Relative $K_0$ and relative cycle class map

TL;DR

This paper investigates the relative $K_0$ groups of exact and triangulated categories and introduces a cycle class map connecting Chow groups with modulus to relative $K_0$, advancing understanding in algebraic K-theory.

Contribution

It constructs a new cycle class map from Chow groups with modulus to relative $K_0$, linking algebraic cycles and K-theory in a relative setting.

Findings

Established properties of relative $K_0$ in exact and triangulated categories

Constructed a cycle class map from Chow groups with modulus to relative $K_0$

Provided applications to algebraic K-theory and cycle theory

Abstract

We study relative $K_0$ of exact categories and triangulated categories. As an application, we construct a cycle class map from Chow groups with modulus to relative $K_0$.

Equations

$$\mathsf{cyc}:{\mathrm{CH}}_k(X)\twoheadrightarrow F_k{K}_0(X)/F_{k-1}{K}_0(X)$$

$$\mathsf{cyc}:{\mathrm{CH}}_k(X|D)\twoheadrightarrow F_k{K}_0(X,D)/F_{k-1}{K}_0(X,D)$$

$$K_0(X)\stackrel{\simeq }{\to }{K}_0^{\mathrm{perf}}(X)$$

$$K_0(X,D)\stackrel{\simeq }{\to }{K}_0^{\mathrm{perf}}(X,D)$$

$$F(P)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$(P^{\prime },{\alpha }^{\prime },Q^{\prime })\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$[X]=[X^{\prime }]+[X^{\prime \prime }].$$

$$[(P,\alpha ,Q)]+[(Q,\beta ,R)]=[(P,\beta \alpha ,R)].$$

$$K_0(F)\simeq {\pi }_0\mathrm{hofib}(K(\mathcal{A})\stackrel{F}{\to }K(\mathcal{B})).$$

$$(P^{\prime },{\alpha }^{\prime },Q^{\prime })\looparrowright (P,\alpha ,Q)$$

$$(P^{\prime },{\alpha }^{\prime },Q^{\prime })\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\left(\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right):P_1\oplus P_2\to Q_1\oplus Q_2,$$

$$\left(\begin{array}{cc}1& a\\ 0& 1\end{array}\right)$$

$$\alpha \oplus 1:P\oplus P\to P\oplus P$$

$$\beta \alpha {\beta }^{-1}=\left(\begin{array}{cc}1& a\\ 0& 1\end{array}\right)$$

$$\left(\begin{array}{ccc}1& a& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$$

$$\left[\left(\begin{array}{ccc}1& 0& a\\ 0& 1& 0\\ 0& 0& 1\end{array}\right),\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 1& 1\end{array}\right)\right]$$

$$[(P,\alpha ,Q)]=[(P,\gamma \alpha ,Q)]$$

$$(P,\alpha ,Q)\looparrowright (P\oplus Q,\alpha \oplus 1,Q\oplus Q)\looparrowleft (P,\gamma \alpha ,Q).$$

$$P^{\prime }\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$K_1^{\mathrm{He}}(\mathcal{A})\stackrel{\simeq }{\to }{K}_1(\mathcal{A}).$$

$$\delta :K_1^{\mathrm{He}}(\mathcal{B})\to K_0(F).$$

$$K_1(\mathcal{A})\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$K_1^{\mathrm{He}}(\mathcal{A})\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 6.30452pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&\crcr}}}\ignorespaces{\hbox{\kern-6.30452pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{$\textstyle{P_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 3.80452pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\hbox{{}{\hbox{\kern 5.0pt\hbox{\ignorespaces\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}}}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 30.30452pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 30.30452pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{$\textstyle{P_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 66.91356pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern-3.0pt\lower 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 66.91356pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{$\textstyle{P_{01}}$}}}}}}}\ignorespaces}}}}\ignorespaces,\quad\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 6.35277pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&\crcr}}}\ignorespaces{\hbox{\kern-6.35277pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{$\textstyle{Q_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 3.85278pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\hbox{{}{\hbox{\kern 5.0pt\hbox{\ignorespaces\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}}}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 30.35277pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 30.35277pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{$\textstyle{Q_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 67.0583pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern-3.0pt\lower 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 67.0583pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{$\textstyle{P_{01}}$}}}}}}}\ignorespaces}}}}\ignorespaces.$$

$$l\colon\quad\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 19.83356pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&\crcr}}}\ignorespaces{\hbox{\kern-19.83356pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{$\textstyle{(P,\alpha,Q)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 17.33356pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\hbox{{}{\hbox{\kern 5.0pt\hbox{\ignorespaces\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}}}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 43.83356pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 43.83356pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{$\textstyle{(R,\beta,S)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 106.53488pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern-3.0pt\lower 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 106.53488pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{$\textstyle{(N,1,N)}$}}}}}}}\ignorespaces}}}}\ignorespaces$$

$${\pi }_0|GF|=K_0(F).$$

$$\theta :K_0(F)\to {\pi }_0\mathrm{hofib}(K(\mathcal{A})\to K(\mathcal{B})).$$

$$\partial :K_1(\mathcal{B})\to {\pi }_0\mathrm{hofib}(K(\mathcal{A})\to K(\mathcal{B})),$$

$$(P_1,{\alpha }_1,Q_1)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

Full text

We apply it to the pair $(X,D)$ of a scheme $X$ and an affine closed subscheme $D$ of $X$, and get a description of the relative $K_{0}$-group $K_{0}(X,D)$ in terms of perfect complexes; it is generated by pairs of two perfect complexes of $X$ together with quasi-isomorphisms along $D$. This description makes it possible to assign a cycle class in $K_{0}(X,D)$ to a cycle on $X$ not meeting $D$ in an intuitive way. When $X$ is a separated regular scheme of finite type over a field and $D$ is an affine effective Cartier divisor on $X$, we prove that the cycle classes induce a surjective group homomorphism from the Chow group with modulus $\operatorname{CH}_{*}(X|D)$ defined by Binda-Saito to a suitable subquotient of $K_{0}(X,D)$.

1. 0 Introduction
2. 1 Relative $K_{0}$ of exact categories
3. 2 Relative $K_{0}$ of triangulated categories
4. 3 Relative cycle class map

0. Introduction

0.1.

Let $X$ be a separated regular scheme of finite type over a field. Then, to every integral closed subscheme $V$ of $X$, we can assign the cycle class $\mathsf{cyc}(V)$ in the Grothendieck group $K_{0}(X)$ of algebraic vector bundles on $X$. Grothendieck has shown that the cycle classes induce surjective group homomorphisms from the Chow groups to subquotients of $K_{0}(X)$

[TABLE]

for all $k\geq 0$, cf. [SGA6, Exp 0, App. Ch II]. Here, $F_{*}$ is the coniveau filtration; $F_{k}K_{0}(X)$ is generated by perfect complexes of $X$ whose supports are of dimension $\leq k$.

The current paper constructs a relative version of the cycle class map (0.1). Let $D$ be an effective Cartier divisor on $X$. We are interested in the relative $K_{0}$-group $K_{0}(X,D)$, which is defined to be the zeroth homotopy group of the homotopy fiber of the canonical map $K(X)\to K(D)$ between $K$-theory spectra. As a cycle theoretical invariant, we use the Chow group with modulus $\operatorname{CH}_{*}(X|D)$ defined by Binda-Saito [BS17]: It is the group generated by cycles on $X$ which do not meet $D$ divided by a variant of rational equivalence (see §3.3 for details). Here is the main theorem, which generalizes (0.1).

Theorem 0.1** (Theorem 3.7, Lemma 3.4).**

Let $X$ be a separated regular scheme of finite type over a field and $D$ an affine effective Cartier divisor on $X$. Then there exist surjective group homomorphisms

[TABLE]

for all $k\geq 0$, where $F_{*}$ is the “coniveau filtration” (Definition 3.3). Furthermore, if $D$ has an affine open neighborhood in $X$, then $F_{\dim X}K_{0}(X,D)=K_{0}(X,D)$.

In [BK18], Binda-Krishna constructed a cycle class map for zero cycles with modulus, namely a map from $\operatorname{CH}_{0}(X|D)$ to $K_{0}(X,D)$, for modulus pairs $(X,D)$ with $X$ smooth quasi-projective over a perfect field. They have also shown that the cycle class map is injective if $X$ is affine and the base filed is algebraically closed. Also, Binda [Bi18] constructed a cycle class map for higher zero cycles with modulus using a slightly different (stronger) modulus condition.

