Isomorphisms up to bounded torsion between relative K 0 -groups and Chow groups with modulus
aa r X i v : . [ m a t h . K T ] N ov ISOMORPHISMS UP TO BOUNDED TORSION BETWEEN RELATIVE K -GROUPS ANDCHOW GROUPS WITH MODULUS RYOMEI IWASA AND WATARU KAI
Contents0.
Introduction 11. A presentation of Chow group with modulus 22. A presentation of relative K -group 43. Adams decomposition 74. Proof of Theorem 0.1 105. Applications 14References 150. Introduction
The purpose of this note is to establish isomorphisms up to bounded torsion between relative K -groups and Chow groups with modulus as defined in [BS17]. Theorem 0.1.
Let X be a separated regular noetherian scheme of dimension d and D an effective Cartierdivisor on X . Assume that D has an affine open neighborhood in X . Then there exists a finite descendingfiltration F ∗ on K ( X, D ) and, for each integer p , there exists a surjective group morphism cyc : CH p ( X | D ) ։ F p K ( X, D ) /F p +1 K ( X, D ) such that its kernel is ( p − N -torsion for some positive integer N depending only on p . Furthermore,the filtration F ∗ coincides with the gamma filtration on K ( X, D ) up to ( d − M -torsion for somepositive integer M depending only on d . The case D = ∅ is a classical theorem of Soulé [So85], which owes its origin to Grothendieck’sRiemann-Roch type formula [SGA6]. The filtration F ∗ and the morphism cyc have been constructedin [Iw19] in a slightly weaker generality. The assumption that D has an affine open neighborhood isessential, see Example 4.8.Let S ( X | D ) be the set of all closed subsets in X not meeting D and S ( X | D, the set of all closedsubsets in X × (cid:3) satisfying the modulus condition along D . It follows easily from the definition of CH ∗ ( X | D ) that there is an exact sequence colim Y ∈S ( X | D, Z ∗ Y ( X × (cid:3) ) / / colim Y ∈S ( X | D ) CH ∗ Y ( X ) / / CH ∗ ( X | D ) / / , where Z ∗ Y ( − ) is the group of cycles with supports in Y and CH ∗ Y ( − ) is the Chow group with supports in Y . The real content of this note is to establish an analogous exact sequence for K -groups. In the secondsection, as Theorem 2.2, we establish an exact sequence colim Y ∈S ( X | D, K Y ( X × (cid:3) ) / / colim Y ∈S ( X | D ) K Y ( X ) / / K ( X, D ) / / . Then, from the classical rational isomorphisms between K -groups and Chow groups, we get a rationalisomorphism between K ( X, D ) and CH ∗ ( X | D ) . The estimate on torsion is obtained by using Adamsdecomposition. Convention.
All rings are noetherian and all schemes are separated noetherian. For a point v of a scheme X , we denote by κ ( v ) the residue field of v .1. A presentation of Chow group with modulus
Chow groups with supports.Notation 1.1.
Let X be a scheme and p an integer.(1) We write X ( p ) for the set of all points of codimension p in X , i.e., points v ∈ X whose closuresin X have codimension p . We understand X ( p ) = ∅ if p < .(2) For a closed subset Y of X , we define Z pY ( X ) to be the free abelian group with the generators [ V ] , one for each v ∈ X ( p ) ∩ Y , with V being the closure of v in X . We write Z p ( X ) = Z pX ( X ) .(3) For a closed subscheme D of pure codimension p in X , we write [ D ] := X x i ∈ D (0) length( O D,x i )[ D i ] ∈ Z pD ( X ) , where D i is the closure of x i in X . Construction 1.2.
Let X be a scheme and p an integer. Let w ∈ X ( p − and write W for its closurein X . For each v ∈ X ( p ) ∩ W , there exists a unique group morphism ν v : κ ( w ) × → Z which sends a ∈ O W,v \ { } to the length of O W,v / ( a ) . For f ∈ κ ( w ) × , we define div( f ) := X v ∈ X ( p ) ∩ W ν v ( f )[ V ] ∈ Z pW ( X ) . Definition 1.3.
Let X be a scheme and p an integer. For a closed subset Y of X , we define CH pY ( X ) := coker (cid:16) M w ∈ X ( p − ∩ Y κ ( w ) × div −−→ Z pY ( X ) (cid:17) . We write CH p ( X ) = CH pX ( X ) . Definition 1.4.
Let X be a topological space with irreducible components { X i } i ∈ I . We say that X is unicodimensional if codim X ( V ) = codim X i ( V ) for any i ∈ I and any irreducible closed subset V of X i . A scheme is unicodimensional if the underlying topological space is unicodimensional. Lemma 1.5.
Let X be a unicodimensional catenary scheme, D a closed subscheme of pure codimension r in X and p an integer. Then D is unicodimensional and D ( p − r ) ⊂ X ( p ) . Furthermore, the inclusion ι : D ֒ → X induces a group morphism ι ∗ : CH p − rD ∩ Y ( D ) → CH pY ( X ) for any closed subset Y of X .Proof. Let { X i } i ∈ I (resp. { D j } j ∈ J ) be the set of irreducible components of X (resp. D ). Take j ∈ J and v ∈ D ( p − r ) ∩ D j . Then codim D j ( v ) + codim X i ( D j ) = codim X i ( v ) = codim X ( v ) for any i ∈ I with X i ⊃ D j . Since codim X i ( D j ) = r regardless of the choices of i, j , we see that D is unicodimensional and that codim X ( v ) = codim D ( v ) + r = p . Hence, D ( p − r ) ⊂ X ( p ) . The laststatement is immediate from this. (cid:3) Lemma 1.6.
