Isomorphisms up to Bounded Torsion between Relative Ko-Groups and Chow Groups with Modulus

The purpose of this note is to establish isomorphisms up to bounded torsion between relative K 0 -groups and Chow groups with modulus as deﬁned by Binda and Saito.

The purpose of this note is to establish isomorphisms up to bounded torsion between relative K 0groups 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 Cartier divisor on X. Assume that D has an affine open neighborhood in X.Then there exists a finite descending filtration F * on K 0 (X, D) and, for each integer p, there exists a surjective group morphism cyc : CH p (X|D) ։ F p K 0 (X, D)/F p+1 K 0 (X, D) such that its kernel is (p − 1)! N -torsion for some positive integer N depending only on p. Furthermore, the filtration F * coincides with the gamma filtration on K 0 (X, D) up to (d − 1)! M -torsion for some positive integer M depending only on d.
The case D = ∅ is a classical theorem of Soulé [So85], which owes its origin to Grothendieck's Riemann-Roch type formula [SGA6].The filtration F * and the morphism cyc have been constructed in [Iw19] in a slightly weaker generality.The assumption that D has an affine open neighborhood is essential, see Example 4.8.
Let S(X|D) be the set of all closed subsets in X not meeting D and S(X|D, 1) the set of all closed subsets in X × 1 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,1) 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 second section, as Theorem 2.2, we establish an exact sequence colim Y ∈S(X|D,1) Then, from the classical rational isomorphisms between K 0 -groups and Chow groups, we get a rational isomorphism between K 0 (X, D) and CH * (X|D).The estimate on torsion is obtained by using Adams decomposition.
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.

A C
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 closures in X have codimension p.We understand X (p) = ∅ if p < 0.
(2) For a closed subset Y of X, we define Z p Y (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 p X (X).
(3) For a closed subscheme D of pure codimension p in X, we write 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−1) and write W for its closure in X.For each v ∈ X (p) ∩ W , there exists a unique group morphism Definition 1.3.Let X be a scheme and p an integer.For a closed subset Y of X, we define We write CH p (X) = CH p X (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 Xi (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 for any i ∈ I with X i ⊃ D j .Since codim Xi (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 last statement is immediate from this.Lemma 1.6.Let X be a unicodimensional catenary scheme, Y a closed subset of X, D an effective Cartier divisor on X and p an integer First of all, note that V × X D is an effective Cartier divisor on V , and thus it is of pure codimension Construction 1.7.Let X be a unicodimensional catenary scheme, Y a closed subset of X, D an effective Cartier divisor on X and p an integer.We define a group morphism 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 , 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 1 (V ) → CH p Y ∩D (D) is the push-forward ensured by Lemma 1.5.Remark 1.8.It is the classical fact that the morphism ι * in Construction 1.7 factors through CH p Y (X) if X is an algebraic scheme, cf., [Fu98,Chapter 2].It would be true more generally, but we do not need such a result for our purpose.
Chow groups with modulus.Notation 1.9.We set 1 := Proj(Z[T 0 , T 1 ]) and let t be the rational coordinate T 0 /T 1 .We write For an integer q and a scheme X, we denote by ι X,q (or simply by ι q ) the inclusion X ֒→ X × 1 defined by t = q.Definition 1.10 (Binda-Saito).Let X be a scheme and D an effective Cartier divisor on X.Let W be a closed subset of X × 1 .Let W N be the normalization of the closure W (with the reduced schemestructure) of W in X × 1 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 × 1 ) ≤ 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 × 1 satisfying the modulus condition along D. Then any closed subset of W satisfies the modulus condition along D.
Here we give an alternative argument which might be useful later.First, since the modulus condition is a local condition, we may assume X is affine X = Spec(A) and D is principal D = (f ).We give W ⊂ X× 1 the reduced scheme structure and consider its restriction to the open subset X×( ).The modulus condition for W is equivalent to the condition that the element 1/tf in the ring of total quotients of A[1/t]/J is integral over this ring, i.e., that there is a relation in A[1/t]/J of the form any closed subset.Then we have the image of the above relation to the coordinate ring of Y ∩ Spec(A[1/t]).This implies the modulus condition for Y .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, 1) is the set of all closed subsets of X × 1 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, 1) is the free abelian group with generators [W ], one for each w ∈ (X × 1 ) (p) whose closure W is dominant over 1 and satisfies the modulus condition along D.
Remark 1.13.We remark that In the latter formula, the difference consists of cycles not dominant over 1 .
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 Z p (X|D, 1) where the morphisms ι * 0 and ι * 1 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 a closed subset of X not meeting D and p an integer.Then the canonical morphism . This prove the lemma.
Proposition 1.16.Let X be a unicodimensional catenary scheme, D an effective Cartier divisor on X and p an integer.Then the sequence Here, ι * 0 , ι * 1 are the morphisms defined in Construction 1.7, and ǫ is the canonical morphism as in Lemma 1.15.
Proof.We only have to show that the composite ǫ 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 canonical morphism K(X) → K(D).For an integer n, we write . 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 admits an affine open neighborhood in X.Then the sequence colim Here, ǫ denotes the obvious morphism.
The surjectivity of ǫ has been observed in [Iw19].
Lemma 2.3.Let X be a scheme and D a closed subscheme of X. Assume that D has an affine open neighborhood in X.Then the canonical morphism Proof.Let U be an affine open neighborhood of D in X.By the localization theorem [TT90, 7.4], the sequence 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].