If $X$ is a smooth quasi-projective scheme over a field, then Grothendieck’s Riemann-Roch type formula implies that the cycle class map (0.1) is a rational isomorphism. In a subsequent paper [IK18], we prove that the relative cycle map (0.2) is a rational isomorphism, at least when $X$ is smooth affine.

0.2.

Let $X$ be a separated regular noetherian scheme and $V$ an integral closed subscheme of $X$. Then the coherent sheaf $\mathcal{O}_{V}$ is a perfect complex of $X$, i.e. quasi-isomorphic to a bounded complex $E_{\bullet}$ of algebraic vector bundles on $X$, and the cycle class is given by $\mathsf{cyc}(V):=\sum(-1)^{i}[E_{i}]\in K_{0}(X)$. Here, it is more natural to consider the group $K^{\mathrm{perf}}_{0}(X)$ generated by perfect complexes of $X$ with the relation $[P]=[P^{\prime}]+[P^{\prime\prime}]$ for each exact triangle $P^{\prime}\to P\to P^{\prime\prime}\to P^{\prime}[1]$. It follows from [SGA6, Exp 1, 6.4] that the canonical map

[TABLE]

is an isomorphism. Under this isomorphism, the cycle class $\mathsf{cyc}(V)$ is just the class of the perfect complex $\mathcal{O}_{V}$ in $K^{\mathrm{perf}}_{0}(X)$.

Now, suppose we are given an affine closed subscheme $D$ of $X$, and we denote the inclusion $D\hookrightarrow X$ by $\iota$. Then the relative $K_{0}$-group $K_{0}(X,D)$ is generated by pairs $(E,E^{\prime})$ of algebraic vector bundles on $X$ together with isomorphisms $E|_{D}\xrightarrow{\simeq}E^{\prime}|_{D}$ along $D$ (Theorem 1.5). As in the absolute case, perfect complexes are more appropriate to construct cycle classes. We define $K^{\mathrm{perf}}_{0}(X,D)$ to be the group generated by pairs $(P,P^{\prime})$ of perfect complexes of $X$ together with isomorphisms $L\iota^{*}P\xrightarrow{\simeq}L\iota^{*}P^{\prime}$ in the derived category of $D$ with suitable relations (Definition 2.1). Then we show that the canonical map

[TABLE]

is an isomorphism (Theorem 3.1). This is a generalization of (0.3).

Let $V$ be an integral closed subscheme of $X$ which does not meet $D$. Then $\mathcal{O}_{V}$ is a perfect complex of $X$ and $L\iota^{*}\mathcal{O}_{V}\simeq 0$. Hence, the pair $(\mathcal{O}_{V},0)$ gives an element of $K^{\mathrm{perf}}_{0}(X,D)$, which we denote by $\mathsf{cyc}(V)$. When $X$ is of finite type over a field and $D$ is a Cartier divisor, we show that $\mathsf{cyc}$ kills the relations that define $\operatorname{CH}_{*}(X|D)$ and get Theorem 0.1.

The hardest part of the above argument is the proof of the isomorphism (0.4). This isomorphism holds more generally for a certain type of exact functors between small exact categories. Actually, in large part of this paper, we discuss relative $K$-theory of exact categories and triangulated categories in general.

Here is a brief summary of the contents of this paper. Let $F\colon\mathcal{A}\to\mathcal{B}$ be an exact functor between small exact categories. We define $K_{0}(F)$ to be the group generated by pairs $(P,Q)$ of two objects in $\mathcal{A}$ together with isomorphisms $F(P)\xrightarrow{\simeq}F(Q)$ in $\mathcal{B}$ with suitable relations (Definition 1.2). Then, under the assumption that $\mathcal{B}$ is split exact and that $F$ is cofinal, $K_{0}(F)$ is isomorphic to the zeroth homotopy group of the homotopy fiber of the map $K(\mathcal{A})\to K(\mathcal{B})$ between the $K$-theory spectra (Theorem 1.5). This essentially follows from Heller’s result in [He65] and we explain it in the first section §1. The second section §2 is the technical heart of this paper. We define a group $K_{0}(T)$ for a triangulated functor $T$ between small triangulated categories (Definition 2.1) as an analogue of the $K_{0}$ for an exact functor between small exact categories. Then, for an exact functor $F\colon\mathcal{A}\to\mathcal{B}$ between small exact categories with $\mathcal{B}$ being split exact, we prove that $K_{0}(F)$ is isomorphic to the $K_{0}$ of the triangulated functor $\mathsf{D}^{b}(\mathcal{A})\to D^{b}(\mathcal{B})$ between the derived categories (Theorem 2.4). This is the general assertion of the isomorphism (0.4). Finally, in the third section §3, we prove Theorem 0.1.

Acknowledgement

I am thankful to Amalendu Krishna for discussions about cycle class maps. Parts of this paper was written when I was in Hausdorff Research Institute for Mathematics as a participant of the trimester program “$K$-theory and Related Fields”. I thank the institute and the organizers for their hospitality. This work was supported by JSPS KAKENHI Grant Number 16J08843, and by the Program for Leading Graduate Schools, MEXT, Japan.

1. Relative $K_{0}$ of exact categories

Let us start from a general construction of a category, which is used throughout this paper.

Definition 1.1**.**

Let $F\colon\mathcal{A}\to\mathcal{B}$ be a functor of categories. We define a category $\mathsf{Rel}(F)$:

• •

Objects are triples $(P,\alpha,Q)$ with $P,Q\in\mathcal{A}$ and $\alpha\colon F(P)\xrightarrow{\simeq}F(Q)$ an isomorphism in $\mathcal{B}$.

• •

Morphisms from $(P,\alpha,Q)$ to $(P^{\prime},\alpha^{\prime},Q^{\prime})$ are pairs $(f,g)$ of morphisms $f\colon P\to P^{\prime}$ and $g\colon Q\to Q^{\prime}$ in $\mathcal{A}$ which make the diagram

[TABLE]

commutative.

1.1. Heller’s theorem

Let $F\colon\mathcal{A}\to\mathcal{B}$ be an exact functor between exact categories. We call a sequence

[TABLE]

in $\mathsf{Rel}(F)$ exact if $P^{\prime}\xrightarrow{f}P\xrightarrow{f^{\prime}}P^{\prime\prime}$ and $Q^{\prime}\xrightarrow{g}Q\xrightarrow{g^{\prime}}Q^{\prime\prime}$ are exact sequences in $\mathcal{A}$. Under this definition, $\mathsf{Rel}(F)$ is an exact category.

Definition 1.2**.**

Let $F\colon\mathcal{A}\to\mathcal{B}$ be an exact functor between small exact categories. We define $K_{0}(F)$ to be the group with the generators $[X]$, one for each $X\in\mathsf{Rel}(F)$, and with the following relations:

1. (a)

For each exact sequence $X^{\prime}\rightarrowtail X\twoheadrightarrow X^{\prime\prime}$ in $\mathsf{Rel}(F)$,

[TABLE] 2. (b)

For each pair $((P,\alpha,Q),(Q,\beta,R))$ of objects in $\mathsf{Rel}(F)$,

[TABLE]

Definition 1.3**.**

An exact category is split exact if every exact sequence is split exact.

Definition 1.4**.**

An additive functor $F\colon\mathcal{A}\to\mathcal{B}$ between additive category is cofinal if for every $B\in\mathcal{B}$ there exists $B^{\prime}\in\mathcal{B}$ and $A\in\mathcal{A}$ such that $F(A)\simeq B\oplus B^{\prime}$.

For a small exact category $\mathcal{A}$, we denote Quillen’s $K$-theory spectrum by $K(\mathcal{A})$. Here is a reinterpretation of Heller’s result in [He65].

Theorem 1.5**.**

Let $F\colon\mathcal{A}\to\mathcal{B}$ be an exact functor between small exact categories. Suppose that $\mathcal{B}$ is split exact and that $F$ is cofinal. Then there exists a natural isomorphism of groups

[TABLE]

We give a proof in §1.4.

1.2. Basic properties

Here, we collect some basic properties of relative $K_{0}$-groups of exact categories (Definition 1.2), whose proof is immediate from the definition.

Lemma 1.6**.**

Let $F\colon\mathcal{A}\to\mathcal{B}$ be an exact functor between small exact categories. Then:

1. (i)

$[0]=0$* in $K_{0}(F)$. If $X,Y\in\mathsf{Rel}(F)$ are isomorphic, then $[X]=[Y]$ in $K_{0}(F)$.* 2. (ii)

If $\gamma\colon P\xrightarrow{\simeq}Q$ is an isomorphism in $\mathcal{A}$, then $[(P,F(\gamma),Q)]=0$ in $K_{0}(F)$. 3. (iii)

For every $(P,\alpha,Q)\in\mathsf{Rel}(F)$, $[(P,\alpha,Q)]+[(Q,\alpha^{-1},P)]=0$ in $K_{0}(F)$. 4. (iv)

Every element of $K_{0}(F)$ has the form $[X]$ for some $X\in\mathsf{Rel}(F)$.

