) → F j ( B )) and n F j is the ﬁltra tion of D ( r ) q ( A/I n /l ). Again b y descending induction on j , it suﬃces to show that for r ≥ 1, q , n 0 ≥ 0 and for 0 ≤ j ≤ r , the map E r +1 j,q + j ( A, B ) → n E r +1 j,q + j 20 AMALENDU KR ISHNA is injectiv e. But this follows again from Lemma 5.1 .  6. Andr ´ e-Quillen to Hochschild Homology In t his section, w e deriv e some consequences of our results of the previous section for the Ho c hsc hild and cyclic homology . Let k b e a ﬁeld and let A b e an essen tia lly of ﬁnite t yp e k -a lg ebra. W e recall that for an y A -algebra B , the Ho c hsc hild ho- mology H H A ∗ ( B ) are the homolo gy group of the pre- simplicial B -mo dule C A ∗ ( B ) giv en b y C A n ( B ) = B ⊗ A · · · ⊗ A B = B ⊗ n +1 and the fa ce maps d i : C A n ( B ) → C A n − 1 ( B ) for 0 ≤ i ≤ n b eing given b y d i ( a 0 , · · · , a n ) = ( a 0 , · · · , a i a i +1 , · · · , a n ) for 0 ≤ i ≤ n − 1 and d n ( a 0 , · · · , a n ) = ( a n a 0 , a 1 , · · · , a n − 1 ) . The asso ciated c hain complex of B -mo dules is calle d the Ho c hsc hild complex of B o v er A . T o deﬁne the cyclic homology of B , o ne uses the natural action of ﬁnite cyclic groups on t he Ho ch sc hild complex to construct the cyclic bicomplex C C A ( B ) and the cyclic homology H C A ∗ ( B ) of B o v er A are deﬁned as the homolo g y of the asso ciated total complex. W e refer the reader to [20] for the details ab out the deﬁnitions of Ho chs c hild and cyclic homology and their prop erties whic h are rel- ev an t to us in this pap er. F or any map B → B ′ of A - algebras, the relative Ho c hsc hild homology H H A ∗ ( B , B ′ ) is deﬁned a s the homolo gy o f the complex Cone  H H A ∗ ( B ) → H H A ∗ ( B ′ )  [ − 1]. F or an ideal I ⊂ B , the relativ e Ho chsc hild homology H H A ∗ ( B , I ) is the relativ e homology of the map B → B /I . F or a map B → B ′ and an ideal I ⊂ B such that I B ′ = I , the double relativ e Ho c hshild homology are deﬁned as the homology of the complex Cone  H H A ∗ ( B , I ) → H H A ∗ ( B ′ , I )  [ − 1]. The relativ e and do uble relativ e cyclic ho- mology are deﬁned in the analogous w a y by taking the cones o v er the total cyclic complexes. If C C A ( B ) 2 denotes the cyclic bicomplex of B consis ting of only the ﬁrst t w o columns of C C A ( B ), then there is a natural short exact sequence ( lo c. cit. , Theorem 2.2 .1) 0 → C C A ( B ) 2 → C C A ( B ) → C C A ( B )[2 , 0] → 0 , whic h giv es the C onnes’ p erio dicity lo ng exact sequenc e (a lso called SBI-sequence ) · · · → H H A n ( B ) I − → H C A n ( B ) S − → H C A n − 2 ( B ) B − → H H A n − 1 ( B ) I − → · · · . T aking the cones ov er the appropr ia te short exact sequences as ab o v e, o ne g ets similar SBI-sequence for the relative and double relativ e Ho c hsc hild and cyclic homology . The most crucial fact whic h will b e useful to us in this pap er is the canonical decomp osition of the Ho c hsc hild homology in terms of t he Andr ´ e-Quillen homology in ch aracteristic zero. W e state it here for t he sak e of reader’s con v enience and refer to lo c. cit. (Theorem 3.5.8 ) f or the pro o f . AN AR TIN -REES THEOREM AN D APPLICA TIONS TO ZER O CYCLES 21 Theorem 6.1. F or a n y ﬂat A -a l g ebr a B , ther e is a c ano nic al de c omp osition H H A n ( B ) ∼ = M r + q = n D ( r ) q ( B / A ) . It is also kno wn tha t this decomp osition is compatible with the Ho dge decom- p osition of the Ho c hsc hild homology . An immediate conse quence of this canonical decomp osition is the following Ho c hsc hild ho molo gy analogue of Prop osition 5.2. As in t he previous section, let A b e an integral domain which is an essen tia lly of ﬁnite t yp e a lgebra ov er a ﬁeld k . Let f : A → B b e the smo oth normalization of B . F or an y subﬁeld l ⊂ k and for i ∈ Z , we denote the k ernel of the map H H l i ( A ) → H H l i ( B ) b y H H l i ( A, B ). Corollary 6.2. L et I ⊂ A b e any g iven c onducting i d e al for the normal i z ation f : A → B a s a b ove. Then for any subﬁeld l ⊂ k and any i ≥ 0 , ( i ) The natur al m ap H H l i ( A/I n ) H H l i ( A ) → H H l i ( A/I n 0 ) H H l i ( A ) is zer o for al l n ≫ n 0 . ( ii ) The natur al map H H l i ( B /I n ) H H l i ( B ) → H H l i ( B /I n 0 ) H H l i ( B ) is zer o for al l n ≫ n 0 . ( iii ) The natur al map H H l i ( A, B ) → H H l i ( A/I n ) is inje c tive fo r al l n ≫ n 0 . Pr o of . This follows immediately from Prop osition 5.2 and the canonical decompo- sition of the Ho c hsc hild homology in Theorem 6.1.  