The rigidity.
Lemma 2.4.Let X be a regular scheme and D an effective Cartier divisor.Let Y ∈ S(X|D, 1) and denote its closure in X × 1 by Y .Assume that D admits an affine open neighborhood in X.Then the two morphisms Proof.First of all, let us fix notation for morphisms of schemes: where p, q are the canonical projections and i is the canonical inclusion.
According to [Iw19, Theorem 3.1], K 0 (X, D) is generated by triples (P, α, Q) where P, Q are perfect complexes of X and α is a quasi-isomorphism First calculation in K 0 (X, D).Recall that we have fixed a rational coordinate t of 1 .We denote by O(−1) the invertible sheaf on 1 generated by t.We write j 0 for the canonical inclusion O(−1) → O 1 sending t to t, and write j 1 for the inclusion O(−1) → O 1 sending t to t − 1.Then we have an exact triangle in K 0 (X, D).We set θ := (t − 1)/t.Then we have a commutative diagram (the vertical arrow is defined after restricting to Since Y satisfies the modulus condition, the multiplication by θ on O Y -modules makes sense in a neighborhood of Y ∩ (D × 1 ).It follows from the above diagram that Adic filtration on F .Let Fil * F be the adic filtration on F with respect to the ideal defining for all l ≥ 0. Hence, we are reduced to showing that Rq * [(Fil l F , θ, Fil l F )] = 0 for some l ≥ 0. We show 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 × {0, 1} and that the restriction q| Y U : Y U → U is finite.We set some notation: The filtration Fil * F on F descends to a filtration Fil * M on M , which is identified with the (I, 1/t)-adic filtration.Observe that Li * Rq * Fil l F = (Li * Fil l M ) ∼ .We claim that there exists n ≥ 0 such that Fil l+1 M = I Fil l M and H k (Li * Fil l M ) = 0 for all l ≥ n and k > 0. If we admit the claim, then on which we know that θ acts as the identity.The claim is a local question on Spec A, and thus we may assume that I is principal, I = (f ) with 0 = f ∈ A. By the modulus condition, we have a relation in A[1/t]/J of the form

Repeated application of this relation gives
In particular, Fil l+1 M = f Fil l M .Furthermore, since the f -power torsion of M has a bounded exponent, Fil l M = f l−n Fil n M has no f -power torsion for l ≫ n.This proves the claim.

End of the proof.
Lemma 2.6.Let X be a scheme and D an effective Cartier divisor on X admitting an affine open neighoborhood in X. Suppose we are given Z ∈ S(X|D) and α ∈ K 1 (X \ Z) whose restriction to K 1 (D) is zero.Then there exist W ∈ S(X|D, 1) and β ∈ K 1 (X × 1 \ W ) such that Proof.By enlarging Z if necessarily, we may assume that X \ Z is affine.Set U = X \ Z.Take a representative of α in GL n (U ), which we also denote by α.By our assumption, the restriction of α to matrices over D is in the group E m (D) of elementary matrices for some m ≥ n.Take a lift , where ǫ is a matrix whose entries are all in the ideal I defining D. We define an (m × m)-matrix over U × 1 by Then the determinant det(α(t)) is an admissible polynomial for D in the sense [BS17, §4], and thus its zero locus W = V (det(α(t))) satisfies the modulus condition along D. Lastly, by definition, α(t) gives an element β in K 1 (X × 1 \ W ) and it satisfies the desired formula.
Proof of Theorem 2.2.By Lemma 2.3 and Corollary 2.5, it remains to show the exactness at the middle term.Suppose we are given Z ∈ S(X|D) and α ∈ K Z 0 (X) which dies in K 0 (X, D) along the obvious morphism.Watch the commutative diagram with exact rows It follows that there exists a lift α ′ ∈ K 1 (X \ Z) of α along the boundary morphism whose restriction to K 1 (D) is zero.Hence, by Lemma 2.6, we find an element β ′ in the group at the left upper corner of the commutative diagram colim Y ∈S(X|D,1) in the upper middle group.This proves the exactness of the lower sequence.