Definition 1.7**.**

Let $F\colon\mathcal{A}\to\mathcal{B}$ be an exact functor between small exact categories. Given two objects $(P,\alpha,Q),(P^{\prime},\alpha^{\prime},Q^{\prime})$ in $\mathsf{Rel}(F)$, we write

[TABLE]

if there exist $N\in\mathcal{A}$ and a commutator $\gamma$ in $\operatorname{Aut}(F(Q))$ which fit into an exact sequence

[TABLE]

Lemma 1.8**.**

Let $F\colon\mathcal{A}\to\mathcal{B}$ be an exact functor between small exact categories and $X,X^{\prime}\in\mathsf{Rel}(F)$. If $X^{\prime}\looparrowright X$, then $[X^{\prime}]=[X]$ in $K_{0}(F)$.

Remark 1.9*.*

In fact, if $\mathcal{B}$ is split exact, then the converse holds, i.e. all relations of $K_{0}(F)$ are generated by $\looparrowright$ (and $\looparrowleft$).

1.3. Elementary transformations

Let $\mathcal{C}$ be an additive category. Let $P,Q\in\mathcal{C}$. Suppose that $P$ and $Q$ have the forms $P=P_{1}\oplus P_{2}$ and $Q=Q_{1}\oplus Q_{2}$. Then a homomorphism from $P$ to $Q$ can be expressed by a matrix

[TABLE]

where $a_{ij}$ is a morphism $P_{j}\to Q_{i}$ in $\mathcal{C}$.

Definition 1.10**.**

An endomorphism $\alpha$ of $P\in\mathcal{C}$ is an elementary transformation if there exists an embedding $\mathcal{C}\hookrightarrow\overline{\mathcal{C}}$ of additive categories and $\alpha$ is isomorphic to an endomorphism of $P_{1}\oplus P_{2}$ of the form

[TABLE]

for some $P_{1},P_{2}\in\overline{\mathcal{C}}$ and $a\colon P_{2}\to P_{1}$. We denote by $E(P)$ the subgroup of $\operatorname{Aut}(P)$ generated by elementary transformations.

Lemma 1.11**.**

Let $P\in\mathcal{C}$ and $\alpha\in E(P)$. Then

[TABLE]

is a commutator of $\operatorname{Aut}(P\oplus P)$.

Proof.

We may assume that $\alpha$ is an elementary transformation, i.e. $\exists\beta\colon P\xrightarrow{\simeq}P_{1}\oplus P_{2}$ and

[TABLE]

for some $a\colon P_{2}\to P_{1}$. Then

[TABLE]

is a commutator of $P_{1}\oplus P_{2}\oplus P_{2}$. Indeed, this is equal to

[TABLE]

in $\operatorname{Aut}(P_{1}\oplus P_{2}\oplus P_{2})$. This implies that $\alpha\oplus 1\colon P\oplus P\to P\oplus P$ is a commutator of $\operatorname{Aut}(P\oplus P)$. ∎

Corollary 1.12**.**

Let $F\colon\mathcal{A}\to\mathcal{B}$ be an exact functor between small exact categories. Let $(P,\alpha,Q)\in\mathsf{Rel}(F)$ and $\gamma\in E(F(Q))$. Then

[TABLE]

in $K_{0}(F)$.

Proof.

According to Lemma 1.11,

[TABLE]

Hence, $[(P,\alpha,Q)]=[(P,\gamma\alpha,Q)]$ by Lemma 1.8. ∎

1.4. Proof of Theorem 1.5

Let $\mathcal{A}$ be a small exact category. In [Ne98], Nenashev provides generators and relations for $K_{1}(\mathcal{A})$; the generators are double exact sequences. In case $\mathcal{A}$ is split exact (Definition 1.3), the generators and relations become simpler and the resulting group coincides with the one considered by Heller in [He65].

Lemma 1.13**.**

Let $\mathcal{A}$ be a small exact category. We define $K_{1}^{\mathrm{He}}(\mathcal{A})$ to be the abelian group with the generators $[h]$, one for each $P\in\mathcal{A}$ and each $h\in\operatorname{Aut}(P)$, and with the following relations:

1. (a)

For a commutative diagram

[TABLE]

with exact rows, $[h]=[h^{\prime}]+[h^{\prime\prime}]$. 2. (b)

$[h_{2}\circ h_{1}]=[h_{2}]+[h_{1}]$* for $h_{1},h_{2}\in\operatorname{Aut}(P)$.*

If $\mathcal{A}$ is split exact, then there exists a natural isomorphism

[TABLE]

Let $F\colon\mathcal{A}\to\mathcal{B}$ be an exact functor between small exact categories. We suppose that $F$ is cofinal (Definition 1.3). Then every element of $K_{1}^{\mathrm{He}}(\mathcal{B})$ is represented by $g\in\operatorname{Aut}(F(P))$ for some $P\in\mathcal{A}$. According to [He65, Proposition 4.2], the class $[(P,g,P)]$ in $K_{0}(F)$ does not depend on the representative and gives a group homomorphism

[TABLE]

We define a map $\iota\colon K_{0}(F)\to K_{0}(\mathcal{A})$ by sending $[(P,\alpha,Q)]$ to $[P]-[Q]$.

Proposition 1.14** (Heller).**

Let $F\colon\mathcal{A}\to\mathcal{B}$ be an exact functor between small exact categories. Suppose that $\mathcal{B}$ is split exact and that $F$ is cofinal. Then the sequence

[TABLE]

is exact.

Proof.

Heller showed the sequence

[TABLE]

is exact [He65, Proposition 5.2]. Now, $K_{1}^{\mathrm{He}}(\mathcal{B})=K_{1}(\mathcal{B})$ and the map $K_{1}^{\mathrm{He}}(\mathcal{A})\to K_{1}^{\mathrm{He}}(\mathcal{B})$ factors through $K_{1}(\mathcal{A})$. This proves the exactness at $K_{1}(\mathcal{B})$. ∎

Proof of Theorem 1.5.

In [GG87], Gillet and Grayson have constructed a simplicial set $G\mathcal{A}$ such that its geometric realization is naturally homotopy equivalent to $K(\mathcal{A})$. The [math]-simplexes of $G\mathcal{A}$ are pairs $(P,Q)$ of objects in $\mathcal{A}$. The $1$-simplexes of $G\mathcal{A}$ are pairs of exact sequences of the forms

[TABLE]

The face maps $G\mathcal{A}_{1}\to G\mathcal{A}_{0}$ send the above to $(P_{0},Q_{0})$ and $(P_{1},Q_{1})$ respectively.

Let $GF_{0}$ be the set of objects in $\mathsf{Rel}(F)$. Let $GF_{1}$ be the set of all pairs $(l,\gamma)$ where $l$ is an exact sequence in $\mathsf{Rel}(F)$ of the form

[TABLE]

and $\gamma$ is a commutator of $\operatorname{Aut}(F(S))$. We define face maps $d_{1},d_{2}\colon GF_{1}\to GF_{0}$ by $d_{1}((l,\gamma)):=(P,\alpha,Q)$ and $d_{2}((l,\gamma)):=(R,\gamma\beta,S)$, and a degeneracy map $s\colon GF_{0}\to GF_{1}$ by $s(X):=(X\xrightarrow{1}X\to 0,1)$. We define $GF$ to be the simplicial set generated by $GF_{1},GF_{0}$. Then

[TABLE]

We have a natural map $GF\to G\mathcal{A}$ which sends $(P,\alpha,Q)\in GF_{0}$ to $(P,Q)$ and $(l,\gamma)\in GF_{1}$ to the underlying pair of exact sequences of $l$. Then the composite $GF\to G\mathcal{A}\to G\mathcal{B}$ is homotopic to zero. Therefore, we obtain a natural map

[TABLE]

According to Proposition 1.14, it remains to show that $\delta\colon K_{1}(\mathcal{B})\to K_{0}(F)$ followed by $\theta$ is equal to the boundary map

[TABLE]

and it is straightforward. ∎

Remark 1.15*.*

The construction of $GF_{0}$ and $GF_{1}$ in the proof suggests that there might be an algebraic construction of $GF_{2},GF_{3},\dotsc$ so that the resulting simplicial set is a model of the double loop space of the relative Waldhausen construction $wS_{\bullet}(S_{\bullet}F)$ in, say, [We13, 8.5.3].