Let X be a unicodimensional catenary scheme, Y a closed subset of X , D an effectiveCartier divisor on X and p an integer. Let v ∈ X ( p ) ∩ Y whose closure V in X is not contained in D .Then [ V × X D ] ∈ Z pD ∩ Y ( D ) .Proof. It suffices to show that codim D ( V × X D ) = p . First of all, note that V × X D is an effectiveCartier divisor on V , and thus it is of pure codimension in V . It follows that codim X ( V × X D ) = p +1 .Since X is catenary, we conclude that codim D ( V × X D ) = codim X ( V × X D ) − codim X ( D ) = p . (cid:3) SOMORPHISMS UP TO BOUNDED TORSION BETWEEN RELATIVE K -GROUPS AND CHOW GROUPS WITH MODULUS 3 Construction 1.7.
Let X be a unicodimensional catenary scheme, Y a closed subset of X , D an effectiveCartier divisor on X and p an integer. We define a group morphism ι ∗ : Z pY ( X ) → CH pY ∩ D ( D ) as follows, where ι refers to the inclusion D ֒ → X . For an integral closed subscheme V of codimension p in X whose support is in Y , ι ∗ ([ V ]) := ( [ V × X D ] if V * Dj ∗ [ O X ( D ) | V ] if V ⊆ D ∈ CH pY ∩ D ( D ) , where the first equation is well-defined by Lemma 1.6 and, for the second, j refers to the inclusion V ֒ → D and j ∗ : CH ( V ) → CH pY ∩ D ( D ) is the push-forward ensured by Lemma 1.5. Remark . It is the classical fact that the morphism ι ∗ in Construction 1.7 factors through CH pY ( X ) if X is an algebraic scheme, cf., [Fu98, Chapter 2]. It would be true more generally, but we do not needsuch a result for our purpose. Chow groups with modulus.Notation 1.9.
We set (cid:3) := Proj( Z [ T , T ]) and let t be the rational coordinate T /T . We write (cid:3) := (cid:3) \ ( t = {∞} ) . For an integer q and a scheme X , we denote by ι X,q (or simply by ι q ) theinclusion X ֒ → X × (cid:3) defined by t = q . Definition 1.10 (Binda-Saito) . Let X be a scheme and D an effective Cartier divisor on X . Let W bea closed subset of X × (cid:3) . Let W N be the normalization of the closure W (with the reduced scheme-structure) of W in X × (cid:3) and denote by φ W the canonical morphism W N → W . We say that W satisfies the modulus condition along D if the following inequality of Cartier divisors on W N holds φ ∗ W ( D × (cid:3) ) ≤ φ ∗ W ( X × {∞} ) . Lemma 1.11.
Let X be a scheme and D an effective Cartier divisor on X . Let W be a closed subset of X × (cid:3) satisfying the modulus condition along D . Then any closed subset of W satisfies the moduluscondition along D .Proof. The proof for [KP12, Proposition 2.4] works, noting that every integral morphism of schemes isclosed (the theorem of Cohen-Seidenberg, [EGA2, (6.1.10)] or [AM69, Theorem 5.10]).Here we give an alternative argument which might be useful later. First, since the modulus conditionis a local condition, we may assume X is affine X = Spec( A ) and D is principal D = ( f ) . We give W ⊂ X × (cid:3) the reduced scheme structure and consider its restriction to the open subset X × ( (cid:3) \{ } ) =Spec( A [1 /t ]) . Let A [ t − ] /J be the coordinate ring of W ∩ Spec( A [1 /t ]) . The modulus condition for W is equivalent to the condition that the element /tf in the ring of total quotients of A [1 /t ] /J is integralover this ring, i.e., that there is a relation in A [1 /t ] /J of the form t n + g t n − + · · · + g n = 0 with g i ∈ f i A [1 /t ] /J . Now let Y ⊂ W be any closed subset. Then we have the image of the aboverelation to the coordinate ring of Y ∩ Spec( A [1 /t ]) . This implies the modulus condition for Y . (cid:3) Notation 1.12.
Let X be scheme, D an effective Cartier divisor on X and p an integer.(1) S ( X | D ) is the set of all closed subsets of X not meeting D .(2) S ( X | D, is the set of all closed subsets of X × (cid:3) satisfying the modulus condition along D .(3) Z p ( X | D ) is the free abelian group with generators [ V ] , one for each v ∈ X ( p ) whose closure V does not meet D .(4) Z p ( X | D, is the free abelian group with generators [ W ] , one for each w ∈ ( X × (cid:3) ) ( p ) whoseclosure W is dominant over (cid:3) and satisfies the modulus condition along D . RYOMEI IWASA AND WATARU KAI
Remark . We remark that Z p ( X | D ) = colim Y ∈S ( X | D ) Z pY ( X ) and Z p ( X | D, ⊂ colim Y ∈S ( X | D, Z pY ( X × (cid:3) ) . In the latter formula, the difference consists of cycles not dominant over (cid:3) . Definition 1.14.
Let X be a unicodimensional catenary scheme, D an effective Cartier divisor on X and p an integer. We define CH p ( X | D ) := coker (cid:0) Z p ( X | D, ι ∗ − ι ∗ −−−−→ Z p ( X | D ) (cid:1) , where the morphisms ι ∗ and ι ∗ are well-defined by Lemma 1.6 and Remark 1.13. Lemma 1.15.
Let X be a unicodimensional catenary scheme, D an effective Cartier divisor on X , Y aclosed subset of X not meeting D and p an integer. Then the canonical morphism Z pY ( X ) → CH p ( X | D ) factors through CH pY ( X ) .Proof. Suppose given w ∈ X ( p − ∩ Y and f ∈ κ ( w ) × . We have to show that div( f ) = 0 in CH p ( X | D ) .Consider the Cartier divisor E on W × (cid:3) defined by f + (1 − f ) t . Then E gives an element [ E ] in Z p ( X | D, and ι ∗ [ E ] − ι ∗ [ E ] = div( f ) in Z p ( X | D ) . This prove the lemma. (cid:3) Proposition 1.16.