Corollary 6.3. L et the notations b e as in Cor ol lary 6 .2. T hen for an y given c onducting ide al I for the normalization and for any i, n 0 ≥ 0 , the natur al maps (6.1) H H l i ( B , I n ) H H l i ( A, I n ) → H H l i ( B , I n 0 ) H H l i ( A, I n 0 ) (6.2) Ker  H H l i ( A, I n ) → H H l i ( B , I n )  → Ker  H H l i ( A, I n 0 ) → H H l i ( B , I n 0 )  ar e zer o for al l n ≫ n 0 . Pr o of . W e consider the follow ing commutativ e diagram of short exact sequences coming from the lo ng ex act relativ e Hochs c hild homolog y se quence. (6.3) 0 / / H H l i +1 ( A/I n ) H H l i +1 ( A ) / /   H H l i ( A, I n ) / /   Ker  H H l i ( A ) → H H l i ( A/I n )  / /   0 0 / / H H l i +1 ( B /I n ) H H l i +1 ( B ) / / H H l i ( B , I n ) / / Ker  H H l i ( B ) → H H l i ( B /I n )  / / 0 22 AMALENDU KR ISHNA Using the naturality of the Ho dge decomp osition of the Hochs c hild ho mology ( lo c. cit. , Theorem 4.5 .10) and smo othness of B , we ha v e the isomorphism Ker  H H l i ( B ) → H H l i ( B /I n )  Ker  H H l i ( A ) → H H l i ( A/I n )  ∼ = − → Ω i ( B ,I n ) /l Ω i ( A,I n ) /l . Using this iden tiﬁcation, the ab o v e diagram give s us f or n ≥ 0, an exact sequence of quotien ts H H l i +1 ( B /I n ) H H l i +1 ( B ) → H H l i ( B , I n ) H H l i ( A, I n ) → Ω i ( B ,I n ) /l Ω i ( A,I n ) /l → 0 . By [15] (L emma 4.1 ), one has that for a n y given i, n 0 ≥ 0, the natural map Ω i ( B,I n ) /l Ω i ( A,I n ) /l → Ω i ( B,I n 0 ) /l Ω i ( A,I n 0 ) /l is zero for all n ≫ n 0 . Corollary 6.2 implies that for i, n 0 ≥ 0, the natura l map H H l i ( B /I n ) H H l i ( B ) → H H l i ( B /I n 0 ) H H l i ( B ) is ze ro for all n ≫ n 0 . Now b y comparing this exact sequenc e for n 0 and n ≥ n 0 and using the doubling trick, w e conclude the pro of of 6.1. T o pro v e 6.2, w e observ e in 6.3 that the right v ertical arro w is injectiv e for all n ≫ n 0 b y Corollary 6.2 (part ( iii )). Hence we can replace the k ernel of the middle v ertical arrow b y the k ernel of the left v ertical arr ow in order to pro v e 6.2, in whic h case it follows from Corollary 6.2 (part ( i )).  Corollary 6.4. L et the notations b e as i n Cor ol lary 6.2. Then for any c onducting ide al I fo r the normalization and for any i ∈ Z and n 0 ≥ 0 , the natur a l map s o f double r elative Ho chschild and cyclic h omolo gy gr oups (6.4) H H l i ( A, B , I n ) → H H l i ( A, B , I n 0 ) (6.5) H C l i ( A, B , I n ) → H C l i ( A, B , I n 0 ) ar e zer o for al l n ≫ n 0 . Pr o of . The double relativ e Ho chs c hild and cyclic homology are classically kno wn to b e zero in negative degrees. So w e assume i ≥ 0. W e ﬁrst prov e the result for the Ho chs c hild homology . The long exact double relativ e Ho chs c hild homology sequence giv es for any i, n ≥ 0, the s hort exact sequence 0 → H H l i +1 ( B , I n ) H H l i +1 ( A, I n ) → H H l i ( A, B , I n ) → Ker  H H l i ( A, I n ) → H H l i ( B , I n )  → 0 . Comparing this exact seque nce for n 0 and n ≥ n 0 , using Corollary 6 .3 and the doubling trick , w e g et the desired result. T o prov e the result for the cyclic homology , w e use induction on i ≥ 0. W e hav e H C l i ( A, B , I n ) = 0 for i < 0 as p oin ted ab ov e, and hence the SBI-sequence giv e isomorphism H H l 0 ( A, B , I n ) ∼ = − → H C l 0 ( A, B , I n ). So the result holds for i ≤ 0. Supp ose the result holds f or all j ≤ i − 1 with i ≥ 1. W e hav e the long exact SBI-sequence H H l i ( A, B , I n ) I − → H C l i ( A, B , I n ) S − → H C l i − 2 ( A, B , I n ) . AN AR TIN -REES THEOREM AN D APPLICA TIONS TO ZER O CYCLES 23 Comparing this exact sequence for n 0 and n ≥ n 0 , using induction on i and 6.4 (whic h w e just prov ed), the doubling tric k gives us the pro of of 6.5.  Pro of of Theorem 1.1 : Let X b e a quasi-pro jectiv e v ariety of dimension d o v er a ﬁeld k and let f : X → X b e the smo oth nor ma lizat io n of X . Let Y ֒ → X b e a give n conducting subs c heme for the normalization. F or i ∈ Z , let K i, ( X, X ,Y ) denote the sheaf of double relativ e K -g roups on X . This is a sheaf whose stalk of at any p o in t x ∈ X is the double relativ e K -group K i ( O X,x , O X ,x , I Y , x ), where I Y is t he ideal sheaf of Y . By [2 2], Prop o sition A.5 (see a lso [27] for more general Br own-Gersten sp ectral sequences), there exists a con v ergent sp ectral se quence n E p,q 2 = H p Zar  X , K q , ( X, X ,nY )  ⇒ K q − p ( X , X , nY ) , with diﬀeren tial d r : n E p,q r → n E p + r,q + r − 1 r . This giv es a ﬁnite ﬁltration K i ( X , X , nY ) = F 0 n ⊇ F 1 n ⊇ · · · ⊇ F d n ⊇ F d +1 n = 0 suc h that for 0 ≤ j ≤ d , F j n /F j +1 n ∼ = n E j,i + j ∞ = n E j,i + j r , where r dep ends only o n d and 0 ≤ j ≤ d . Th us b y the desc ending induction on j and using the dou- bling tric k, it suﬃces to sho w that for 0 ≤ j ≤ d and n 0 ≥ 0, the natural map n E j,i + j 2 → n 0 E j,i + j 2 is ze ro for all n ≫ n 0 . This will b e pro v ed if w e show t ha t for i ∈ Z and n 0 ≥ 0, the natural map of shea v es K i, ( X, X ,nY ) → K i, ( X,X ,n 0 Y ) is zero for a ll n ≫ n 0 . F or i ≤ 0 , these shea v es are classically know n to b e zero ( [4]). So w e assume i ≥ 1. Since X is a k -v ariety , it is enough to sho w this when X is aﬃne. Th us w e need to sho w that if A is an essen tially of ﬁnite type k -alg ebra and B is the smo o th normalization of A , then fo r any conducting ideal I and i, n 0 ≥ 0 , the natural map K i ( A, B , I n ) → K i ( A, B , I n 0 ) is zero for all n ≫ n 0 . But this follows immediately from Corollary 6.4 (with l = Q ) and C ortinas’ result ([8], Corollary 0.2 ) that these double relativ e K - groups are in fact rational v ector spaces a nd the Chern c haracter map K i ( A, B , I n ) → H C Q i − 1 ( A, B , I n ) is an iso- morphism. This completes the pro of of Theorem 1 .1 .  