Example 1.16**.**

Here are examples of exact functors $F\colon\mathcal{A}\to\mathcal{B}$ between exact categories such that $\mathcal{B}$ is split exact and that $F$ is cofinal.

1. (1)

A base change functor $\mathbf{P}(A)\to\mathbf{P}(B)$ induced from a ring homomorphism $A\to B$. Here, $\mathbf{P}(-)$ is the category of finitely generated projective modules. 2. (2)

A base change functor $\mathsf{Vec}(X)\to\mathsf{Vec}(Y)$ induced from a morphism of schemes $Y\to X$ with $Y$ affine. Here, $\mathsf{Vec}(-)$ is the category of algebraic vector bundles.

2. Relative $K_{0}$ of triangulated categories

The goal in this section is to prove Theorem 2.4.

2.1. The definition and basic properties

Let $F\colon\mathcal{A}\to\mathcal{B}$ be a triangulated functor between triangulated categories. Refer to Definition 1.1 for the definition of the category $\mathsf{Rel}(F)$. We denote by $[1]$ the shift functor of $\mathcal{A}$ or $\mathcal{B}$, and we define an endofunctor of $\mathsf{Rel}(F)$ by $(P,\alpha,Q)[1]:=(P[1],\alpha[1],Q[1])$. We call a sequence

[TABLE]

in $\mathsf{Rel}(F)$ an exact triangle if

[TABLE]

are exact triangles in $\mathcal{A}$. Under this definition, $\mathsf{Rel}(F)$ is an additive category which satisfies the first two axioms (TR1) and (TR2) of triangulated category in [Ve77, 1.1], but may not satisfy the other axioms (TR3) nor (TR4).

Definition 2.1**.**

Let $F\colon\mathcal{A}\to\mathcal{B}$ be a triangulated functor between small triangulated categories. We define $K_{0}(F)$ to be the group with the generators $[X]$, one for each $X\in\mathsf{Rel}(F)$, and with the following relations:

1. (a)

For each exact triangle $X^{\prime}\to X\to X^{\prime\prime}\to X^{\prime}[1]$ in $\mathsf{Rel}(F)$,

[TABLE] 2. (b)

For each pair $((P,\alpha,Q),(Q,\beta,R))$ of objects in $\mathsf{Rel}(F)$,

[TABLE]

The same properties as in Lemma 1.6 also hold for triangulated categories. Again, the proof is immediate.

Lemma 2.2**.**

Let $F\colon\mathcal{A}\to\mathcal{B}$ be a triangulated functor between small triangulated categories. Then:

1. (i)

$[0]=0$* in $K_{0}(F)$. If $X,Y\in\mathsf{Rel}(F)$ are isomorphic, then $[X]=[Y]$ in $K_{0}(F)$.* 2. (ii)

For every $X\in\mathsf{Rel}(F)$ and every integer $n$, $[X[n]]=(-1)^{n}[X]$ in $K_{0}(F)$. 3. (iii)

If $\gamma\colon P\xrightarrow{\simeq}Q$ is an isomorphism in $\mathcal{A}$, then $[(P,F(\gamma),Q)]=0$ in $K_{0}(F)$. 4. (iv)

For every $(P,\alpha,Q)\in\mathsf{Rel}(F)$, $[(P,\alpha,Q)]+[(Q,\alpha^{-1},P)]=0$ in $K_{0}(F)$. 5. (v)

Every element of $K_{0}(F)$ has the form $[X]$ for some $X\in\mathsf{Rel}(F)$.

Lemma 2.3**.**

Let $F\colon\mathcal{A}\to\mathcal{B}$ be a triangulated functor between small triangulated categories. Let $\mathcal{A}_{0}$ be a thick triangulated subcategory of $\mathcal{A}$ such that $F(\mathcal{A}_{0})=0$. Then $F$ factors through the Verdier quotient $\mathcal{A}/\mathcal{A}_{0}$, and there is an exact sequence of abelian groups

[TABLE]

Proof.

Note that $\mathsf{Ob}(\mathsf{Rel}(F))=\mathsf{Ob}(\mathsf{Rel}(\bar{F}))$. An exact triangle in $\mathsf{Rel}(\bar{F})$ is a sequence

[TABLE]

in $\mathsf{Rel}(\bar{F})$ such that

[TABLE]

are isomorphic in $\mathcal{A}/\mathcal{A}_{0}$ to exact triangles in $\mathcal{A}$. It follows that $K_{0}(\bar{F})$ is the group with the generators $[X]$, one for each $X\in\mathsf{Rel}(F)$, and with the relations (a) (b) of Definition 2.1 and an additional relation

1. (c)

For $X=(P,0,Q)$ with $P,Q\in\mathcal{A}_{0}$, $[X]=0$.

This says that $K_{0}(\bar{F})$ is the quotient of $K_{0}(F)$ by the image of $K_{0}(\mathcal{A}_{0})$. ∎

2.2. Comparison theorem

For an additive category $\mathcal{A}$, we use the following notation:

1. (1)

$\mathsf{Ch}^{b}(\mathcal{A})$ is the category of bounded chain complexes in $\mathcal{A}$. 2. (2)

$\mathsf{K}^{b}(\mathcal{A})$ is the bounded homotopy category, i.e. the same object with $\mathsf{Ch}^{b}(\mathcal{A})$ and morphisms up to homotopy. We regard $\mathsf{K}^{b}(\mathcal{A})$ as a triangulated category in the standard way (cf. [Ve77]).

Here is the main theorem in this section.

Theorem 2.4**.**

Let $\mathcal{A}$ be a small exact category which is closed under the kernels of surjections. Let $\mathcal{B}$ be a small split exact category and $F\colon\mathcal{A}\to\mathcal{B}$ an exact functor. We define $\mathsf{K}^{b,\emptyset}(\mathcal{A})$ to be the full subcategory of $\mathsf{K}^{b}(\mathcal{A})$ consisting acyclic complexes. Then $F$ induces a triangulated functor

[TABLE]

and the canonical map

[TABLE]

is an isomorphism.

Here, we have chosen an embedding of $\mathcal{A}$ into an abelian category (such an embedding exists by Freyd-Mitchell theorem). The terms “kernels of surjections” and “acyclic complexes” are understood in this abelian category. The theorem implies that $K_{0}(\mathsf{D}(F))$ does not depend on the choice of the embedding.

The difficulty in the proof of Theorem 2.4 is how to define the inverse. In absolute case, i.e. $\mathcal{B}=0$, the inverse is clear, which is given by $E_{\bullet}\mapsto\sum(-1)^{i}[E_{i}]$. We cannot imitate this directly because we have to keep track of homotopy equivalences in $\mathcal{B}$. However, a variant does still work. We call it the Euler characteristic and study in §2.3 and §2.4. Using this machinery, the proof of Theorem 2.4 is completed in §2.5.

We fix $\mathcal{A}$, $\mathcal{B}$ and $F$ as in Theorem 2.4 until the end of this section.

2.3. Euler characteristic I

Let $C$ be a bounded complex in $\mathcal{B}$ which is homotopic to zero. Then there exists $s\colon C_{n}\to C_{n+1}$ such that $sd+ds=1$ and $ss=0$, which we call a strict split of $C$. Then the direct sum of the maps

[TABLE]

gives an isomorphism

[TABLE]

Lemma 2.5**.**

If $s^{\prime}$ is another strict split of $C$, then there exists an elementary transformation (Definition 1.10) $\gamma$ of $C_{2*}$ such that $\Phi_{C,s}=\gamma\Phi_{C,s^{\prime}}$.

Proof.

Take an embedding of $\mathcal{B}$ into an abelian category $\overline{\mathcal{B}}$ by Freyd-Mitchell theorem. Let $Z_{n}\in\overline{\mathcal{B}}$ be the kernel of $d\colon C_{n}\to C_{n-1}$. Then $C$ decomposes into short exact sequences

[TABLE]

and $s\epsilon\colon Z_{n-1}\to C_{n}$ or $\delta s\colon C_{n}\to Z_{n}$ give splits of these short exact sequences. Hence, the map

[TABLE]

is an isomorphism with the inverse $\phi_{s,n}^{-1}=(\epsilon,s\epsilon)$.

Now, it is clear that there is an elementary transformation $\gamma_{n}$ of $Z_{n}\oplus Z_{n-1}$ such that $\phi_{s,n}=\gamma_{n}\phi_{s^{\prime},n}$.