Let X be a unicodimensional catenary scheme, D an effective Cartier divisor on X and p an integer. Then the sequence colim Y ∈S ( X | D, Z pY ( X × (cid:3) ) ι ∗ − ι ∗ / / colim Y ∈S ( X | D ) CH pY ( X ) ǫ / / CH p ( X | D ) / / is exact. Here, ι ∗ , ι ∗ are the morphisms defined in Construction 1.7, and ǫ is the canonical morphism asin Lemma 1.15.Proof. We only have to show that the composite ǫ ◦ ( ι ∗ − ι ∗ ) is zero. Let Y ∈ S ( X | D, and v ∈ ( X × (cid:3) ) ( p ) ∩ Y . Then the closure V of v satisfies the modulus condition along D by Lemma 1.11. If v / ∈ X × { , } , then it is immediate from the definition of CH p ( X | D ) that ( ǫ ◦ ( ι ∗ − ι ∗ ))([ V ]) = 0 .If v ∈ X × { , } , say v ∈ X × { } , then ι ∗ [ V ] = 0 and ι ∗ [ V ] = j ∗ [ O X × (cid:3) ( X ) | V ] = 0 since X is aprincipal divisor in X × (cid:3) . This completes the proof. (cid:3) A presentation of relative K -group For a scheme X , we denote by K ( X ) Thomason-Trobaugh’s K -theory spectrum [TT90, 3.1]. Thespectrum K ( X ) is contravariant functorial in X . Definition 2.1.
Let X be a scheme.(1) Let D be a closed subscheme of X . We define K ( X, D ) to be the homotopy fiber of the canonicalmorphism K ( X ) → K ( D ) . For an integer n , we write K n ( X, D ) = π n K ( X, D ) .(2) Let Y be a closed subset of X . We define K Y ( X ) to be the homotopy fiber of the canonicalmorphism K ( X ) → K ( X \ Y ) . For an integer n , we write K Yn ( X ) = π n K Y ( X ) .The goal of this section is to prove the following theorem. Theorem 2.2.
Let X be a regular scheme and D an effective Cartier divisor on X . Assume that D admitsan affine open neighborhood in X . Then the sequence colim Y ∈S ( X | D, K Y ( X × (cid:3) ) ι ∗ − ι ∗ / / colim Y ∈S ( X | D ) K Y ( X ) ǫ / / K ( X, D ) / / is exact. Here, ǫ denotes the obvious morphism. The surjectivity of ǫ has been observed in [Iw19]. SOMORPHISMS UP TO BOUNDED TORSION BETWEEN RELATIVE K -GROUPS AND CHOW GROUPS WITH MODULUS 5 Lemma 2.3.
Let X be a scheme and D a closed subscheme of X . Assume that D has an affine openneighborhood in X . Then the canonical morphism ǫ : colim Y ∈S ( X | D ) K Y ( X ) → K ( X, D ) is surjective.Proof. Let U be an affine open neighborhood of D in X . By the localization theorem [TT90, 7.4], thesequence K X \ U ( X ) / / K ( X, D ) / / K ( U, D ) is exact. Hence, we may replace X by U and reduce to the case X is affine. Then the result follows from[Iw19, Lemma 3.4]. (cid:3) The rigidity.Lemma 2.4.
Let X be a regular scheme and D an effective Cartier divisor. Let Y ∈ S ( X | D, anddenote its closure in X × (cid:3) by Y . Assume that D admits an affine open neighborhood in X . Then thetwo morphisms ι ∗ , ι ∗ : K Y ( X × (cid:3) ) → K ( X, D ) coincide.Proof. First of all, let us fix notation for morphisms of schemes: X ι / / ι / / X × (cid:3) p (cid:15) (cid:15) q / / X D i o o (cid:3) where p, q are the canonical projections and i is the canonical inclusion.According to [Iw19, Theorem 3.1], K ( X, D ) is generated by triples ( P, α, Q ) where P, Q are perfectcomplexes of X and α is a quasi-isomorphism Li ∗ P ∼ −→ Li ∗ Q . The morphism ι ∗ a : K Y ( X × (cid:3) ) → K ( X, D ) a ∈ { , } sends [ P ] to [( Lι ∗ a P, , , where P is a perfect complex of X × (cid:3) whose support lies in Y . Since X is regular, K Y ( X × (cid:3) ) is generated by coherent O Y -modules. Hence, it suffices to show that, for anycoherent O Y -modules F , [( Lι ∗ F , , Lι ∗ F , , in K ( X, D ) . In the sequel, we denote by F a coherent O Y -module. First calculation in K ( X, D ) . Recall that we have fixed a rational coordinate t of (cid:3) . We denote by O ( − the invertible sheaf on (cid:3) generated by t . We write j for the canonical inclusion O ( − → O (cid:3) sending t to t , and write j for the inclusion O ( − → O (cid:3) sending t to t − . Then we have an exacttriangle Lp ∗ O ( − ⊗ LX × (cid:3) F p ∗ j a / / F / / ι a ∗ Lι ∗ a F + / / of perfect complexes of X × (cid:3) for a ∈ { , } . Consequently, [( Lι ∗ a F , , − Rq ∗ [( Lp ∗ O ( − ⊗ LX × (cid:3) F , p ∗ j a , F )] RYOMEI IWASA AND WATARU KAI in K ( X, D ) . We set θ := ( t − /t . Then we have a commutative diagram (the vertical arrow is definedafter restricting to (cid:3) \ { } ) O (cid:3) multiplication by θ (cid:15) (cid:15) O ( − j ♣♣♣♣♣♣♣ j ' ' ◆◆◆◆◆◆ O (cid:3) . Since Y satisfies the modulus condition, the multiplication by θ on O Y -modules makes sense in a neigh-borhood of Y ∩ ( D × (cid:3) ) . It follows from the above diagram that [( Lp ∗ O ( − ⊗ LX × (cid:3) F , p ∗ j , F )] + [( F , θ, F )] = [( Lp ∗ O ( − ⊗ LX × (cid:3) F , p ∗ j , F )] . Hence, it remains to show that Rq ∗ [( F , θ, F )] = 0 in K ( X, D ) . Adic filtration on F . Let
Fil ∗ F be the adic filtration on F with respect to the ideal defining D × {∞} in X × (cid:3) . Since (1 /t ) Fil l F ⊂