A
Notation 3.1.For i ≥ 0, we set where B n denotes the n-th Bernoulli number.

Lemma 3.2.
(i) If a prime p divides w i , then (p − 1) divides i.The converse is true if i is even.
(ii) For i ≥ 1 and for N large enough than i, 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 > 0 which commute with each other.Suppose that there is a finite filtration with 0 ≤ i ≤ j consisting of subgroups preserved by ψ k such that ψ k acts on F p /F p+1 by the multiplication by k p .For p ≥ 0, let A (p) be the subgroup of A consisting of elements x ∈ A such that ψ k x = k p x for all k > 0. Then: Proof.We follow [So85, 2.8].Take integers A pqk (p = q) so that We fix i ≤ p ≤ j.The morphism sends A to F p , and on A (p) it is the multiplication by p−1 q=i w p−q .On the other hand, the morphism sends F p to A (p) , and modulo F p+1 it is the multiplication by j q=p+1 w q−p .Moreover, this morphism kills F p+1 .To summarize, ( p−1 q=i w p−q )A (p) is in F p and the induced morphism

Next, we prove (ii). Let us consider the commutative diagram
with exact rows.The left vertical arrow is a ( 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) , q p=i+1 A (p) /A (p) ∩ F i+1 is killed by j p=i+1 w p−i .Combining it with (i), we see that the right vertical arrow is a ( j p=i w p−i )2 -isomorphism.Consequently, the middle vertical arrow is a ( i≤p,q≤j w |p−q| )-isomorphism.
Example I: γ-filtration.We refer to [AT69] for the definition of (non-unital) special λ-rings, γ-filtrations and Adams operations.Definition 3.6.Let X be a scheme.We define the global sections over X of a Zariski-fibrant replacement of (a functorial model of) B GL + .Let Y be a closed subset of X and D a closed subscheme of X.We define K Y (X, D) to be the iterated homotopy fiber of the square For a non-negative integer n, we write Lemma 3.7.Let X be a scheme, Y a closed subset of X and D a closed subscheme of X.Then K Y n (X, D) is naturally a special λ-ring for each n ≥ 0. The first grading of the γ-filtration is where H * Zar,Y ((X, D), O × ) denotes the homology of the iterated homotopy fiber of the square Proof.Refer to [Le97, Corollary 5.6] for the fact K Y n (X, D) is a special λ-ring.The first grading is calculated by the determinant det : B GL + → BO × .Indeed, we have Example II: coniveau filtration.
Definition 3.8.Let X be a scheme and Y a closed subset of X.For each p ≥ 0, we define where Z runs over all closed subset of Y whose codimension in X is greater or equal to p.We call the filtration the coniveau filtration.We write Gr p K Y 0 (X) for the p-th grading of the coniveau filtration.
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 = 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 * the pushfoward K 0 (Y ) → K Y 0 (X).Then, by [So85, Théorème 3], we have Since ψ k is the identity on K 0 (Y ) ≃ Z and j * is an isomorphism, we conclude that ψ k acts by the multiplication by 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 , where Z runs over all closed subsets of Y whose codimension in X is greater or equal to p + 1, is surjective.By induction, the Adams operation ψ k acts by the multiplication by k q on the left term, and so on the right.It remains to show that the Adams operation ψ k acts by the multiplication by . This follows from the exact sequence [GS87, Lemma 5.2] Note that the Adams operations act on the sequence and we have seen that ψ k acts on the right term by the multiplication by k p .This completes the proof.
Definition 3.10.Let X be a scheme and D a closed subscheme of X.For each p ≥ 0, we define where Z runs over all closed subset in X not meeting D of codimension greater or equal to p.We call the filtration the relative coniveau filtration.We write Gr p K 0 (X, D) for the p-th grading of the relative coniveau filtration.Lemma 3.11.Let X be a scheme of dimension d with an ample family of line bundles and D a closed subschme of X. Assume that X \ D is regular and that D has an affine open neighborhood in X.Then the relative coniveau filtration together with the Adams operations satisfies the condition of Proposition 3.4.
is surjective.The morphism is compatible with the Adams operations, and thus the result follows from Lemma 3.9.