On the other hand, $\phi_{s,n}$’s give isomorphisms

[TABLE]

and the composite is equal to $\Phi_{C,s}$ because $\epsilon\delta=d$ and $s\epsilon\delta s=sds=s$. Therefore, there exists an elementary transformation $\gamma$ such that $\Phi_{C,s}=\gamma\Phi_{C^{\prime},s}$. ∎

Let $P,Q$ be bounded complexes in $\mathcal{A}$ and $\alpha$ a homotopy equivalence $F(P)\xrightarrow{\sim}F(Q)$. We apply the above construction to $\operatorname{cone}\alpha$. Set $\Phi:=\Phi_{\operatorname{cone}\alpha,s}$ for some strict split $s$ of $\operatorname{cone}\alpha$. Thanks to Lemma 2.5 and Corollary 1.12, the following is well-defined.

Definition 2.6**.**

We define the Euler characteristic of $(P,\alpha,Q)$ by

[TABLE]

Here are first properties of the Euler characteristic.

Lemma 2.7**.**

Let $P,Q$ be bounded complexes in $\mathcal{A}$ with a homotopy equivalence $\alpha\colon F(P)\xrightarrow{\sim}F(Q)$.

1. (i)

Let $P^{\prime},Q^{\prime}\in\mathsf{Ch}^{b}(\mathcal{A})$ with isomorphisms of complexes $\gamma\colon F(P)\xrightarrow{\simeq}F(P^{\prime})$ and $\delta\colon F(Q)\xrightarrow{\simeq}F(Q^{\prime})$. If $[(P_{n},\gamma_{n},P_{n}^{\prime})]=[(Q_{n},\delta_{n},Q^{\prime}_{n})]=0$ for all $n$, then

[TABLE] 2. (ii)

For every integer $n$, $\chi(P[n],\alpha[n],Q[n])=(-1)^{n}\chi(P,\alpha,Q)$.

Proof.

(i) $\gamma:=(\gamma,\delta)$ gives an isomorphism of complexes $\operatorname{cone}\alpha\xrightarrow{\simeq}\operatorname{cone}(\delta\alpha\gamma^{-1})$. Hence, $\Phi_{\operatorname{cone}(\delta\alpha\gamma^{-1})}$ is equal to the composite

[TABLE]

It follows from the assumption that

[TABLE]

and thus $\chi(P^{\prime},\delta\alpha\gamma^{-1},Q^{\prime})=\chi(P,\alpha,Q)$.

(ii) By the construction,

[TABLE]

By Lemma 2.2 (iv), the right hand side equals to

[TABLE]

which equals to $-\chi(P,\alpha,Q)$ by (i). ∎

We will use the following lemma on homological algebra.

Lemma 2.8**.**

Let $\mathcal{C}$ be an additive category. Suppose given a commutative diagram

[TABLE]

in $\mathcal{C}$ such that the rows are split exact sequences and that $d^{\prime},d^{\prime\prime}$ are split epimorphisms. Let $s^{\prime},s^{\prime\prime}$ be splits of $d^{\prime},d^{\prime\prime}$. Then there exists $\tilde{s}\colon B\to A$ such that $\tilde{s}f_{2}=f_{1}s^{\prime}$, $g_{1}\tilde{s}=s^{\prime\prime}g_{2}$ and $\gamma:=d\tilde{s}$ is an elementary transformation of $B$. In particular, $s:=\tilde{s}\gamma^{-1}$ is a split of $d$.

Proof.

Let $a$ and $b$ are splits of $g_{1}$ and $f_{2}$ respectively;

[TABLE]

We define

[TABLE]

Then $\tilde{s}f_{2}=f_{1}s^{\prime}$ and $g_{1}\tilde{s}=s^{\prime\prime}g_{2}$. Set $\gamma:=d\tilde{s}$. Then $\gamma f_{2}=f_{2}$ and $g_{2}\gamma=g_{2}$, which implies that $\gamma$ is an elementary transformation of $B$. ∎

2.4. Euler characteristic II

In this subsection, we prove that the Euler characteristic defined in Definition 2.6 gives a group homomorphism $\chi\colon K_{0}(\mathsf{D}(F))\to K_{0}(F)$.

Lemma 2.9**.**

Suppose we are given exact sequences

[TABLE]

in $\mathsf{Ch}^{b}(\mathcal{A})$ and homotopy equivalences $\alpha,\alpha^{\prime},\alpha^{\prime\prime}$ fitting into a commutative diagram

[TABLE]

Then

[TABLE]

Proof.

We have a commutative diagram

[TABLE]

with exact rows and columns, where $Z_{*},Z_{*}^{\prime},Z_{*}^{\prime\prime}$ are the kernels of the differentials of $\operatorname{cone}\alpha,\operatorname{cone}\alpha^{\prime},\operatorname{cone}\alpha^{\prime\prime}$. We take splits $s^{\prime}$ and $s^{\prime\prime}$ of $\delta^{\prime}$ and $\delta^{\prime\prime}$. Let $\tilde{s}$ be a map $Z_{n-1}\to F(P_{n-1})\oplus F(Q_{n})$ as in Lemma 2.8, and set $\gamma:=\delta\tilde{s}$, $s:=\tilde{s}\gamma$. Then $\gamma$ is an elementary transformation and $s$ is a split of $\delta$.

Now, we have isomorphisms $\tilde{\phi}_{n},\phi_{n}^{\prime},\phi_{n}^{\prime\prime}$ fitting into a commutative diagram

[TABLE]

Set

[TABLE]

Then we obtain a sequence in $\mathsf{Rel}(F)$

[TABLE]

and it is an exact sequence since the given exact sequences are degree-wise exact.

On the other hand, $\tilde{\Phi}$ is equal to $\Phi:=\phi_{2*}^{-1}\phi_{2*+1}$, $\phi_{n}:=(\epsilon,s)$, modulo elementary transformations. Therefore,

[TABLE]

∎

Lemma 2.10**.**

Let $P,Q$ be bounded complexes in $\mathcal{A}$ with a homotopy equivalence $\alpha\colon F(P)\xrightarrow{\sim}F(Q)$ such that $H_{*}X,H_{*}Y\in\mathcal{A}$. Then we have

[TABLE]

Proof.

Since $\chi(P[1],\alpha[1],Q[1])]=-\chi(P,\alpha,Q)$ by Lemma 2.7 (ii), we may assume that $P_{i}=Q_{i}=0$ for $i<0$. It is easy to see from our assumptions that the kernels and the images of $d_{P}\colon P_{n}\to P_{n-1}$ and $d_{Q}\colon Q_{n}\to Q_{n-1}$ are in $\mathcal{A}$. Now, we have an exact sequence

[TABLE]

By Lemma 2.9 and by induction, we may assume that $P_{i}=Q_{i}=0$ for $i\geq 2$ and that $d_{P}\colon P_{1}\to P_{0}$ and $d_{Q}\colon Q_{1}\to Q_{0}$ are admissible monomorphisms. Now, the cone of $\alpha$ has the form

[TABLE]

which fits into a commutative diagram

[TABLE]

with exact rows and columns. We take a split $s^{\prime\prime}$ of $\bar{\alpha}$, and take $\tilde{s}\colon F(Q_{0})\to F(P_{0})\oplus F(Q_{1})$ as in Lemma 2.8. Then we have a commutative diagram

[TABLE]

Since the rows lift canonically to exact sequences in $\mathcal{A}$, we have

[TABLE]

Finally, it follows from the exact sequence

[TABLE]

that

[TABLE]

∎

Corollary 2.11**.**

1. (i)

Let $f\colon P\to P^{\prime}$ and $g\colon Q\to Q^{\prime}$ be quasi-isomorphisms of bounded complexes in $\mathcal{A}$ with homotopy equivalences $\alpha\colon F(P)\xrightarrow{\sim}F(Q)$ and $\alpha^{\prime}\colon F(P^{\prime})\xrightarrow{\sim}F(Q^{\prime})$ such that $\alpha^{\prime}F(f)=F(g)\alpha$. Then

[TABLE] 2. (ii)

Let $P,Q$ be bounded complexes in $\mathcal{A}$ with a homotopy equivalence $\alpha\colon F(P)\xrightarrow{\sim}F(Q)$. Suppose that $\alpha$ is homotopic to another homotopy equivalence $\beta\colon F(P)\xrightarrow{\sim}F(Q)$. Then

[TABLE]

Proof.