Fil l +1 F in a neighborhood of Y ∩ ( D × (cid:3) ) , we see that [(Fil l F / Fil l +1 F , θ, Fil l F / Fil l +1 F )] = 0 for all l ≥ . Hence, we are reduced to showing that Rq ∗ [(Fil l F , θ, Fil l F )] = 0 for some l ≥ . Weshow that θ acts on Li ∗ Rq ∗ Fil l F as the identity for sufficiently large l .Take an affine open neighborhood U of D in X such that Y U := Y × X U misses X × { , } and thatthe restriction q | Y U : Y U → U is finite. We set some notation: A = O ( U ) A/I = O ( D ) A [1 /t ] /J = O ( Y U ) M = F ( Y U ) The filtration
Fil ∗ F on F descends to a filtration Fil ∗ M on M , which is identified with the ( I, /t ) -adicfiltration. Observe that Li ∗ Rq ∗ Fil l F = ( Li ∗ Fil l M ) ∼ .We claim that there exists n ≥ such that Fil l +1 M = I Fil l M and H k ( Li ∗ Fil l M ) = 0 for all l ≥ n and k > . If we admit the claim, then Li ∗ Fil l M = Fil l M/I
Fil l M = Fil l M/ Fil l +1 M on which we know that θ acts as the identity. The claim is a local question on Spec A , and thus we mayassume that I is principal, I = ( f ) with = f ∈ A . By the modulus condition, we have a relation in A [1 /t ] /J of the form t n + g t n − + · · · + g n − t + g n = 0 for some n ≥ and g k ∈ f k A [1 /t ] /J with ≤ k ≤ n . Repeated application of this relation gives Fil l M = f l M + f l − t M + · · · + f l − ( n − t n − M for l ≥ n . In particular, Fil l +1 M = f Fil l M . Furthermore, since the f -power torsion of M has abounded exponent, Fil l M = f l − n Fil n M has no f -power torsion for l ≫ n . This proves the claim. (cid:3) Corollary 2.5.
Under the situation in Theorem 2.2, ǫ ◦ ( ι ∗ − ι ∗ ) = 0 .Proof. Since X is regular, the restriction morphism K Y ( X × (cid:3) ) → K Y ( X × (cid:3) ) is surjective, where Y ∈ S ( X | D, and Y is the closure of Y in X × (cid:3) . Hence, the result follows fromLemma 2.4. (cid:3) SOMORPHISMS UP TO BOUNDED TORSION BETWEEN RELATIVE K -GROUPS AND CHOW GROUPS WITH MODULUS 7 End of the proof.Lemma 2.6.
Let X be a scheme and D an effective Cartier divisor on X admitting an affine openneighoborhood in X . Suppose we are given Z ∈ S ( X | D ) and α ∈ K ( X \ Z ) whose restriction to K ( D ) is zero. Then there exist W ∈ S ( X | D, and β ∈ K ( X × (cid:3) \ W ) such that β | t =0 − β | t =1 = α in colim Y ∈S ( X | D ) K ( X \ Y ) . Proof.
By enlarging Z if necessarily, we may assume that X \ Z is affine. Set U = X \ Z . Take arepresentative of α in GL n ( U ) , which we also denote by α . By our assumption, the restriction of α tomatrices over D is in the group E m ( D ) of elementary matrices for some m ≥ n . Take a lift α ′ ∈ E m ( U ) of α | D . Then α = α ′ + ǫ in GL m ( U ) , where ǫ is a matrix whose entries are all in the ideal I defining D . We define an ( m × m ) -matrix over U × (cid:3) by α ( t ) := α ′ + (1 − t ) ǫ. Then the determinant det( α ( t )) is an admissible polynomial for D in the sense [BS17, §4], and thus itszero locus W = V (det( α ( t ))) satisfies the modulus condition along D . Lastly, by definition, α ( t ) givesan element β in K ( X × (cid:3) \ W ) and it satisfies the desired formula. (cid:3) Proof of Theorem 2.2.
By Lemma 2.3 and Corollary 2.5, it remains to show the exactness at the middleterm. Suppose we are given Z ∈ S ( X | D ) and α ∈ K Z ( X ) which dies in K ( X, D ) along the obviousmorphism. Watch the commutative diagram with exact rows K ( X \ Z, D ) / / K ( X \ Z ) / / (cid:15) (cid:15) K ( D ) (cid:15) (cid:15) K ( X \ Z, D ) / / K Z ( X ) / / K ( X, D ) . It follows that there exists a lift α ′ ∈ K ( X \ Z ) of α along the boundary morphism whose restriction to K ( D ) is zero. Hence, by Lemma 2.6, we find an element β ′ in the group at the left upper corner of thecommutative diagram colim Y ∈S ( X | D, K ( X × (cid:3) \ Y ) ι ∗ − ι ∗ / / (cid:15) (cid:15) colim Y ∈S ( X | D ) K ( X \ Y ) (cid:15) (cid:15) colim Y ∈S ( X | D, K Y ( X × (cid:3) ) ι ∗ − ι ∗ / / colim Y ∈S ( X | D ) K Y ( X ) ǫ / / K ( X, D ) such that ι ∗ β ′ − ι ∗ β ′ = α ′ in the upper middle group. This proves the exactness of the lower sequence. (cid:3) Adams decomposition
Notation 3.1.
For i ≥ , we set w i = if i is zero if i is odddenominator of | B i/ | / i if i is positive evenwhere B n denotes the n -th Bernoulli number. Lemma 3.2. (i)
If a prime p divides w i , then ( p − divides i . The converse is true if i is even. (ii) For i ≥ and for N large enough than i , w i = gcd k ≥ k N ( k i − . Proof.
See [MS74, Appendix B]. (cid:3)
RYOMEI IWASA AND WATARU KAI
Definition 3.3.
Let N be a positive integer. We say that a morphism f : A → B of abelian groups is an N -monomorphism (resp. N -epimorphism ) if the kernel (resp. cokernel) of f is killed by N . We say that f is an N -isomorphism if it is an N -monomorphism and an N -epimorphism. Proposition 3.4.