7. F ormula f or The Cho w group of Zero Cycles Let X b e a quasi-pro jective v ariety of dimension d ≥ 2 ov er a ﬁeld k . W e assume in this section that X is Cohen-Macaulay (all lo cal rings are Cohen-Macaulay) and it has only isolated singularities. Note that this automatically implies that X is normal. Our aim in this section is to pro v e Theorem 1 .2. So let p : e X → X b e a go o d resolution o f singularities o f X and let E denote the reduced exceptional divisor on e X . L et S ⊂ X b e the singular lo cus of X . W e giv e S the reduced induced subsc heme structure and denote b y nS , the n th inﬁnitesimal thic k ening of S in X . Let Y ֒ → X be a closed subsc heme of X suc h that p is the blo w-up of X alo ng Y . Then S = Y r ed . Let I denote the ideal sheaf for Y . Then o ne has e X = Pro j X ( ⊕ n ≥ 0 I n ) and E = (Pro j Y ( ⊕ n ≥ 0 I n / I n +1 )) r ed . 24 AMALENDU KR ISHNA Putting e Y = Pro j Y ( ⊕ n ≥ 0 I n / I n +1 ), w e get S ⊂ Y ⊂ nS and E ⊂ e Y ⊂ nE for all suﬃcien tly large n . Lemma 7.1. F or al l n ≥ 1 , the map F d K 0 ( X , nS ) → F d K 0 ( X ) is an isomor- phism. In p articular, ther e ar e natur a l maps F d K 0 ( X ) → F d K 0 ( e X , nE ) such that Diagr am 1.1 c o mmutes a nd al l maps ther e ar e surje ctive. Pr o of . Since S is zero-dimensional, the map F d K 0 ( X , nS ) → F d K 0 ( X ) is an iso- morphism by [14] (L emma 3.1). On the other hand, there are natural maps F d K 0 ( X , nS ) → F d K 0 ( e X , nE ) b y the deﬁnition of relat ive K -groups (see Sec- tion 1). The isomorphism ab ov e no w shows that this map factors through a map F d K 0 ( X ) → F d K 0 ( e X , nE ). The surjectivit y assertion follo ws from Lemmas 3.1 and 3.2 of [1 4].  Pro of of Theorem 1.2 : The Northcott-Rees theory giv es a minimal reduc tion of ideal sheaf J ⊂ I of I in the sense that J I n = I n +1 for all suﬃcien tly large n . F urthermore, since X is Cohen-Macaula y and S is a ﬁnite set of closed p oin ts, w e can c ho ose ( cf. [29]) J to be a lo cal complete in tersection ideal sheaf on X . No w w e follo w the pro of of Theorem 1.1 of [16] to get a factorization e X f & & M M M M M M M M M M M M M p   X ′ p ′ x x p p p p p p p p p p p p p X where p ′ is the blo w-up of X along J and f is the normalization morphism. Let Y 1 denote the lo cal complete inte rsection subsc heme of X deﬁned b y J . Since J is a reduction for I , w e see that Y 1 r ed = Y r ed = S and hence I n ⊂ J ⊂ I for a ll large n . Let Y ′ = Y × X X ′ , Y ′ 1 = Y 1 × X X ′ , e Y = Y × X e X , e Y 1 = Y 1 × X e X , and S ′ = (S × X X ′ ) red . Let Z ′ ⊂ X ′ b e a conducting subsc heme for the normalizat io n map f . Put e Z ′ = Z ′ × X ′ e X . Then w e see that Z ′ r ed ⊂ S ′ and e Z ′ r ed ⊂ E . In particular, giv en an y m > 0, w e ha v e mZ ′ ⊂ nS ′ and m e Z ′ ⊂ nE for all lar ge n . Hence for a given m > 0, w e ha ve the followin g commutativ e diagram for all suﬃcien t ly larg e n with all maps surjectiv e. AN AR TIN -REES THEOREM AN D APPLICA TIONS TO ZER O CYCLES 25 (7.1) F d K 0 ( e X , nE ) / / / / F d K 0 ( e X , m e Z ′ ) / / / / F d K 0 ( e X ) F d K 0 ( X ′ , nS ′ ) / / / / O O O O F d K 0 ( X ′ , mZ ′ ) / / / / O O O O F d K 0 ( X ′ ) O O O O F d K 0 ( X , nS ) O O O O / / / / F d K 0 ( X ) 5 5 5 5 j j j j j j j j j j j j j j j j j j j U U U U U U U U U U U U U U U U U The surjectivit y of all maps follow s fr o m Lemma 3.2 of [14 ], and the bottom hor- izon tal map is an isomorphism by Lemma 7 .1. Now since p ′ is a blo w-up along a lo cal complete intersec tion subsc heme, the map F d K 0 ( X ) → F d K 0 ( X ′ ) is also injectiv e by [16] (Corollary 2.5). Com bining this with the surjectivit y of arrow s in the ab o v e diagram, we get another diagram below with all the arrows b eing isomorphisms. F d K 0 ( X ′ , nS ′ ) / / F d K 0 ( X ′ , mZ ′ ) / / F d K 0 ( X ′ ) F d K 0 ( X , nS ) O O / / F d K 0 ( X ) 5 5 j j j j j j j j j j j j j j j j j U U U U U U U U U U U U U U U U U Next w e study the relation b etw een F d K 0 ( X ′ , mZ ′ ) and F d K 0 ( e X , m e Z ′ ) for ﬁxed m > 0. By the long exact sequence o f double relativ e K -theory , one has an exact sequence K 0 ( X ′ , e X , mZ ′ ) → K 0 ( X ′ , mZ ′ ) → K 0 ( e X , m e Z ′ ) . W e compare this exact sequence for m = 1 and m > > 0 to get a diagr am K 0 ( X ′ , e X , mZ ′ ) / /   K 0 ( X ′ , mZ ′ ) / /   K 0 ( e X , m e Z ′ )   K 0 ( X ′ , e X , Z ′ ) / / K 0 ( X ′ , Z ′ ) / / K 0 ( e X , e Z ′ ) The left v ertical map is zero for m ≫ 0 b y Theorem 1.1. Put A m = Ker  F d K 0 ( X ′ , mZ ′ ) → F d K 0 ( e X , m e Z ′ )  . Then the ab o v e diagram giv es another diagram of short exact sequence s 0 / / A m / /   F d K 0 ( X ′ , mZ ′ ) / /   F d K 0 ( e X , m e Z ′ ) / /   0 0 / / A 1 / / F d K 0 ( X ′ , Z ′ ) / / F d K 0 ( e X , e Z ′ ) / / 0 where the left v ertical map is zero fo r m ≫ 0. By mapping Diagram 7.1 to a similar diagram with m = 1, w e see that the middle v ertical map a b o v e is an isomorphism. 26 AMALENDU KR ISHNA A diagram c hase sho ws that F d K 0 ( X ′ , mZ ′ ) → F d K 0 ( e X , m e Z ′ ) is an isomorphism. This, together with the isomorphism F d K 0 ( X ′ , nS ′ ) → F d K 0 ( X ′ , mZ ′ ) no w sho ws that the map F d K 0 ( X , nS ) → F d K 0 ( e X , nE ) is an isomorphism for all lar ge n . This giv es the desired isomorphisms F d K 0 ( e X , nE ) → F d K 0 ( e X , ( n − 1) E ) a nd F d K 0 ( X ) → F d K 0 ( e X , nE ) for all large n . Finally , for X aﬃne or pro jectiv e, w e get C H d ( X ) ∼ = lim ← − n F d K 0 ( e X , nE ) if k is algebraically closed.  