Let $C(F(P))$ be the mapping cylinder of the identity map of $F(P)$. Then $\alpha$ and $\beta$ extend to a homotopy equivalence $C(F(P))\to F(Q)$. Since the canonical map $F(P)\to C(F(P))$ lifts to a quasi-isomorphism in $\mathcal{A}$, (ii) follows from (i).

For (i), we have an exact sequence

[TABLE]

According to Lemma 2.9, we have

[TABLE]

Since $H_{*}\operatorname{cone}f=H_{*}\operatorname{cone}g=0$, it follows from Lemma 2.10 that

[TABLE]

∎

Lemma 2.12**.**

Let $P,Q,R$ be bounded complexes in $\mathcal{A}$ with homotopy equivalences $\alpha\colon F(P)\xrightarrow{\sim}F(Q)$ and $\beta\colon F(Q)\xrightarrow{\sim}F(R)$. Then

[TABLE]

Proof.

From the exact sequence

[TABLE]

it follows that

[TABLE]

Let $\beta^{-1}$ be a homotopy inverse of $\beta$. Then we have homotopy equivalences

[TABLE]

where $\gamma$ is a product of elementary transformations. Hence, by Corollary 2.11 (ii) and Lemma 2.7 (i), we have

[TABLE]

∎

Proposition 2.13**.**

The Euler characteristic defined in Definition 2.6 gives a group homomorphism

[TABLE]

Proof.

Let $X=(P,\bar{\alpha},Q)$ be an object of $\mathsf{Rel}(\mathsf{K}^{b}(\mathcal{A})/\mathsf{K}^{b,\emptyset}(\mathcal{A})\xrightarrow{\mathsf{D}(F)}\mathsf{K}^{b}(\mathcal{B}))$; $P,Q$ are bounded complexes in $\mathcal{A}$ and $\bar{\alpha}$ is the homotopy equivalent class of a homotopy equivalence $\alpha\colon F(P)\xrightarrow{\sim}F(Q)$. Hence, by Corollary 2.11 (ii), the Euler characteristic of $X$

[TABLE]

is well-defined. It remains to show that $\chi$ kills the relations (a) (b) of $K_{0}(\mathsf{D}(F))$ in Definition 2.1.

For the relation (b), let $(P,\bar{\alpha},Q),(Q,\bar{\beta},R)\in\mathsf{Rel}(\mathsf{D}(F))$. Then $\bar{\beta}\bar{\alpha}$ is a homotopy equivalent class of $\beta\alpha$. Hence, by Lemma 2.12, we have

[TABLE]

For the relation (a), let

[TABLE]

be an exact triangle in $\mathsf{Rel}(\mathsf{D}(F))$. According to Corollary 2.11 (i), we may assume that $f,g$ are maps of complexes.

Now, there are isomorphism $\beta\colon P_{3}\xrightarrow{\simeq}\operatorname{cone}f$ and $\gamma\colon Q_{3}\xrightarrow{\simeq}\operatorname{cone}g$ in $\mathsf{K}^{b}(\mathcal{A})/\mathsf{K}^{b,\emptyset}(\mathcal{A})$ which make the diagrams

[TABLE]

commutative. We have a homotopy equivalence $\alpha_{3}^{\prime}\colon F(\operatorname{cone}f)\xrightarrow{\sim}F(\operatorname{cone}g)$ fitting into the commutative diagram

[TABLE]

By Corollary 2.11 (i), we have

[TABLE]

By Lemma 2.9, we conclude that

[TABLE]

∎

2.5. Proof of Theorem 2.4

We prove that the Euler characteristic $\chi\colon K_{0}(\mathsf{D}(F))\to K_{0}(F)$ is the inverse of the canonical map $\iota\colon K_{0}(F)\to K_{0}(\mathsf{D}(F))$. It is clear that, for $X\in\mathsf{Rel}(F)$, $\chi\iota[X]=[X]$. Hence, it remains to prove the following.

Lemma 2.14**.**

For $(P,\alpha,Q)\in\mathsf{Rel}(\mathsf{D}(F))$,

[TABLE]

in $K_{0}(\mathsf{D}(F))$.

Proof.

We may assume that $P_{i}=Q_{i}=0$ for $i<0$. We prove the lemma by induction on $N:=\min\{n\,|\,P_{i}=Q_{i}=0\quad\forall i>n\geq 0\}$. The case $N=0$ is clear.

We use the following notation: Set

[TABLE]

so that $\chi(P,\alpha,Q)=[(\Omega_{1},\Phi,\Omega_{2})]$. According to Freyd-Mitchell theorem, $\mathcal{A}$ has an embedding into an abelian category of modules, which allows us talking about elements of objects in $\mathcal{A}$. In principle, we shall denote elements of $P_{i}$ (resp. $Q_{i}$) by $x_{i}$ (resp. $y_{i}$).

First of all, we construct $(P^{\prime},\alpha^{\prime},Q^{\prime})\in\mathsf{Rel}(\mathsf{D}(F))$ with morphisms

[TABLE]

Here, $Q^{\prime}:=Q$ and

[TABLE]

The quasi-isomorphism $\alpha^{\prime}\colon F(P^{\prime})\to F(Q)$ is given by

[TABLE]

The canonical inclusion $P\to P^{\prime}$ is a quasi-isomorphism, and it yields an isomorphism $(P,\alpha,Q)\xrightarrow{\simeq}(P^{\prime},\alpha^{\prime},Q)$ in $\mathsf{Rel}(\mathsf{D}(F))$. The map $\theta$ is given by

[TABLE]

We show that $[\operatorname{cone}\theta]=0$ in $K_{0}(\mathsf{D}(F))$, which proves the lemma. First few degrees of $\operatorname{cone}\theta$ look like

[TABLE]

It follows that the class of $\operatorname{cone}\theta$ in $K_{0}(\mathsf{D}(F))$ is equal to $-[(R,\beta,S)]$ where

[TABLE]

$\beta_{i}=\alpha_{i+1}$ for $i\geq 1$ and $\beta_{1}$ is given by

[TABLE]

The induction hypothesis implies that $[(R,\beta,S)]=\iota\chi(R,\beta,S)$. We show that $\chi(R,\beta,S)=0$.

We write

[TABLE]

and denote the projections $\Omega_{l}\to\Omega_{l}^{\prime}$ by $\mathrm{pr}_{l}$. Then we have

[TABLE]

where $\Phi^{\prime}$ is given by

[TABLE]

Observe that we have $\chi(R,\beta,S)\looparrowright(\Omega_{1}\oplus\Omega_{2},\Psi,\Omega_{2}\oplus\Omega_{1})$ (see Definition 1.7 for “$\looparrowright$”) with

[TABLE]

Since the class of $(\Omega_{2}\oplus\Omega_{1},-\Phi^{-1}\oplus\Phi,\Omega_{2}\oplus\Omega_{1})$ is zero, we have

[TABLE]

Now, $\Psi(-\Phi^{-1}\oplus\Phi)$ is given by

[TABLE]

where $A:=s((d_{P}s(x_{0},y_{1})_{P_{1}},0))_{P_{1}}$, $B:=s((d_{P}s(x_{0},y_{1})_{P_{1}},0))_{Q_{2}}$ and $s$ is the split $Q_{0}\to P_{0}\oplus Q_{1}$ or $P_{0}\oplus Q_{1}\to P_{1}\oplus Q_{2}$. Hence,

[TABLE]

where $\Psi^{\prime}$ is the restriction of $\Psi(-\Phi^{-1}\oplus\Phi)$. We calculate the class of the left hand side in $K_{0}(F)$ and show that it is zero.

We set $M_{0}:=Q_{0}$, $M_{2}:=P_{1}\oplus Q_{2}$, $M_{1}:=P_{0}\oplus Q_{1}$, and denote by $\delta\colon M_{l}\to M_{l-1}$ the differential of the cone of $\alpha$. Since $-d_{P}s(x_{0},y_{1})_{P_{1}}+(s\delta(x_{0},y_{1}))_{P_{0}}=x_{0}$, we have

[TABLE]

Let $p$ and $q$ be the projections $M_{1}\to P_{0}$ and $M_{1}\to Q_{1}$ respectively. Then $\Psi^{\prime}$ is expressed by the matrix (an endomorphism of $F(M_{0})\oplus F(M_{2})\oplus F(M_{1})$)

[TABLE]

and we have

[TABLE]

Therefore, $\Psi^{\prime}$ lifts to an automorphism of $M_{0}\oplus M_{2}\oplus M_{1}$ modulo elementary transformations, and thus

[TABLE]

∎

3. Relative cycle class map

3.1. Relative $K$-theory of schemes

For a scheme $X$, we use the following notation:

1. (1)

$\mathsf{Vec}(X)$ is the category of algebraic vector bundles on $X$. 2. (2)

$K(X)$ is Quillen’s $K$-theory spectrum of the exact category $\mathsf{Vec}(X)$. 3. (3)

$\mathsf{D}^{b}(X)$ is the derived category of bounded complexes of $\mathcal{O}_{X}$-modules. 4. (4)

$\mathsf{D}^{\mathrm{perf}}(X)\subset\mathsf{D}^{b}(X)$ is the full subcategory of perfect complexes.