Let A be an abelian group equipped with endomorphisms ψ k for k > which commutewith each other. Suppose that there is a finite filtration A = F i ⊃ F i +1 ⊃ · · · ⊃ F j ⊃ F j +1 = 0 with ≤ i ≤ j consisting of subgroups preserved by ψ k such that ψ k acts on F p /F p +1 by the multipli-cation by k p . For p ≥ , let A ( p ) be the subgroup of A consisting of elements x ∈ A such that ψ k x = k p x for all k > . Then: (i) For i ≤ p ≤ j , ( Q p − q = i w p − q ) A ( p ) is in F p and the induced morphism p − Y q = i w p − q : A ( p ) → F p /F p +1 is a ( Q jq = i w | p − q | ) -isomorphism. (ii) The canonical morphism j M p = i A ( p ) → A is a ( Q i ≤ q,p ≤ j w | p − q | ) -isomorphism.Proof. We follow [So85, 2.8]. Take integers A pqk ( p = q ) so that w | p − q | = X A pqk ( k p − k q ) . We fix i ≤ p ≤ j . The morphism Φ p := p − Y q = i (cid:16)X k> A pqk ( ψ k − k q ) (cid:17) sends A to F p , and on A ( p ) it is the multiplication by Q p − q = i w p − q . On the other hand, the morphism Ψ p := j Y q = p +1 (cid:16)X k> A pqk ( ψ k − k q ) (cid:17) sends F p to A ( p ) , and modulo F p +1 it is the multiplication by Q jq = p +1 w q − p . Moreover, this morphismkills F p +1 . To summarize, ( Q p − q = i w p − q ) A ( p ) is in F p and the induced morphism p − Y q = i w p − q : A ( p ) → F p /F p +1 is a ( Q jq = i w | p − q | ) -isomorphism.Next, we prove (ii). Let us consider the commutative diagram / / q M p = i +1 A ( p ) ∩ F i +1 / / (cid:15) (cid:15) q M p = i A ( p ) / / (cid:15) (cid:15) (cid:16) q M p = i +1 A ( p ) /A ( p ) ∩ F i +1 (cid:17) ⊕ A ( i ) (cid:15) (cid:15) / / / / F i +1 / / A / / A/F i +1 / / with exact rows. The left vertical arrow is a ( Q i +1 ≤ p,q ≤ j w | p − q | ) -isomorphism by induction. Since Φ i +1 sends A to F i +1 and it is the multiplication by w p − i on A ( p ) , L qp = i +1 A ( p ) /A ( p ) ∩ F i +1 is killed by SOMORPHISMS UP TO BOUNDED TORSION BETWEEN RELATIVE K -GROUPS AND CHOW GROUPS WITH MODULUS 9 Q jp = i +1 w p − i . Combining it with (i), we see that the right vertical arrow is a ( Q jp = i w p − i ) -isomorphism.Consequently, the middle vertical arrow is a ( Q i ≤ p,q ≤ j w | p − q | ) -isomorphism. (cid:3) Example I: γ -filtration. We refer to [AT69] for the definition of (non-unital) special λ -rings, γ -filtrationsand Adams operations. Lemma 3.5.
Let I be a non-unital special λ -ring. Assume that the γ -filtration on I is finite. Then the γ -filtration on I together with the Adams operations satisfies the condition of Proposition 3.4.Proof. This follows from [loc. cit., Proposition 5.3]. (cid:3)
Definition 3.6.
Let X be a scheme. We define e K ( X ) := H Zar ( X, B GL + ) the global sections over X of a Zariski-fibrant replacement of (a functorial model of) B GL + . Let Y bea closed subset of X and D a closed subscheme of X . We define e K Y ( X, D ) to be the iterated homotopyfiber of the square e K ( X ) / / (cid:15) (cid:15) e K ( D ) (cid:15) (cid:15) e K ( X \ Y ) / / e K ( D \ Y ) . For a non-negative integer n , we write e K Yn ( X, D ) := π n e K Y ( X, D ) . When D = ∅ (resp. Y = X ), wewrite e K Y ( X ) (resp. e K ( X, D ) ) for e K Y ( X, D ) . Lemma 3.7.
Let X be a scheme, Y a closed subset of X and D a closed subscheme of X . Then e K Yn ( X, D ) is naturally a special λ -ring for each n ≥ . The first grading of the γ -filtration is F γ e K Yn ( X, D ) /F γ e K Yn ( X, D ) ≃ H ,Y (( X, D ) , O × ) n = 0 H ,Y (( X, D ) , O × ) n = 10 n > where H ∗ Zar ,Y (( X, D ) , O × ) denotes the homology of the iterated homotopy fiber of the square H Zar ( X, O × ) / / (cid:15) (cid:15) H Zar ( D, O × ) (cid:15) (cid:15) H Zar ( X \ Y, O × ) / / H Zar ( D \ Y, O × ) . Proof.
Refer to [Le97, Corollary 5.6] for the fact e K Yn ( X, D ) is a special λ -ring. The first gradingis calculated by the determinant det : B GL + → B O × . Indeed, we have γ + γ + · · · = 0 in SK Y ∗ ( X, D ) = H −∗ Zar ,Y (( X, D ) , B SL + ) as in [So85, p524] , and thus F γ SK Y ∗ ( X, D ) = SK Y ∗ ( X, D ) . (cid:3) Example II: coniveau filtration.Definition 3.8.
Let X be a scheme and Y a closed subset of X . For each p ≥ , we define F p K Y ( X ) := colim Z image( K Z ( X ) → K Y ( X )) where Z runs over all closed subset of Y whose codimension in X is greater or equal to p . We call thefiltration the coniveau filtration . We write Gr p K Y ( X ) for the p -th grading of the coniveau filtration. In [So85, Théoremè 4], it is asserted that K ( X ) = H ( X, Z ) ⊕ Pic( X ) ⊕ F γ K ( X ) , but it is not true. The proof shows Gr γ K ( X ) ≃ Pic( X ) , but Pic( X ) may not split. Lemma 3.9.