8. Cohomology of Milnor K -shea ves It is by no w a well known f a ct that algebraic cycle s are closely conne cted to the cohomology of Quillen K -theory shea ves . This is true ev en for certain classes o f singular v arieties. Ho w ev er, the Quillen K -theory gro ups are of t en ve ry diﬃcult to compute. On the other hand, one also has the shea ves of Milnor K -theory on sc hemes which are relativ ely simpler lo o king ob jects. Our a pplicatio ns of Theo- rem 1.2 in this pap er ar e based on t he observ ation that for the purp oses of zero cycles, the appro priate cohomology of Quillen K -theory shea v es can often b e ap- pro ximated by the cohomo lo gy of Milnor K -theory sheav es, whic h can b e computed b y some other means. In this section, w e pro v e certain general reduction steps in order t o use Theorem 1.2 for s tudying the Chow group of zero cyc les on v arieties with isolated singularities. In pa rticular, w e giv e a v ery precise suﬃcien t condition for the Cho w gro up of zero cycles on a v ariety X with Cohen-Macaula y isolated singularities to b e isomorphic to the sim ilar group on a resolution of singularities of X . W e b egin with the follo wing result. F or any v ariety X ov er a ﬁeld k , let K M m,X denote the sheaf of Milnor K -groups on X . This is a sheaf whose stalk at an y p oin t x of X is the Milnor K -group of the lo cal ring O X,x . F or an y closed em b edding i : Y ֒ → X , let K M m, ( X,Y ) b e the sheaf of relativ e Milnor K -groups de ﬁned so that the seque nce of shea v es (8.1) 0 → K M m, ( X,Y ) → K M m,X → i ∗ ( K M m,Y ) → 0 is exact. Note that the map K M m,X → i ∗ ( K M m,Y ) is alw a ys surjectiv e. F rom now on, w e shall assume our base ﬁeld k to algebraically closed unless men tioned otherwise. Lemma 8.1. L et X b e an aﬃne or pr oje ctive variety of dimension d over k . Then ther e ar e natur a l isomo rp h isms C H d ( X ) ∼ = F d K 0 ( X ) ∼ = H d ( X , K d,X ) ∼ = H d ( X , K M d,X ) . Pr o of . The isomorphism C H d ( X ) ∼ = F d K 0 ( X ) w as show n b y Lev ine ([17], Corol- lary 2.7 a nd Theorem 3.2). In this case, Barvieri-Viale has sho wn ([2], Corollary A) that there is a natural surjection C H d ( X ) ։ H d ( X , K d,X ) with ﬁnite ke rnel. If X is aﬃne, then this k ernel must b e zero by [17] (Theorem 2.6). If X is pro j ec- tiv e, t hen there is an albanese map C H d ( X ) → H d ( X , K d,X ) → H d ( e X , K d, e X ) → Alb ( e X ), where e X is a resolution of singularities of X . Now the isomorphism AN AR TIN -REES THEOREM AN D APPLICA TIONS TO ZER O CYCLES 27 Alb ( X ) ∼ = Alb ( e X ) (since X is normal) and the Roitman torsion theorem implies that C H d ( X ) → H d ( X , K d,X ) m ust b e an isomorphism. Finally , the isomorphism H d ( X , K d,X ) ∼ = H d ( X , K M d,X ) is sho wn in [1 4] (Corollary 4.2 ) .  Prop osition 8.2. L et e X b e a smo oth quasi - pr oje ctive v a riety of dimension d + 1 over a ﬁeld k . L et E ֒ → e X b e a strict no rmal cr ossing divisor. Then the na tur al cup p r o duct map H d ( E , K M m,E ) ⊗ k ∗ → H d ( E , K M m +1 ,E ) is surje ctive for al l m ≥ d . Pr o of . W e prov e this b y induction o n d ≥ 1 and divide the pro o f into sev era l cases . Case I: d a rbitrary and E smo oth. In this case one ha s the Gersten resolution ([10], Prop osition 4.3 ) K M m,E → i ∗  K M m ( k ( E ))  → ⊕ x ∈ E (1) i ∗  K M m − 1 ( k ( x ))  → · · · → ⊕ x ∈ E ( d − 1) i ∗  K M m − d +1 ( k ( x ))  → ⊕ x ∈ E ( d ) i ∗  K M m − d ( k ( x ))  → 0 , where t he ﬁrst map is generically injectiv e (in fact ev erywhere injectiv e b y a recen t result of Kerz [13]). This resolution giv es a comm utativ e diagram  ⊕ x ∈ E ( d ) i ∗  K M m − d ( k ( x ))   ⊗ k ∗ / / / /   H d ( E , K M m,E ) ⊗ k ∗   ⊕ x ∈ E ( d ) i ∗  K M m +1 − d ( k ( x ))  / / / / H d ( E , K M m +1 ,E ) . The horizon tal arro ws in this diagram are surjec tiv e by the abov e resolution. The left ve rtical arrow is surjectiv e since k ( x ) = k for x ∈ E ( d ) as k is algebraically closed and the map K M i ( k ) ⊗ k M j ( k ) → K M i + j ( k ) is surjectiv e. A dia g ram c hase pro v es the result. Case I I: d = 1 and E not smo oth. Let E = E 1 ∪ · · · ∪ E r with r ≥ 2 and put E ′ = E 1 ∪ · · · ∪ E r − 1 , F r = E ′ ∩ E r . Since E is a strict normal crossing divisor on e X , w e see that F r is a strict normal crossing divisor on E r . Th us F r is ﬁnite set of closed p oin ts for whic h the prop osition is ob vious. Let I E ′ (resp I E r ) denote the ideal sheaf of E ′ (resp E r ) on e X . L et E b e the closed subsc heme of e X deﬁned b y the sheaf of ideals I E ′ ∩ I E r . Then one has (8.2) K M m,E ։ K M m, E and this map is g enerically an isomorphism. In particular, w e ha v e (8.3) H d ( E , K M m,E ) ∼ = − → H d ( E , K M m, E ) ∀ m. 28 AMALENDU KR ISHNA Note here t hat E is the subsc heme of e X lo cally deﬁned b y the product of I E ′ and I E r . By [14] (Lemma 4.5 ) , there is a short exact sequ ence of shea ves (8.4) 0 → K M m, E → i ∗  K M m,E ′  ⊕ i ∗  K M m,E r  → i ∗  K M m,F r  → 0 . T aking the cohomology