For the most part in this section, we shall assume that a scheme has an ample family of line bundles, cf. [TT90, 2.1.1]. For example, any scheme quasi-projective over an affine scheme has an ample family of line bundles. Also, any separated regular noetherian scheme has an ample family of line bundles [SGA6, II 2.2.7.1]. If a scheme $X$ has an ample family of line bundles, then Quillen’s $K$-theory spectrum $K(X)$ behaves well, that is, it is equivalent to the $K$-theory spectrum of the Waldhausen category of perfect complexes of $X$, cf. [TT90, 3.9].

The following theorem is a consequence of the results in §1 and §2.

Theorem 3.1**.**

Let $X$ be a scheme with an ample family of line bundles, $Y$ an affine scheme and $f\colon Y\to X$ a morphism of schemes. Then there exists a natural isomorphism

[TABLE]

See Definition 2.1 for the definition of the right group.

Proof.

Since $X$ has an ample family of line bundles, every perfect complex is quasi-isomorphic to a bounded complex of algebraic vector bundles, and thus there is an equivalence of triangulated categories

[TABLE]

Since $Y$ is affine, $\mathsf{Vec}(Y)$ is split exact and $\mathsf{D}^{\mathrm{perf}}(Y)\simeq\mathsf{K}^{b}(\mathsf{Vec}(Y))$. Now, the triangulated functor $Lf^{*}\colon\mathsf{D}^{\mathrm{perf}}(X)\to\mathsf{D}^{\mathrm{perf}}(Y)$ is identified with the functor $D(f^{*})$ induced from the exact functor $f^{*}\colon\mathsf{Vec}(X)\to\mathsf{Vec}(Y)$, cf. §2.2. Therefore, by Theorem 2.4, we have an isomorphism

[TABLE]

By Theorem 1.5, the left hand side is isomorphic to $\pi_{0}\operatorname{hofib}(K(X)\xrightarrow{f^{*}}K(Y))$, and we get the theorem. ∎

Suppose we are given two schemes $X,Y$ and a morphism of schemes $f\colon Y\to X$ between them. If the morphism $f$ is obvious from the context, we denote by $K(X,Y)$ the homotopy fiber of $f^{*}\colon K(X)\to K(Y)$ and write $K_{0}(X,Y):=\pi_{0}K(X,Y)$. We adapted this notation because our main interest is the case $Y$ is a closed subscheme of $X$; in this case, the map $f\colon Y\to X$ is the canonical inclusion.

It is clear from the definition that $K_{0}(X,Y)$ is contravariant functorial, i.e. a commutative diagram

[TABLE]

induces a group homomorphism $K_{0}(X^{\prime},Y^{\prime})\to K_{0}(X,Y)$.

According to the base change theorem [SGA6, IV 3.1.1]111“La conjecture de finitude [loc. cit., III 2.1]” assumed there has been proved by Kiehl in [Ki72]., we have a proper transfer of $K_{0}(X,Y)$ in the following case.

Proposition 3.2**.**

Suppose we are given a cartesian diagram

[TABLE]

of schemes. Assume that:

1. (i)

$X,X^{\prime}$* have ample family of line bundles and $Y,Y^{\prime}$ are affine.* 2. (ii)

$f$* and $g$ are Tor-independent over $X$ [loc. cit., III 1.5].* 3. (iii)

$g$* is proper and perfect [loc. cit., III 4.1].*

Then there is a map

[TABLE]

which sends $(P,\alpha,Q)\in\mathsf{Rel}(Lf^{\prime*})$ to $(Rg_{*}P,Rg^{\prime}_{*}\alpha,Rg_{*}Q)\in\mathsf{Rel}(Lf^{*})$.

3.2. Coniveau filtration

Let $X$ be a scheme of dimension $d$ which has an ample family of line bundles, $Y$ an affine closed subscheme of $X$, and we denote the inclusion $Y\hookrightarrow X$ by $\iota$. We assume that $X\setminus Y$ is regular. This is a standing assumption that prevails in §3.2.

By Theorem 3.1, we identify $K_{0}(X,Y)$ with the $K_{0}$ of the triangulated functor $Lf^{*}\colon\mathsf{D}^{\mathrm{perf}}(X)\to\mathsf{D}^{\mathrm{perf}}(Y)$.

Definition 3.3**.**

1. (i)

For $\mathfrak{A}=(P,\alpha,Q)\in\mathsf{Rel}(L\iota^{*})$, let $S_{\mathfrak{A}}$ be the set of open neighborhoods $U$ of $Y$ in $X$ such that there exits an isomorphism $\tilde{\alpha}\colon P|_{U}\xrightarrow{\simeq}Q|_{U}$ in $\mathsf{D}^{\mathrm{perf}}(U)$ which lifts $\alpha$. 2. (ii)

For $-1\leq i\leq d$, we define $F_{i}K_{0}(X,Y)$ to be the subgroup of $K_{0}(X,Y)$ generated by elements $\mathfrak{A}\in\mathsf{Rel}(L\iota^{*})$ for which there exists $U\in S_{\mathfrak{A}}$ with $\dim(X\setminus U)\leq i$.

By the definition, $F_{i}K_{0}(X,Y)\subset F_{i+1}K_{0}(X,Y)$, and $F_{-1}K_{0}(X,Y)$ is generated by $(P,\alpha,Q)$ for which there exists $\tilde{\alpha}\colon P\xrightarrow{\simeq}Q$ such that $L\iota^{*}\tilde{\alpha}=\alpha$. Hence, it follows from Lemma 2.2 that $F_{-1}K_{0}(X,Y)=0$. In general, $F_{d}K_{0}(X,Y)$ may not be equal to $K_{0}(X,Y)$. However, we have:

Lemma 3.4**.**

If $Y$ has an affine open neighborhood in $X$, then $F_{d}K_{0}(X,Y)=K_{0}(X,Y)$.

Proof.

Let $\mathfrak{A}\in\mathsf{Rel}(L\iota^{*})$. According to Theorem 2.4, the class of $\mathfrak{A}$ in $K_{0}(X,Y)$ is equal to the one of some $(P,\alpha,Q)$ with $P,Q\in\mathsf{Vec}(X)$. It suffices to show that $S_{(P,\alpha,Q)}\neq\emptyset$, i.e. there exists an open neighborhood $U$ of $Y$ in $X$ and an isomorphism $P|_{U}\xrightarrow{\simeq}Q|_{U}$ which lifts $\alpha$.

By our assumption, we may assume that $X$ is affine, say $X=\operatorname{Spec}A$ and $Y=\operatorname{Spec}A/I$. Since $Q$ is a projective $A$-module, we have an $A$-homomorphism $\gamma\colon P\to Q$ which fits into the commutative diagram

[TABLE]

Let $K$ and $L$ be the kernel and the cokernel of $\gamma$ respectively. We claim that for every $y\in Y$, $K_{y}=L_{y}=0$. Since $-\otimes_{A_{y}}A_{y}/\mathfrak{m}_{y}$ ($\mathfrak{m}_{y}$ is the maximal ideal) is right exact and $\gamma_{y}\otimes_{A_{y}}A_{y}/\mathfrak{m}_{y}$ is an isomorphism, we have $L_{y}\otimes_{A_{y}}A_{y}/\mathfrak{m}_{y}=0$. By Nakayama’s lemma, $L_{y}=0$. Since $Q_{y}$ is projective, the exact sequence

[TABLE]

is split exact. Hence, $K_{y}\otimes_{A_{y}}A_{y}/\mathfrak{m}_{y}=0$ and $K_{y}=0$.

Since the supports of $K$ and $L$ are closed, it follows from the claim that there exists some open neighborhood of $Y$ on which $\gamma$ is an isomorphism. This completes the proof. ∎

Let $C_{k}(X|Y)$ be the set of integral closed subschemes of $X$ of dimension $k$ which do not meet $Y$, and $Z_{k}(X|Y)$ the free abelian group generated by $C_{k}(X|Y)$. For $V\in C_{k}(X|Y)$, the triple $(\mathcal{O}_{V},0,0)$ defines an element of $F_{k}K_{0}(X,Y)$, which we denote by $\mathsf{cyc}(V)$.