Let X be a regular scheme of dimension d and Y a closed subset of X of codimension p .Then the coniveau filtration K Y ( X ) = F p ⊃ F p +1 ⊃ · · · ⊃ F d ⊃ F d +1 = 0 together with the Adams operations satisfies the condition of Proposition 3.4.Proof. First, we prove the case Y is a closed point of X . Let j be the inclusion Y ֒ → X and j ∗ thepushfoward K ( Y ) → K Y ( X ) . Then, by [So85, Théorème 3], we have ψ k ◦ j ∗ = k d ( j ∗ ◦ ψ k ) . Since ψ k is the identity on K ( Y ) ≃ Z and j ∗ is an isomorphism, we conclude that ψ k acts by the multiplicationby k d on K Y ( X ) . This proves the case Y is a closed point.We prove the remaining case by descending induction on p . We have seen the case p = d . Let p < d .If q > p , then the canonical morphism colim Z Gr q K Z ( X ) → Gr q K Y ( X ) , where Z runs over all closed subsets of Y whose codimension in X is greater or equal to p + 1 , issurjective. By induction, the Adams operation ψ k acts by the multiplication by k q on the left term, andso on the right. It remains to show that the Adams operation ψ k acts by the multiplication by k p on Gr p K Y ( X ) = K Y ( X ) /F p +1 K Y ( X ) . This follows from the exact sequence [GS87, Lemma 5.2] / / F p +1 K Y ( X ) / / K Y ( X ) / / M y ∈ Y ∩ X ( p ) K y ( X y ) / / . Note that the Adams operations act on the sequence and we have seen that ψ k acts on the right term bythe multiplication by k p . This completes the proof. (cid:3) Example III: relative coniveau filtration.Definition 3.10.
Let X be a scheme and D a closed subscheme of X . For each p ≥ , we define F p K ( X, D ) := colim Z image( K Z ( X ) → K ( X, D )) where Z runs over all closed subset in X not meeting D of codimension greater or equal to p . We callthe filtration the relative coniveau filtration . We write Gr p K ( X, D ) for the p -th grading of the relativeconiveau filtration. Lemma 3.11.
Let X be a scheme of dimension d with an ample family of line bundles and D a closedsubschme of X . Assume that X \ D is regular and that D has an affine open neighborhood in X . Thenthe relative coniveau filtration K ( X, D ) = F ⊃ F ⊃ · · · ⊃ F d ⊃ F d +1 = 0 together with the Adams operations satisfies the condition of Proposition 3.4.Proof. By Lemma 2.3, F K ( X, D ) = K ( X, D ) . By definition, the canonical morphism colim Y ∈S ( X | D ) Gr p K Y ( X ) → Gr p K ( X, D ) is surjective. The morphism is compatible with the Adams operations, and thus the result follows fromLemma 3.9. (cid:3) Proof of Theorem
Cycle class morphisms with supports.Definition 4.1.
Let X be a regular scheme, Y a closed subset of X and p an integer. The cycle classmorphism is a group morphism cyc : Z pY ( X ) → K Y ( X ) defined by sending the closure V of a point v ∈ X ( p ) ∩ Y to O V . SOMORPHISMS UP TO BOUNDED TORSION BETWEEN RELATIVE K -GROUPS AND CHOW GROUPS WITH MODULUS 11 Theorem 4.2 (Gillet-Soulé) . Let X be a regular scheme, Y a closed subset of X and p an integer. Thenthe cycle class morphism induces a surjective group morphism cyc : CH pY ( X ) ։ F p K Y ( X ) /F p +1 K Y ( X ) and its kernel is ( Q p − i =1 ω i ) -torsion.Proof. This is essentially a consequence of results in [So85] as observed in [GS87, Theorem 8.2]. Sinceour formulation claims a little bit stronger than the original one, we give a sketch of the proof here.Consider the Gersten-Quillen spectral sequence E p,q = M x ∈ X ( p ) ∩ Y K − p − q ( k ( x )) ⇒ K Y − p − q ( X ) . The Adams operations act on this spectral sequence, and by the Riemann-Roch type formula [So85,Théorème 3] ψ k acts by the multiplication by k p + i on E p, − p − ir for i = 0 , and r ≥ . It follows thatthe differential d pr : E p − r, − p + r − r → E p, − pr is killed by ω r − for p ≥ r > . In fact, d rr = 0 for r > ,because E , − ≃ M x ∈ X (0) ∩ Y O ( { x } ) × ≃ E , − ∞ . Consequently, the kernel of the canonical surjection E p, − p ։ E p, − p ∞ is killed by Q p − i =1 ω i . On the otherhand, we have E p, − p ≃ CH pY ( X ) and E p, − p ∞ ≃ Gr p K Y ( X ) . Hence, we get the result. (cid:3) Lemma 4.3.
Let X be a regular scheme, Y a closed subset of X , D a principal effective Cartier divisoron X and p an integer. We denote the inclusion D ֒ → X by ι . Then the diagram K Y ( X ) ι ∗ / / K Y ∩ D ( D ) /F p +1 K Y ∩ D ( D ) Z pY ( X ) ι ∗ / / cyc O O CH pY ∩ D ( D ) cyc O O commutes. Here, the bottom horizontal map ι ∗ is the one defined in Construction 1.7.Proof. Suppose that we are given v ∈ X ( p ) ∩ Y and denote its closure in X by V . If V * D , then thecommutativity is clear, i.e., ( ι ∗ ◦ cyc )[ V ] = ( cyc ◦ ι ∗ )[ V ] . Suppose that V ⊆ D . Then ι ∗ [ V ] = 0 in CH pY ∩ D ( D ) . Also, ι ∗ ( O V ) = [ O V f −→ O V ] = 0 in K Y ∩ D ( D ) , where f denotes the defining equationof D . This proves the lemma. (cid:3) Corollary 4.4.
Under the situation in Lemma 4.3, the restriction morphism K Y ( X ) → K Y ∩ D ( D ) preserves the coniveau filtration. On codimension one.Lemma 4.5.