exact sequences, w e get a comm utativ e diagra m of exact sequence s (8.5) H d − 1 ( F r , K M m,F r ) ⊗ k ∗   / / H d ( E , K M m,E ) ⊗ k ∗   / / ⊕ H d ( E ′ , K M m,E ′ ) ⊗ k ∗ H d ( E r , K M m,E r ) ⊗ k ∗ / / 0 H d − 1 ( F r , K M m +1 ,F r ) / / H d ( E , K M m +1 ,E ) / / ⊕ H d ( E ′ , K M m +1 ,E ′ ) H d ( E r , K M m +1 ,E r ) / / 0 ,   where the last terms in b o th row s a re zero b ecause dim( F r ) ≤ d − 1. W e ha ve al- ready observ ed that the left vertical arro w is surjectiv e. The map H d ( E r , K M m,E r ) ⊗ k ∗ → H d ( E r , K M m +1 ,E r ) is surjectiv e b y Case I. The map b etw een the other sum- mand of the righ t v ertical arrow is surjec tiv e b y induction on the num b er of com- p onen ts since n ( E ′ ) < n ( E ). W e complete the pro o f o f Case I I by a diagra m chase and 8.3. Case I I I: d ≥ 2. Assume b y induction tha t the prop o sition holds fo r whenev er dimension of the normal crossing divisor is d ′ < d . Let n ( E ) denote the n um b er of irreducible comp onen ts of E . W e no w induct on n ( E ). The case n ( E ) = 1 is already prov ed abov e. So assume n ( E ) = n ≥ 2 and assume w e hav e pro ved Case I I for all normal crossing divisors E ′ with n ( E ′ ) < n . Let E ′ , E r , F r and E b e as in Case II. Then E ′ is a strict normal crossing divisor on e X with n ( E ′ ) < n . More- o v er, E r is smo oth a nd F r is a strict normal crossing divisor on E r (if not empty ) and E r is a smo oth v ariet y of smaller dimension than e X . So the propo sition holds for E ′ , F r and E r b y induction and smo oth case. A diagram c hase again in 8.5 and 8.3 no w complete the pro of.  Prop osition 8.3. L et Z b e a quasi-pr o je ctive variety of dime nsion d over a ﬁeld k and let W = Z red . F or i ≥ 0 , let Ω i ( Z,W ) / Q denote the kerne l of the map Ω i Z/ Q → Ω i W / Q . Then ther e is a natur al isomorphism H d  Z , K M d +1 , ( Z,W )  → H d   Z , Ω d ( Z,W ) / Q ∂  Ω d − 1 ( Z,W ) / Q    , wher e ∂ : Ω i Z/ Q → Ω i +1 Z/ Q is the diﬀer e ntial m ap. AN AR TIN -REES THEOREM AN D APPLICA TIONS TO ZER O CYCLES 29 Pr o of . Let φ Z : K M m,Z → K m,Z denote the natural map from Milnor K -theory to Quillen K -theory . Since w e are in c haracteristic 0, there exists by [26 ](Theorem 12 .3) a natural map ψ Z : K m,Z → K M m,Z suc h t ha t ψ Z ◦ φ Z = (( − 1) m − 1 ( m − 1)!)Id ∀ m ≥ 1. F urthermore, with resp ect to the γ -ﬁltra tion and Adams op eration on the Quillen K -theory ([18]), one has F d +1 K d +1 ,Z = K ( d +1) d +1 ,Z and the map φ Z : K M d +1 ,Z → K ( d +1) d +1 ,Z is isomorphism mo dulo d !. In particular, w e get na tural maps (8.6) K M d +1 ,Z φ Z − → K ( d +1) d +1 ,Z ψ Z − → K M d +1 ,Z whic h are isomorphisms mo dulo d !. W e no w consider the comm utative diagram 0 / / K M d +1 , ( Z,W ) / / φ Z W   K M d +1 ,Z φ Z   / / K M d +1 ,W φ W   / / 0 0 / / K ( d +1) d +1 , ( Z,W ) / / ψ Z W   K ( d +1) d +1 ,Z / / ψ Z   K ( d +1) d +1 ,W / / ψ W   0 0 / / K M d +1 , ( Z,W ) / / K M d +1 ,Z / / K M d +1 ,W / / 0 , where the top and the b ottom ro ws are exact b y 8.1 and the group K ( d +1) d +1 , ( Z,W ) is deﬁned to mak e the middle r o w exact. A diagram c hase giv es maps ψ Z W and φ Z W whic h are in v erses to eac h other mo dulo d !. Since the map K ( d +1) d +1 , ( Z,W ) ։ K ( d +1) d +1 , ( Z,W ) is isomorphism on the smoot h lo cus of W , w e get H d  Z , K ( d +1) d +1 , ( Z,W )  ∼ = H d  Z , K ( d +1) d +1 , ( Z,W )  , whic h in turn sh o ws that the map (8.7) H d  Z , K ( d +1) d +1 , ( Z,W )  → H d  Z , K M d +1 , ( Z,W )  is an isomorphism mo dulo d !. How ev er, W b eing the reduced part of Z , K ( d +1) d +1 , ( Z,W ) is a sheaf of Q -vec tor spaces ([5], Section 1) and K M d +1 , ( Z,W ) is a sheaf of divisib le groups b y [14] (Sublemma 4.8). In particular, the left hand side in 8.7 is a Q -v ector space and the rig h t hand sid e is a divisible group (since H d is righ t ex act on Z ). This sho ws that the map in 8.7 m ust b e an isomorphism. By [5] (Theorem 1), there exists a functorial isomorphism of ﬁltered shea v es of Q -vec tor spaces K d +1 , ( Z,W ) ∼ = − → H C d, ( Z,W ) , where the ﬁltratio n is giv en b y the γ -ﬁltratio n on b oth sides, and HC denote the shea v es of cyclic homology ov er the base Q . This give s an isomorphism K ( d +1) d +1 , ( Z,W ) ∼ = − → HC ( d ) d, ( Z,W ) and b y 8.7, w e get 30 AMALENDU KR ISHNA H d  Z , K M d +1 , ( Z,W )  ∼ = H d  Z , HC ( d ) d, ( Z,W )  . Thus w e are r educed to sho wing that there is a natural isomorphism (8.8) H d  Z , HC ( d ) d, ( Z,W )  ∼ = − → H d   Z , Ω d ( Z,W ) / Q ∂  Ω d − 1 ( Z,W ) / Q    . W e ha v e an exact sequence of shea ves HC ( d ) d, ( Z,W ) → HC ( d ) d,Z → HC ( d ) d,W → 0 , where the last term is zero since HC ( d ) d,Z ∼ = Ω d Z / Q ∂  Ω d − 1 Z / Q  ։ Ω d W / Q ∂  Ω d − 1 W / Q  ∼ = HC ( d ) d,W b y [20] (Theorem 4.6.8). F urthermore, since O Z ։ O W is lo cally split on the smo oth lo cus of W , w e see that the ﬁrst map in the ab ov e exact sequence is injectiv e on the smo oth locus of W . By the same reason, there is a natural surjection Ω d ( Z,W ) / Q ∂  Ω d − 1 ( Z,W ) / Q  ։ Ker Ω d Z / Q ∂  Ω d − 1 Z / Q  ։ Ω d W / Q ∂  Ω d − 1 W / Q  ! , whic h is an isomorphism on t he smoo th lo cus of W . In particular, we get surjectiv e maps HC ( d ) d, ( Z,W ) / / / / Ker  Ω d Z / Q ∂ ( Ω d − 1 Z / Q ) ։ Ω d W / Q ∂ ( Ω d − 1 W / Q )  Ω d ( Z,W ) / Q ∂ “ Ω d − 1 ( Z,W ) / Q ” o o o o whic h are isomorphisms on the smo oth locus of W , a nd hence they induce isomor- phisms on the top cohomology H d . This prov es 8.8 and hence the prop osition.  