1.1 Modulus pairs and cycle complex presheaves

We begin by the definition of the categories of modulus pairs.

We would like to have a variant $\mathsf{MSm}^{\ast }$ of it on which cycle complexes with modulus are functorial. Our specific choice below is not crucial in this work. As another possible choice of $\mathsf{MSm}^{\ast }$ , one could probably take a version of Levine’s ${\mathcal{L}}(\text{Sm})$ [Reference LevineLev98, page 9].

Since the dependence on $F$ does not play a major role in this article, we do not make it explicit in the notation. Note that we have an obvious forgetful functor $\mathsf{MSm}^{\ast }\rightarrow \mathsf{MSm}$ given by $((X,D),\unicode[STIX]{x1D706},f)\mapsto (X,D)$ .

We refer the reader to [Reference Binda and SaitoBS17] for the definition of Binda–Saito’s cycle complex with modulus $z^{i}(X|D,\bullet )$ .

Definition 1.5. Let $i\geqslant 0$ be a non-negative integer. For each $((X,D),\unicode[STIX]{x1D706},f)\in \mathsf{MSm}^{\ast }$ , denote by

$$\begin{eqnarray}z_{\mathsf{rel}}^{i}((X,D),\unicode[STIX]{x1D706},f;\bullet )\subset z^{i}(X|D,\bullet )\end{eqnarray}$$

the subcomplex of cycles $V\in z^{i}(X|D,n)$ such that for every morphism

$$\begin{eqnarray}(g,\unicode[STIX]{x1D711})\,:\,((X^{\prime },D^{\prime }),\unicode[STIX]{x1D706}^{\prime },f^{\prime })\rightarrow ((X,D),\unicode[STIX]{x1D706},f)\quad \text{ in }\mathsf{MSm}^{\ast },\end{eqnarray}$$

its pull-back $g^{\ast }V\in z^{i}(X^{\prime }|D^{\prime },n)$ is well defined. This defines a presheaf $z_{\mathsf{rel}}^{i}$ of complexes on $\mathsf{MSm}^{\ast }$ , which we call the codimension $i$ cycle complex presheaf on $\mathsf{MSm}^{\ast }$ . We remark that by a moving lemma with modulus (Theorem B.1), the inclusion $z_{\mathsf{rel}}^{i}((X,D),\unicode[STIX]{x1D706},f;\bullet )$ ${\hookrightarrow}z^{i}(X|D,\bullet )$ is a quasi-isomorphism locally in the Nisnevich topology on each $X$ .

Next, we want to define a presheaf of complexes $p_{\ast }z_{\mathsf{rel}}^{r}$ which serves as “the cycle complex of the projective bundle associated to the universal vector bundle on $B\text{GL}_{r}$ ”. Defining it requires some more notation which we now introduce.

For a non-negative integer $n\geqslant 0$ , set $[n]=\{0,\ldots ,n\}$ and endow it with the usual order. For a non-negative integer $r\geqslant 0$ , let us recall (or adopt the convention) that $B\text{GL}_{r}$ is the simplicial $k$ -scheme $B_{n}\text{GL}_{r}:=(\text{GL}_{r})^{n}$ with the structure morphism associated to each ordered map $\unicode[STIX]{x1D703}\,:\,[m]\rightarrow [n]$ :

$$\begin{eqnarray}(\text{GL}_{r})^{n}\rightarrow (\text{GL}_{r})^{m};\quad (\unicode[STIX]{x1D6FC}_{1},\ldots ,\unicode[STIX]{x1D6FC}_{n})\mapsto (\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D703}(j-1)+1}\cdots \unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D703}(j)})_{1\leqslant j\leqslant m}.\end{eqnarray}$$

The simplicial scheme $B\text{GL}_{r}$ defines a simplicial presheaf on $\mathsf{MSm}^{\ast }$ by $((X,D),\unicode[STIX]{x1D706},f)\mapsto B\text{GL}_{r}(X)$ .

Recall that the simplicial $k$ -scheme $\mathbb{P}(E\text{GL}_{r})$ has $\mathbb{P}(E_{n}\text{GL}_{r})=\mathbb{P}^{r-1}\times (\text{GL}_{r})^{n}$ as its $n$ th component, and the structure morphism corresponding to $\unicode[STIX]{x1D703}\,:\,[m]\rightarrow [n]$ is defined by

$$\begin{eqnarray}(z,\unicode[STIX]{x1D6FC}_{1},\ldots ,\unicode[STIX]{x1D6FC}_{n})\mapsto (z\unicode[STIX]{x1D6FC}_{1}\ldots \unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D703}(0)},\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D703}(0)+1}\ldots \unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D703}(1)},\ldots ,\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D703}(m-1)+1}\ldots \unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D703}(m)}),\end{eqnarray}$$

where the expression $z\unicode[STIX]{x1D6FC}$ for $z\in \mathbb{P}^{r-1}$ and $\unicode[STIX]{x1D6FC}\in \text{GL}_{r}$ denotes the right action of matrices on row vectors. Write also $[\unicode[STIX]{x1D6FC}]\,:\,\mathbb{P}^{r-1}\rightarrow \mathbb{P}^{r-1}$ for this action, so that $[\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}]=[\unicode[STIX]{x1D6FD}][\unicode[STIX]{x1D6FC}]$ holds, and $[\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}]^{\ast }=[\unicode[STIX]{x1D6FC}]^{\ast }[\unicode[STIX]{x1D6FD}]^{\ast }$ for pull-back operations.