Let X be a regular scheme and Y a closed subset of X not containing any irreduciblecomponents of X . Then there are natural isomorphisms K Y ( X ) ≃ CH Y ( X ) ⊕ F K Y ( X ) , CH Y ( X ) ≃ H ,Y ( X, O × ) and F K Y ( X ) ≃ F γ K Y ( X ) . Proof.
Since K Y ( X ) = F K Y ( X ) , we get an isomorphism cyc : CH Y ( X ) ∼ −→ K Y ( X ) /F K Y ( X ) by Theorem 4.2. Since CH Y ( X ) is the free abelian group generated by the irreducible componentsof Y of codimension one in X , the above isomorphism factors through K Y ( X ) . This proves the firstisomorphism. The second isomorphism follows from the quasi-isomorphism Z ( X, • ) ≃ O × [1] ([Bl86,Theorem 6.1]) and Lemma 3.7. The last isomorphism follows from the first two isomorphisms. (cid:3) Corollary 4.6.
Let X be a regular scheme and D an effective Cartier divisor on X . Assume that D hasan affine open neighborhood in X . Then there are exact sequences colim Y ∈S ( X | D, F K Y ( X × (cid:3) ) ι ∗ − ι ∗ / / colim Y ∈S ( X | D ) F K Y ( X ) ǫ / / F K ( X, D ) / / Y ∈S ( X | D, Gr K Y ( X × (cid:3) ) ι ∗ − ι ∗ / / colim Y ∈S ( X | D ) Gr K Y ( X ) ǫ / / Gr K ( X, D ) / / and the same for the γ -filtration.Proof. The morphisms are well-defined by Corollary 4.4. We may assume that X is connected. If D isempty, then the result is clear. If D is not empty, then it follows from Theorem 2.2 and Lemma 4.5. (cid:3) Theorem 4.7.
Let X be a regular scheme and D an effective Cartier divisor on X . Assume that D hasan affine open neighborhood in X . Then there are natural isomorphisms CH ( X | D ) ≃ Gr K ( X, D ) ≃ Gr γ K ( X, D ) ≃ Pic(
X, D ) . Proof.
The first isomorphism follows from the commutative diagram colim Y ∈S ( X | D, Gr K Y ( X × (cid:3) ) ι ∗ − ι ∗ / / colim Y ∈S ( X | D ) Gr K Y ( X ) ǫ / / Gr K ( X, D ) / / Y ∈S ( X | D, Z Y ( X × (cid:3) ) ι ∗ − ι ∗ / / cyc O O O O colim Y ∈S ( X | D ) CH Y ( X ) ǫ / / cyc ≃ O O CH ( X | D ) / / O O . This is indeed commutative by Lemma 4.3, and the rows are exact by Proposition 1.16 and Corollary 4.6.The middle vertical arrow is an isomorphism by Theorem 4.2.The second isomorphism follows from Lemma 4.5 and Corollary 4.6. The last isomorphism has beenobserved in Lemma 3.7. (cid:3)
Example . Let k be a field, X = P k × k P and D = P k which we regard as a Cartier divisor on X bythe diagonal embedding. Then CH ( X | D ) = 0 , but Gr γ K ( X, D ) = Pic(
X, D ) = Z . On higher codimension.Lemma 4.9.
Let X be a regular scheme, D an effective Cartier divisor on X and p an integer. Assumethat D has an affine open neighborhood in X . Suppose we are given α ∈ ker (cid:16) colim Y ∈S ( X | D ) Gr p K Y ( X ) ǫ −→ Gr p K ( X, D ) (cid:17) . Then there exists β ∈ Gr p K W ( X × (cid:3) ) for some W ∈ S ( X | D, such that ι ∗ β − ι ∗ β = (cid:16) p − Y i =2 w p − i (cid:17)(cid:16) Y ≤ i,j ≤ p w | i − j | (cid:17) α. SOMORPHISMS UP TO BOUNDED TORSION BETWEEN RELATIVE K -GROUPS AND CHOW GROUPS WITH MODULUS 13 Proof.
We may assume that D is non-empty. The case p ≤ is true by Theorem 2.2 and Lemma 4.6.Let p > . We consider the diagram colim Y ∈S ( X | D, p M i =2 ( F K Y ( X × (cid:3) ) /F p +1 ) ( i ) (cid:15) (cid:15) / / colim Y ∈S ( X | D ) p M i =2 ( F K Y ( X ) /F p +1 ) ( i ) (cid:15) (cid:15) colim Y ∈S ( X | D, F K Y ( X × (cid:3) ) /F p +1 / / colim Y ∈S ( X | D ) F K Y ( X ) /F p +1 colim Y ∈S ( X | D, Gr p K Y ( X × (cid:3) ) / / ?(cid:31) O O colim Y ∈S ( X | D ) Gr p K Y ( X ) ?(cid:31) O O Suppose given α as in the statement. According to Corollary 4.6, there exists ˜ β ∈ F K W ( X × (cid:3) ) /F p +1 for some W ∈ S ( X | D, such that ι ∗ β − ι ∗ β = α in colim Y ∈S ( X | D ) F K Y ( X ) /F p +1 . By Lemma 3.4, ( Q ≤ i,j ≤ p w | i − j | ) ˜ β lifts to β (2) + β (3) + · · · + β ( p ) ∈ p M i =2 ( F K W ( X × (cid:3) ) /F p +1 ) ( i ) . By Lemma 3.4 again, we see that (cid:16) Y ≤ i,j ≤ p w | i − j | (cid:17) ( ι ∗ β ( p ) − ι ∗ β ( p ) ) = (cid:16) Y ≤ i,j ≤ p w | i − j | (cid:17) α. Since ( Q p − i =2 w p − i ) β ( p ) is in Gr p K W ( X × (cid:3) ) , we are done. (cid:3) Proof of Theorem 0.1.