Corollary 8.4. L et X b e either an aﬃne or a pr oje ctive variety of dimension d ≥ 2 over a ﬁeld k . Assume that X is Cohen-Mac aulay and has only iso late d singularities. L et p : e X → X b e a g o o d r esolution of singularities of X such that the exc eptional divisor E is a strict normal cr os sings div isor (such a r esolution always exists). Then the map C H d ( X ) → C H d ( e X ) is an isomorp h ism if the fol lowing two c onditions hold. ( i ) The map H d − 1  e X , K d − 1 , e X  ⊗ k ∗ → H d − 1 ( E , K d − 1 ,E ) ⊗ k ∗ is surje ctive. ( ii ) H d − 1  nE , Ω d − 1 ( nE ,E ) / Q ∂ “ Ω d − 2 ( nE ,E ) / Q ”  = 0 for al l n ≥ 1 . Pr o of . By Theorem 1.1, we need to sho w that the map F d K 0 ( e X , nE ) → F d K 0 ( e X ) is an isomorphism for all large n . By [14] (Prop osition 4.3), this further reduces to sho wing that the map H d  e X , K M d, ( e X ,nE )  → H d  e X , K M d, e X  is an isomorphism for AN AR TIN -REES THEOREM AN D APPLICA TIONS TO ZER O CYCLES 31 all n ≥ 1. Considering the long exact cohomology se quence corresp onding to 8.1, H d − 1  e X , K M d, e X  → H d − 1  nE , K M d,nE  → H d  e X , K M d, ( e X ,nE )  → H d  e X , K M d, e X  → 0 , w e need to sho w tha t the ﬁrst map on the left is surjectiv e for a ll n . First w e consider the case n = 1. W e ha ve s een in the pro of of Prop osition 8.3 that there is a natural map ψ E : K d − 1 ,E → K M d − 1 ,E whose cokerne l is of ﬁxed exp o nen t ( d − 2)!. In particular, the cokerne l of the map H d − 1 ( E , K d − 1 ,E ) → H d − 1  E , K M d − 1 ,E  is of ﬁnite exp onen t (since H d − 1 is right exact on E ). How ev er, as k ∗ is a divisible group ( k = k ), we m ust ha v e (8.9) H d − 1 ( E , K d − 1 ,E ) ⊗ k ∗ ։ H d − 1  E , K M d − 1 ,E  ⊗ k ∗ . Next w e hav e a comm utative diagram H d − 1  e X , K M d − 1 , e X  ⊗ k ∗   / / H d − 1  e X , K M d, e X    H d − 1  E , K M d − 1 ,E  ⊗ k ∗ / / H d − 1  E , K M d,E  . The ﬁrst condition of the corollar y and 8.9 t o gether imply that the left ve rtical arro w is su rjectiv e. The b ottom horizontal arrow is surjectiv e b y Prop osition 8 .2. A diagram c ha se sho ws that the righ t ve rtical arrow is also surjectiv e. This ﬁnishes the case n = 1. No w assume n ≥ 2. Then 8.1 for t he pair E ֒ → nE gives an exact sequence H d − 1  nE , K M d, ( nE , E )  → H d − 1  nE , K M d,nE  → H d − 1  E , K M d,E  → 0 . The second condition of the corollary and Prop o sition 8.3 together imply that the group on the left is zero. This complete s the proo f .  9. Cho w gr oup of affine cones Let Y ֒ → P N k b e a smo oth pro jectiv e v ariety of dimension d o v er a ﬁeld k . Let X = C ( Y ) b e the aﬃne cone ov er Y and let X ֒ → P N +1 k b e the pro jectiv e cone o v er Y . Let P ∈ X ֒ → X denote the vertex of the cone. W e assume that X is Cohen-Macaula y . Since P is the o nly singular p oint of both X and X , w e see that X is also Cohen-Macaula y . Let p : e X → X b e the blo w-up of X along the ve rtex P and let E = p − 1 ( P ) be the exceptional divisor for the blo w-up. This situation giv es rise to the fo llo wing comm utativ e diag ram. (9.1) E   i / /   2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 e X p / /  k   1 1 1 1 1 1 π   ( ( ( ( ( ( ( ( ( ( ( ( ( ( X  k   2 2 2 2 2 2 Z π   p / / X Y 32 AMALENDU KR ISHNA The map p is the blo w-up of X along P . The map π is the A 1 -bundle ov er Y asso ciated to the ample line bundle O Y (1) with t he zero-section E and π is the P 1 - bundle ov er Y asso ciated to the v ector bundle O Y ⊕ O Y (1). In particular, e X (resp Z ) is a go o d resolution of singularities of X (resp X ) suc h that t he exceptional divisor E ֒ → e X ֒ → Z is smo oth a nd the inclusion of E in e X and Z has sections giv en b y the maps π and π . Our aim in this section is to prov e Theorem 1.5 for whic h w e need the following preliminary results. Lemma 9.1. L et e X b e a smo oth quasi-pr oje ctive variety of dimensio n d + 1 over a ﬁeld k and let E ֒ → e X b e a smo oth divis o r such that H d  E , Ω i E /k ⊗ E I n I n +1  = 0 for i ≥ 0 and n ≥ 1 , wher e I is the ide al she a f o f E on e X . Th en H d  nE , Ω i ( nE , E ) /k  = 0 for i ≥ 0 and n ≥ 1 . Pr o of . W e ﬁrst claim that (9.2) H d  nE , Ω i nE /k ⊗ I I n  = 0 ∀ i ≥ 0 , n ≥ 1 . T o pro v e the claim, we consider for i ≥ 0 and n ≥ 1 the compatible maps 0 = Ω i nE /k ⊗ I n I n → Ω i nE /k ⊗ I n − 1 I n → · · · → Ω i nE /k ⊗ I I n and put F i j = Image  Ω i nE /k ⊗ I j I n → Ω i nE /k ⊗ I I n  for 1 ≤ j ≤ n . This giv es a ﬁnite ﬁltration { F i j } 1 ≤ j ≤ n of Ω i nE /k ⊗ I I n suc h that (9.3) Ω i nE /k ⊗ nE I j I j +1 ։ F i j F i j +1 ∀ 1 ≤ j ≤ n. Since H d is righ t exact on nE , this ﬁltratio n also giv es exact sequence H d  nE , F n j +1  → H d  nE , F n j  → H d  nE , F i j F i j +1  → 0 for 1 ≤ j ≤ n . Th us b y a descending induction on j and by 9.3, it suﬃces to sh o