Lemma 3.5**.**

Under the standing assumption in §3.2, the map

[TABLE]

is surjective for all $k\geq 0$.

Proof.

Suppose we are given $\mathfrak{A}\in\mathsf{Rel}(L\iota^{*})$ whose class is in $F_{k}K_{0}(X,Y)$ but not in $F_{k-1}$, so that there exists $U\in S_{\mathfrak{A}}$ such that $\dim(X\setminus U)=k$. Set $V:=X\setminus U$ equipped with the reduced scheme structure.

Let $K^{V}(X)$ be the $K$-theory spectrum of $X$ with support in $V$, i.e. the homotopy fiber of the canonical map $K(X)\to K(U)$. Since $V$ does not meet $Y$, $K^{V}(X)$ is equivalent to the homotopy fiber of $K(X,Y)\to K(U,Y)$. Also, since $V$ does not meet $Y$ and $X\setminus Y$ is regular, $K^{V}(X)$ is equivalent to the $G$-theory spectrum $G(V)$ of $V$. Hence, we have an exact sequence

[TABLE]

Now, the class of $\mathfrak{A}$ dies in $K_{0}(U,Y)$, and thus it comes from $G_{0}(V)$.

Since the usual cycle map

[TABLE]

is surjective, the class of $\mathfrak{A}$ is in the image of

[TABLE]

This proves the lemma. ∎

3.3. Chow group with modulus

Let $X$ be a scheme separated of finite type over a field $k$ and $D$ an effective Cartier divisor on $X$. We denote the inclusion $D\hookrightarrow X$ by $\iota$. We recall the definition of the Chow group with modulus by Binda-Saito [BS17].

Let $k\geq 0$. Let $R_{k}(X|D)$ be the set of integral closed subschemes $V$ of $X\times\mathbb{P}^{1}$ of dimension $k+1$ which are dominant over $\mathbb{P}^{1}$ and satisfy the following condition (modulus condition): Let $V^{N}$ be the normalization of $V$ and $\phi$ the canonical map $V^{N}\to X\times\mathbb{P}^{1}$, then we have an inequality of Cartier divisors

[TABLE]

For each $V\in R_{k}(X|D)$, the inverse images $V_{t}$ of $t\in\mathbb{P}^{1}$ are purely of dimension $k$ and do not meet $D$, and thus define elements of $Z_{k}(X|D)$ in the standard way.

Definition 3.6** (Binda-Saito [BS17]).**

The Chow group with modulus $\operatorname{CH}_{k}(X|D)$ is defined to be the quotient of $Z_{k}(X|D)$ by the relations $[V_{0}]=[V_{1}]$ for all $V\in R_{k}(X|D)$.

Theorem 3.7**.**

Suppose that $X$ is regular and that $D$ is affine. Then there is a surjective group homomorphism

[TABLE]

for every $k\geq 0$.

Proof.

Since $X$ is separated regular noetherian, it has an ample family of line bundles [SGA6, II 2.2.7.1]. Hence, the assumption in §3.2 is satisfied, and by Lemma 3.5, we have a surjective homomorphism

[TABLE]

We show this map factors through $\operatorname{CH}_{k}(X|D)$.

We have to show that, for all $V\in R_{k}(X|D)$, $\mathsf{cyc}(V_{0})=\mathsf{cyc}(V_{1})$ in $F_{k}K_{0}(X,D)/F_{k-1}K_{0}(X,D)$. Let $V\in R_{k}(X|D)$ and consider the diagram

[TABLE]

where $\pi$ is the normalization and the other maps are the obvious ones. We fix a parameter $t$ of $\mathbb{P}^{1}\setminus\{\infty\}$. Let $j_{0}\colon\mathcal{O}(-1)\to\mathcal{O}_{\mathbb{P}^{1}}$ be the canonical inclusion (sending $t$ to $t$) and $j_{1}\colon\mathcal{O}(-1)\to\mathcal{O}_{\mathbb{P}^{1}}$ the map sending $t$ to $t-1$. Now, $V_{t}$ ($t=0,1$) are the closed subschemes of $V$ defined by $p^{*}j_{t}$, and these are regarded as closed subschemes of $X$ by the restriction of $q$.

By Theorem 3.1, we identify $K_{0}(V^{N},q_{N}^{*}D)$ with the $K_{0}$ of the triangulated functor $L\iota^{\prime*}\colon\mathsf{D}^{\mathrm{perf}}(V^{N})\to\mathsf{D}^{\mathrm{perf}}(q_{N}^{*}D)$, where $\iota^{\prime}\colon q_{N}^{*}D\hookrightarrow V^{N}$ is the canonical inclusion. Let $V_{t}^{\prime}$ be the scheme theoretic inverse image of $V_{t}$ by $\pi\colon V^{N}\to V$. Then the triple $(\mathcal{O}_{V_{t}^{\prime}},0,0)$ gives an element of $\mathsf{Rel}(L\iota^{\prime*})$.

First, we show that

[TABLE]

in $K_{0}(V^{N},q_{N}^{*}D)$. Now, we have an exact sequence

[TABLE]

of $\mathcal{O}_{V^{N}}$-modules. Hence, the class of $(\mathcal{O}_{V_{t}^{\prime}},0,0)$ in $K_{0}(V^{N},q^{*}D)$ is equal to the one of

[TABLE]

Let $\theta$ be the multiplication of $(t-1)/t$, which is defined on $\mathbb{P}^{1}\setminus\{0\}$ and an automorphism on $\mathbb{P}^{1}\setminus\{0,1\}$; $\theta$ fits into the commutative diagram

[TABLE]

It follows that

[TABLE]

It is clear that the restriction of $p_{N}^{*}\theta$ on $q_{N}^{*}(\{\infty\})=\phi^{*}(X\times\{\infty\})$ is the identity. Hence, by the modulus condition, the restriction of $p_{N}^{*}\theta$ on $\phi^{*}(D\times\mathbb{P}^{1})=q_{N}^{*}D$ is the identity. This implies that the second term of the above equation is zero; in other words,

[TABLE]

This proves $[(\mathcal{O}_{V_{0}^{\prime}},0,0)]=[(\mathcal{O}_{V_{1}^{\prime}},0,0)]$ in $K_{0}(V^{N},q_{N}^{*}D)$.

Now, $\iota\colon D\hookrightarrow X$ and $q_{N}\colon V^{N}\to X$ are Tor-independent, and $q_{N}$ is proper and perfect since $X$ is regular. Hence, by Proposition 3.2, we have a transfer map

[TABLE]

Consequently, we have

[TABLE]

in $K_{0}(X,D)$.

Finally, we claim that

[TABLE]

modulo $F_{k-1}K_{0}(X,D)$, which completes the proof. The first term is $\sum_{i}m_{i}[(\mathcal{O}_{V_{t,i}},0,0)]$ by definition, where $V_{t,i}$ are irreducible components of $V_{t}$ and $m_{i}$ are their multiplicity. By Lemma 3.8 below, it suffices to compare the length of $\mathcal{O}_{V_{t}}$ and $q_{N*}\mathcal{O}_{V_{t}^{\prime}}$ at the generic point of $V_{t,i}$. This is clear because $V_{t}\hookrightarrow V$ and $V_{t}^{\prime}\hookrightarrow V^{N}$ are defined by the same rational function. ∎

Lemma 3.8**.**

Let $\mathcal{F}$ be a coherent sheaf on $X$ whose support is of dimension $k$ and disjoint from $D$. Then

[TABLE]

in $F_{k}K_{0}(X,D)/F_{k-1}K_{0}(X,D)$. Here, $V$ runs over all integral closed subschemes of $X$ of dimension $k$ and $m_{V}(\mathcal{F})$ is the length of the stalk of $\mathcal{F}$ at the generic point of $V$.

Proof.

Let $j\colon Z\hookrightarrow X$ be the scheme theoretic support of $\mathcal{F}$, i.e. $\mathcal{F}=j_{*}\mathcal{G}$ for some coherent module $\mathcal{G}$ of $Z$. The map $G_{0}(Z)\to K_{0}(X,D)$ sending $[\mathcal{G}]$ to $[(j_{*}\mathcal{G},0,0)]$ is compatible with the coniveau filtration. Hence, it suffices to show that

[TABLE]

in $F_{k}G_{0}(Z)/F_{k-1}G_{0}(Z)$. This is easily verified by induction. ∎