Let X be a regular scheme, D an effective Cartier divisor on X and p an integer.Let us consider the commutative diagram colim Y ∈S ( X | D, Gr p K Y ( X × (cid:3) ) / / colim Y ∈S ( X | D ) Gr p K Y ( X ) / / Gr p K ( X, D ) / / Y ∈S ( X | D, CH pY ( X × (cid:3) ) / / cyc O O O O colim Y ∈S ( X | D ) CH pY ( X ) / / cyc O O O O CH p ( X | D ) cyc rel O O O O / / . Since the bottom row is exact (Proposition 1.16) and the composite of the first two morphisms in theupper row is zero (Theorem 2.2), a morphism cyc rel is induced and it is surjective.Suppose we are given α ∈ CH p ( X | D ) such that cyc rel ( α ) = 0 . By Lemma 4.9 and by a simplediagram chase, there exists β ∈ ker( cyc ) which lifts (cid:16) p − Y i =2 w p − i (cid:17)(cid:16) Y ≤ i,j ≤ p w | i − j | (cid:17) α. By Theorem 4.2, ( Q p − i =1 w i ) β = 0 .The last statement (comparison between F ∗ and the gamma filtration) follows from Proposition 3.4and Theorem 4.7. (cid:3) Applications
Multiplicative structure on Chow groups with modulus.
As an application of Theorem 0.1, we provethat there is a natural multiplicative structure on CH ∗ ( X | D ) up to torsion. We formulate it keeping trackof the changes of D . Note that the Chow groups with modulus yield a contravariant functor CH ∗ ( X |− ) : Div + ( X ) op → GrAb from the category of effective Cartier divisors on X to the category of graded abelian groups. Notation 5.1.
Let A be an additive category and l a positive integer. We define a category A Z [1 /l ] havingthe same objects as A and Hom A [1 /l ] ( − , − ) = Hom A ( − , − ) ⊗ Z Z [1 /l ] . For an object M in A , wedenote its image in A Z [1 /l ] by M Z [1 /l ] . Theorem 5.2.
Let X be a regular scheme of dimension d . Let Div +aff ( X ) be the full subcategory of Div + ( X ) consisting of divisors admitting affine open neighborhoods in X . Then CH ∗ ( X |− ) Z [1 / ( d − ∈ Fun ( Div +aff ( X ) op , GrAb ) Z [1 / ( d − has a natural commutative monoid structure.Proof. By Theorem 0.1, it suffices to show that Gr ∗ γ K ( X, − ) has a commutative monoid structure, butit is obvious from its definition. (cid:3) Chow groups with topological modulus.
Here, we show that if D is K -regular then the Chow groupwith modulus CH ∗ ( X | D ) becomes much simpler (at least up to torsion). Compare the following defini-tion with Notation 1.12 and Definition 1.14. Definition 5.3.
Let X be a unicodimensional catenary scheme, D an effective Cartier divisor and p aninteger.(1) S ( X | D top ) := S ( X | D ) and Z p ( X | D top ) := Z p ( X | D ) .(2) S ( X | D top , is the set of all closed subsets of X × (cid:3) not meeting D × (cid:3) .(3) Z p ( X | D top , is the free abelian group with generators [ W ] , one for each w ∈ ( X × (cid:3) ) ( p ) whose closure W is dominant over (cid:3) and not meeting D × (cid:3) .(4) We define CH p ( X | D top ) := coker (cid:0) Z p ( X | D top , ι ∗ − ι ∗ −−−−→ Z p ( X | D top ) (cid:1) . Remark . The groups CH p ( X | D top ) and its higher variant have been studied in [Mi17, IK] by thename of naïve Chow groups with modulus and Chow groups with topological modulus respectively.
Lemma 5.5.
Let X be a unicodimensional catenary scheme, D an effective Cartier divisor on X and p an integer. Then the sequence colim Y ∈S ( X | D top , Z pY ( X × (cid:3) ) ι ∗ − ι ∗ / / colim Y ∈S ( X | D top ) CH pY ( X ) ǫ / / CH p ( X | D top ) / / is exact.Proof. Same as Proposition 1.16. (cid:3)
The following is a variant of Theorem 2.2, which is a special case of [IK, Lemma xx].
Lemma 5.6.
Let X be a scheme and D an effective Cartier divisor on X admitting an affine openneighborhood in X . Assume that X is K -regular and that D is K -regular. Then the sequence colim Y ∈S ( X | D top , K Y ( X × (cid:3) ) ι ∗ − ι ∗ / / colim Y ∈S ( X | D top ) K Y ( X ) ǫ / / K ( X, D ) / / is exact. SOMORPHISMS UP TO BOUNDED TORSION BETWEEN RELATIVE K -GROUPS AND CHOW GROUPS WITH MODULUS 15 Proof.
By the assumption, K ( X, D ) ≃ K ( X × (cid:3) , D × (cid:3) ) , from which it follows that the composite ǫ ◦ ( ι ∗ − ι ∗ ) is zero. The surjectivity of ǫ follows from Lemma 2.3, and the exactness at the middle termfollows from Lemma 2.6. (cid:3) Proposition 5.7.
Let X be a regular scheme and D an effective Cartier divisor on X . Assume that D is K -regular and admits an affine open neighborhood in X . Then, for each integer p , there exists asurjective group morphism CH p ( X | D top ) ։ F p K ( X, D ) /F p +1 K ( X, D ) such that its kernel is ( p − N -torsion for some positive integer N depending only on p .Proof. This follows from Lemma 5.5 and Lemma 5.6 as in §4. (cid:3)
Theorem 5.8.
Let X be a regular scheme and D an effective Cartier divisor on X . Assume that D is K -regular and admits an affine open neighborhood in X . Then, for each integer p , the canonicalmorphism CH p ( X | D ) → CH p ( X | D top ) is a ( p − N -isomorphism for some positive integer N depending only in p .Proof. This follows from Theorem 0.1 and Proposition 5.7. (cid:3)
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Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø.
E-mail address : [email protected] Mathematical Institute, Tohoku University. Aza-Aoba 6-3, Sendai 980-8578, Japan.
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