w that (9.4) H d  nE , Ω i nE /k ⊗ nE I j I j +1  = 0 ∀ 1 ≤ j ≤ n in o r der to pro v e the claim. If n = 1 o r i = 0 , this is our assumption. So assume n ≥ 2 and i ≥ 1. Then w e hav e Ω i nE /k ⊗ nE I j I j +1 ∼ = Ω i nE /k ⊗ nE  O E ⊗ E I j I j +1  ∼ = Ω i nE /k ⊗ E I j I j +1 , AN AR TIN -REES THEOREM AN D APPLICA TIONS TO ZER O CYCLES 33 where Ω i nE /k = Ω i nE /k ⊗ nE O E . Next w e ha v e short exact sequence (9.5) 0 → I I 2 → Ω 1 nE /k → Ω 1 E /k → 0 , where the ﬁrst term is zero b ecause the long exact sequence of Andr ´ e-Quillen homology w ould tell us that this term w ould otherwise b e D 1 ( E /k ) whic h v anishes as E is smo oth. Since E is a smo oth divisor on a smo oth v a riet y e X , L = I I 2 is a line bundle o n E and Ω 1 E /k is clearly a vec tor bundle on E . This implies that Ω 1 nE /k is also a vec tor bundle on E and for i ≥ 1, the obv ious ﬁltration of ∧ i E  Ω 1 nE /k  = Ω i nE /k in terms of the tensor pro duct of the exterior p ow ers of L and Ω 1 E /k giv es us an exact se quence (9.6) 0 → Ω i − 1 E /k ⊗ E L → Ω i nE /k → Ω i E /k → 0 . T ensoring this with L j and observing that L j = I j I j +1 for j ≥ 1, we get exact sequence (9.7) 0 → Ω i − 1 E /k ⊗ E L j +1 → Ω i nE /k ⊗ E L j → Ω i E /k ⊗ E L j → 0 . Using the righ t exactne ss of H d as b efore and our assumption, w e g et 9.4 and hence 9.2. Finally , to prov e the lemma, we consider the comm utativ e diagram of exact sequence s 0 / / Ω i ( nE , E ) /k / /   Ω i nE /k     / / Ω i E /k / / 0 0 / / Ω i − 1 E /k ⊗ E L / / Ω i nE /k / / Ω i E /k / / 0 , where the b ottom row is the exact seque nce of 9 .6. Observing that the k ernel of the middle vertical arrow is a quotien t of Ω i nE /k ⊗ I I n , a diagram c hase ab ov e giv es an exact sequence Ω i nE /k ⊗ I I n → Ω i ( nE , E ) /k → Ω i − 1 E /k ⊗ E L → 0 . No w the lemma follo ws from the right exactness of H d on nE , the ab ov e claim 9.2 and our assumption.  Lemma 9.2. Under the c ondition s of L e m ma 9.1, the natur al map H d  nE , Ω i e X /k ⊗ e X O nE  → H d  E , Ω i E /k  is an isom orphism fo r i ≥ 0 and n ≥ 1 . 34 AMALENDU KR ISHNA Pr o of . The pro of follow s f rom the argumen ts very similar to that in the pro of of Lemma 9.1. W e giv e a sk etch. First w e sho w this when n = 1. In this case, the argumen t used in the pro of of the exact seq uence 9.6 also gives the exact seq uence 0 → Ω i − 1 E /k ⊗ E L → Ω i e X /k ⊗ e X O E → Ω i E /k → 0 . T aking the ex act sequence of H d on E , our h yp othesis no w prov es the case n = 1. F or n ≥ 2, we can use the n = 1 case to reduce the pro of of the lemma to sho wing that for n ≥ 2 the nat ur a l map (9.8) H d  nE , Ω i e X /k ⊗ e X O nE  → H d  E , Ω i e X /k ⊗ e X O E  is an isomorphism. W e hav e the short exact sequence o f shea v es on e X 0 → I / I n → O nE → O E → 0 , whic h in turn giv es the exact seque nce 0 → Ω i e X /k ⊗ e X I / I n → Ω i e X /k ⊗ e X O nE → Ω i e X /k ⊗ e X O E → 0 , as Ω i e X /k is a v ector bundle on e X . T aking the exact sequence of H d on nE , w e only need to sho w that H d  nE , Ω i e X /k ⊗ e X I / I n  = 0 for n ≥ 2. But this is prov ed exactly in the same w ay as w e pro v ed 9.2.  Lemma 9.3. Under the c ondition s of L e m ma 9.1, one has H d  nE , Ω i ( nE , E ) / Q  = 0 for i ≥ 0 and n ≥ 1 . Pr o of . The spectral seque nce of 5.1 giv es a ﬁnite ﬁltra tion Ω i nE / Q = F 0 n ⊇ F 1 n ⊇ · · · ⊇ F i n ⊇ F i +1 n = 0 of Ω i nE / Q and a morphism o f ﬁltered mo dules Ω i nE / Q ։ Ω i ( n − 1) E / Q suc h tha t Ω i k / Q ⊗ k Ω i − j nE /k ։ F j n /F j +1 n for 0 ≤ j ≤ i . W e also see immediately fro m the spectral sequence 5.1 a nd b y a desc ending induction on j that F j n ։ F j 1 for 0 ≤ j ≤ i and n ≥ 1. In particular, Ω i ( nE , E ) / Q = Ker  Ω i nE / Q ։ Ω i E / Q  has a ﬁltration Ω i ( nE , E ) / Q = F 0 ( n, 1) ⊇ F 1 ( n, 1) ⊇ · · · F i ( n, 1) ⊇ F i +1 ( n, 1) = 0 with F j ( n, 1) = Ker  F j n ։ F j 1  and F j ( n, 1) /F j +1 ( n, 1) = Ker  F j n /F j +1 n ։ F j 1 /F j +1 1  for 0 ≤ j ≤ i. On the other hand, t he sp ectral sequence 5.1 also sho ws that Ω i k / Q ⊗ k Ω i − j E /k ∼ = F j 1 /F j +1 1 for 0 ≤ j ≤ i as E is smo oth, and this giv es Ω i k / Q ⊗ k Ω i − j ( nE , E ) /k ։ F j ( n, 1) /F j +1 ( n, 1) . AN AR TIN -REES THEOREM AN D APPLICA TIONS TO ZER O CYCLES 35 This in turn g ives a surjection (9.9) Ω i k / Q ⊗ k H d  nE , Ω i − j ( nE , E ) /k  ։ H d  nE , F j ( n, 1) /F j +1 ( n, 1)  for 0 ≤ j ≤ i and n ≥ 1 . No w the lemma follows b y using the exact sequence H d  nE , F j +1 ( n, 1)  → H d  nE , F j ( n, 1)  → H d  nE , F j ( n, 1) /F j +1 ( n, 1)  → 0 , 9.9, Lemma 9.1 and a descending induction o n j .  