P T .
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 class morphism is a group morphism cyc : Theorem 4.2 (Gillet-Soulé).Let X be a regular scheme, Y a closed subset of X and p an integer.Then the cycle class morphism induces a surjective group morphism Proof.This is essentially a consequence of results in [So85] as observed in [GS87, Theorem 8.2].Since our formulation claims a little bit stronger than the original one, we give a sketch of the proof here.
Consider the Gersten-Quillen spectral sequence 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−i r for i = 0, 1 and r ≥ 1.It follows that the differential Consequently, the kernel of the canonical surjection . Hence, we get the result.
Lemma 4.3.Let X be a regular scheme, Y a closed subset of X, D a principal effective Cartier divisor on X and p an integer.We denote the inclusion D ֒→ X by ι.Then the diagram 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 the commutativity is clear, i.e., (ι

On codimension one.
Lemma 4.5.Let X be a regular scheme and Y a closed subset of X not containing any irreducible components of X.Then there are natural isomorphisms is the free abelian group generated by the irreducible components of Y of codimension one in X, the above isomorphism factors through K Y 0 (X).This proves the first isomorphism.The second isomorphism follows from the quasi-isomorphism Z 1 (X, •) ≃ O × [1] ([Bl86, Theorem 6.1]) and Lemma 3.7.The last isomorphism follows from the first two isomorphisms.
Proof.The morphisms are well-defined by Corollary 4.4.We may assume that X is connected.If D is empty, then the result is clear.If D is not empty, then it follows from Theorem 2.2 and Lemma 4.5.
Theorem 4.7.Let X be a regular scheme and D an effective Cartier divisor on X. Assume that D has an affine open neighborhood in X.Then there are natural isomorphisms Proof.The first isomorphism follows from the commutative diagram colim Y ∈S(X|D,1) 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 been observed in Lemma 3.7.
Example 4.8.Let k be a field, X = P 1 k × k P 1 and D = P 1 k which we regard as a Cartier divisor on X by the diagonal embedding.Then CH 1 (X|D) = 0, but Gr 1 γ K 0 (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.Assume that D has an affine open neighborhood in X. Suppose we are given Then there exists β ∈ Gr p K W 0 (X × 1 ) for some W ∈ S(X|D, 1) such that Proof.We may assume that D is non-empty.The case p ≤ 1 is true by Theorem 2.2 and Lemma 4.6.Let p > 1.We consider the diagram colim Y ∈S(X|D,1) Suppose given α as in the statement.According to Corollary 4.6, there exists β ∈ F 2 K W 0 (X × 1 )/F p+1 for some W ∈ S(X|D, 1) such that By Lemma 3.4, ( 2≤i,j≤p w |i−j| ) β lifts to By Lemma 3.4 again, we see that 2≤i,j≤p ), we are done.
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,1) Since the bottom row is exact (Proposition 1.16) and the composite of the first two morphisms in the upper 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 simple diagram chase, there exists β ∈ ker(cyc 0 ) which lifts

A
Multiplicative structure on Chow groups with modulus.As an application of Theorem 0.1, we prove that there is a natural multiplicative structure on CH * (X|D) up to torsion.We formulate it keeping track of 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.Proof.By Theorem 0.1, it suffices to show that Gr * γ K 0 (X, −) has a commutative monoid structure, but it is obvious from its definition.
Chow groups with topological modulus.Here, we show that if D is K 1 -regular then the Chow group with modulus CH * (X|D) becomes much simpler (at least up to torsion).Compare the following definition 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 an integer.
(2) S(X|D top , 1) is the set of all closed subsets of X × 1 not meeting D × 1 .
(3) Z p (X|D top , 1) is the free abelian group with generators [W ], one for each w ∈ (X × 1 ) (p) whose closure W is dominant over 1 and not meeting D × 1 .(4) We define CH p (X|D top ) := coker Z p (X|D top , 1) Remark 5.4.The groups CH p (X|D top ) and its higher variant have been studied in [Mi17,IK] by the name 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

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].

0(
D), where f denotes the defining equation of D. This proves the lemma.Corollary 4.4.Under the situation in Lemma 4.3, the restriction morphism K Y 0 (X) → K Y ∩D 0 (D) preserves the coniveau filtration.

Corollary 4. 6 .
Let X be a regular scheme and D an effective Cartier divisor on X. Assume that D has an affine open neighborhood in X.Then there are exact sequences colim Y ∈S(X|D,1) By Theorem 4.2, ( p−2 i=1 w i )β = 0.The last statement (comparison between F * and the gamma filtration) follows from Proposition 3.4 and Theorem 4.7.
Same as Proposition 1.16.The following is a variant of Theorem 2.2, which is a special case of[IK, Lemma xx].Let X be a scheme and D an effective Cartier divisor on X admitting an affine open neighborhood in X. Assume that X is K 0 -regular and that D is K 1 -regular.Then the sequence colim ǫ / / CH p (X|D top ) / / 0 is exact.Proof.Y ∈S(X|Dtop,1) K Y 0 (X × 1 ) ι * 0 −ι * 1 / / colim Y ∈S(X|Dtop) K Y 0 (X) ǫ / / K 0 (X, D) / / 0 is exact.