Pro of of Theorem 1.5: W e ﬁrst sho w at once tha t ( i ) implies ( iv ) and ( v ). W e fo llow the diagr a m 9.1 and the notations as in the b eginning of this section. Since e X (resp Z ) is an A 1 -bundle (resp a P 1 -bundle) o v er Y , it is easy to sho w that C H d +1 ( e X ) = 0 and C H d +1 ( Z ) ∼ = C H d ( Y ). Th us it suﬃces to show that the natural maps (9.10) C H d +1 ( X ) → C H d +1 ( e X ) and C H d +1 ( X ) → C H d +1 ( Z ) are isomorphisms. W e prov e the ﬁrst isomorphism. The pro of o f the second isomor- phism is exactly the same, once w e observ e that E ֒ → e X ֒ → Z is the exce ptional divisor fo r b oth p and p . W e only need to v erify the t w o conditions of Corollary 8.4. The ﬁr st condition is ob vious since the inclusion E ֒ → e X has a section. T o pro ve the second condition, it is enough to show tha t H d  nE , Ω i ( nE , E ) / Q  = 0 , since dim( nE ) = d . By Lemma 9.3, this further reduces to sho wing t ha t (9.11) H d  E , Ω i E /k ⊗ E I n I n +1  = 0 for i ≥ 0 and n ≥ 1 . Let T ⊂ Y b e a hy p erplane section of Y for the g iven embedding Y ֒ → P N k . Then for n ≥ 1, one has as short exact sequence 0 → O Y ( n − 1) → O Y ( n ) → O T ( n ) → 0 . Since T is ( d − 1)-dimensional, o ur assumption no w immediately implies that H d ( Y , O Y ( n )) = 0 ∀ n ≥ 1. Ho w ev er, since π is an A 1 -bundle ov er Y asso ciated to the line bundle O Y (1) with a section E , w e ha v e I n / I ( n +1) ∼ = O Y ( n ) f o r a ll n ≥ 1. Hence w e get (9.12) H d  E , I n / I ( n +1)  = 0 ∀ n ≥ 1 . This prov es 9.11 for i = 0. F or i ≥ 1, w e hav e H d  E , Ω i E /k ⊗ E I n / I ( n +1)  ∼ = H d  Y , Ω i Y /k ( n )  under the isomorphism E ∼ = − → Y . But this last group is zero for n ≥ 1 and i ≥ 1 b y the Akizuki-Nak ano v anishing theorem ( cf. [11], Theorem 1.3). This pro v es 9.11 a nd hence ( i ) implies ( iv ) a nd ( v ). No w w e assume that k is a unive rsal do main. In t his case, the implication ( iv ) ⇒ ( i ) w a s show n b y Sriniv as in [25] (Corollary 2). Before we pro v e the other implications, we ﬁrst observ e that as X has only isolated singularities, the map 36 AMALENDU KR ISHNA H d +1  X , Ω i X /k  → H d +1  X , p ∗  Ω i Z/k  is an isomorphism for i ≥ 0. No w the Lera y sp ectral sequence giv es us for i ≥ 0 , a n exact sequence (9.13) H d  Z , Ω i Z/k  / / lim ← − n H d  nE , Ω i Z/k ⊗ Z O nE  / / H d +1  X , Ω i X /k  / / H d +1  Z , Ω i Z/k  / / 0 . F urthermore, since π is a P 1 -bundle, w e can use the L eray sp ectral sequence again to get (9.14) H i ( Z , O Z ) ∼ = H i ( Y , O Y ) ∼ = H i ( E , O E ) ∀ i ≥ 0 and H i  Z , Ω r Z/k  ։ H i  E , Ω r E /k  ∀ i ≥ 0 r ≥ 1 . Pro of of ( i ) ⇔ ( ii ). Supp ose H d +1  X , O X  = 0. Then ( i ) follows from 9.13 for i = 0 and 9.14, once w e observ e that E ֒ → nE ha s a section for all n ≥ 1. If ( i ) holds, then w e hav e already seen in 9.2 that H d ( nE , I / I n ) = 0 for n ≥ 1 and hence H d ( nE , O nE ) ∼ = H d ( E , O E ) for all n ≥ 1. No w ( ii ) follo ws from 9.13 for i = 0 a nd 9.14. Pro of of ( ii ) ⇔ ( iii ). W e only need to sho w tha t ( ii ) ⇒ ( iii ). Since H d +1  W , Ω i W /k  are biratio na l inv ariants of W , we can replace W by Z ev erywhere b elo w. It suﬃces then to sho w that H d +1  X , Ω i X /k  ∼ = H d +1  Z , Ω i Z/k  for i ≥ 1. By 9.13 and 9.14, this reduces to show ing that H d  nE , Ω i nE /k ⊗ nE I / I n  = 0 for n ≥ 1. But this follo ws from the Akizuk i-Nak ano v anishing theorem and the claim 9.2 in the proof of Lemma 9.1. Pro of of ( iii ) ⇔ ( iv ). W e hav e already sho wn that ( iii ) ⇔ ( i ) a nd ( i ) ⇔ ( iv ) . So w e get ( iii ) ⇔ ( iv ). Pro of of ( iv ) ⇔ ( v ). W e hav e seen before that ( iv ) ⇔ ( i ) ⇒ ( v ). So we a re only left to show that ( v ) ⇒ ( iv ). F or t his, it suﬃces to sho w that Ker  C H d +1 ( X ) → C H d +1 ( Z )  ։ C H d +1 ( X ). Let H ֒ → X b e the hyperplane at inﬁnity , i.e., H is the complemen t of X in X and is isomorphic to Y . Moreov er, the inclusion H ֒ → X fa cto r s through the inclusion H j − → Z and the natural map C H d ( H ) j ∗ − → C H d +1 ( Z ) is an isomorphism. No w w e ha v e the following comm uta tiv e diag ram F / / / /   C H d ( H ) j ∗   0 / / Ker ( p ∗ )   / / C H d +1 ( X )     / / C H d +1 ( Z ) / /   0 0 / / Ker ( p ∗ ) / / C H d +1 ( X ) / / C H d +1 ( e X ) / / 0 , AN AR TIN -REES THEOREM AN D APPLICA TIONS TO ZER O CYCLES 37 where F is the free ab elian group on the closed p oints o f H . Note that as H is con tained in the smo oth lo cus of X , the map F → H d +1 ( X ) is w ell deﬁned and the comp osite F → H d +1 ( X ) → C H d +1 ( X ) is clearly zero. W e already kno w that C H d +1 ( e X ) = 0. Since j ∗ is an isomorphism, a diagram c hase sho ws that Ker ( p ∗ ) ։ C H d +1 ( X ). This completes the proof o f Theorem 1.5.  Reference s [1] M. Andr´ e, Homologie des alg` e br es co mm utatives , Die Gr undlehren der mathema tis chen Wissenschaften, Ba nd 2 06, Spr inger-V erlag, Be r lin-New Y ork, 1974 . [2] L. Ba rvieri-Via le , Zero- cycles on singular v arie ties: torsion and Blo ch’s formula , J. P ure Appl. Alg., 78 , (1 992), 1-13. [3] L. Barbieri-Viale, C. P edrini, C. W eib el, Roitman’s Theorem for singula r V ar ieties , Duk e Math. J., 84 , (1996), 1 55-19 0. [4] H. Bass, Algebraic K -theor y , W. A. Benjamin, Inc., New Y ork - Amsterdam 1 9 68. [5] J. Cathelineau, Lambda structures in algebraic K -theor y , K -Theory , 4 , (1991 ), 591 -606. [6] C. 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W eib el, Ne gative K -the ory of normal surfac es , Duk e Math. J., 108 , (2 001), 1-35. Sc ho ol of Mathematics T a ta Institute Of F undamen tal Researc h Homi Bhabha Road Mum bai,400005, India email : amal@math.tifr.res.in

An Artin-Rees Theorem and applications to zero cycles

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