Denote by $p\,:\,\mathbb{P}(E\text{GL}_{r})\rightarrow B\text{GL}_{r}$ the projection. It is a projective bundle with fiber $\mathbb{P}^{r-1}$ and defines a projective bundle (= a presheaf locally isomorphic to the constant presheaf $\mathbb{P}^{r-1}$ ) on the category $\mathsf{MSm}^{\ast }/B\text{GL}_{r}$ , which motivates the following definition.

Definition 1.7. For each $i\geqslant 0$ , we define the presheaf $p_{\ast }z_{\mathsf{rel}}^{i}$ of complexes on $\mathsf{MSm}^{\ast }/B\text{GL}_{r}$ as follows: for each object $((X,D),\unicode[STIX]{x1D706},f;n,\unicode[STIX]{x1D6FC})\in \mathsf{MSm}^{\ast }/B\text{GL}_{r}$ , denote by

$$\begin{eqnarray}p_{\ast }z_{\mathsf{rel}}^{i}((X,D),\unicode[STIX]{x1D706},f;n,\unicode[STIX]{x1D6FC};\bullet )\subset z^{i}(\mathbb{P}^{r-1}\times X|\mathbb{P}^{r-1}\times D,\bullet )\end{eqnarray}$$

the subcomplex of cycles $V\in z^{i}(\mathbb{P}^{r-1}\times X|\mathbb{P}^{r-1}\times D,m)$ such that for every morphism $(g,\unicode[STIX]{x1D711})\,:\,((X^{\prime },D^{\prime })\unicode[STIX]{x1D706}^{\prime },f^{\prime })\rightarrow ((X,D),\unicode[STIX]{x1D706},f)$ in $\mathsf{MSm}^{\ast }$ , its pull-back $(\text{id}_{\mathbb{P}^{r-1}}\times g)^{\ast }V\in z^{i}(\mathbb{P}^{r-1}\times X^{\prime }|\mathbb{P}^{r-1}\times D^{\prime },m)$ is well defined. (This does not depend on the data $(n,\unicode[STIX]{x1D6FC})$ .)

Given a morphism $(g,\unicode[STIX]{x1D711},\unicode[STIX]{x1D703})\,:\,((X^{\prime },D^{\prime }),\unicode[STIX]{x1D706}^{\prime },f^{\prime };n^{\prime },\unicode[STIX]{x1D6FC}^{\prime })\rightarrow ((X,D),\unicode[STIX]{x1D706},f;n,\unicode[STIX]{x1D6FC})$ in $\mathsf{MSm}^{\ast }/B\text{GL}_{r}$ , define the pull-back map

$$\begin{eqnarray}(g,\unicode[STIX]{x1D711},\unicode[STIX]{x1D703})^{\ast }\,:\,p_{\ast }z_{\mathsf{rel}}^{i}((X,D),\unicode[STIX]{x1D706},f;n,\unicode[STIX]{x1D6FC};\bullet )\rightarrow p_{\ast }z_{\mathsf{rel}}^{i}((X^{\prime },D^{\prime }),\unicode[STIX]{x1D706}^{\prime },f^{\prime };n^{\prime },\unicode[STIX]{x1D6FC}^{\prime };\bullet )\end{eqnarray}$$

to be the pull-back along the morphism (which depends on the data $(n,\unicode[STIX]{x1D6FC})$ ):

$$\begin{eqnarray}\begin{array}{@{}ccccc@{}}\mathbb{P}^{r-1}\times X^{\prime } & \xrightarrow[{}]{{\sim}} & \mathbb{P}^{r-1}\times X^{\prime } & \xrightarrow[{}]{\text{id}\times g} & \mathbb{P}^{r-1}\times X.\\ (z,x^{\prime }) & \mapsto & (z\unicode[STIX]{x1D6FC}_{1}\cdots \unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D703}(0)},x^{\prime }) & \end{array}\end{eqnarray}$$

Last, denote by $p^{\ast }\,:\,z_{\mathsf{rel}}^{i}\rightarrow p_{\ast }z_{\mathsf{rel}}^{i}$ the map of presheaves on $\mathsf{MSm}^{\ast }/B\text{GL}_{r}$ given by the pull-back along the projections $\mathbb{P}^{r-1}\times X\rightarrow X$ .

The presence of $p_{\ast }z_{\mathsf{rel}}^{i}$ is the main reason why we work in $\mathsf{MSm}^{\ast }/B\text{GL}_{r}$ rather than $\mathsf{MSm}^{\ast }$ . This presheaf has the information of $p_{\ast }z_{\mathbb{P}(E)|\mathbb{P}(E)\times _{X}D}^{i}$ for all objects $((X,D),\unicode[STIX]{x1D706},f)\in \mathsf{MSm}^{\ast }$ and vector bundles $E\rightarrow X$ in the following sense.

Let us suppose $\unicode[STIX]{x1D6EC}=\{\ast \}$ for simplicity, so that $\mathsf{MSm}^{\ast }\simeq (X,D)_{\text{Nis}}$ . Given a vector bundle $E\rightarrow X$ of rank $r\geqslant 0$ , let $p\,:\,\mathbb{P}(E)\rightarrow X$ be the associated projective bundle $\mathbb{P}(E):=\mathbf{Proj}(\operatorname{Sym}_{{\mathcal{O}}_{X}}E^{\vee })$ . We can consider the cycle complex $z_{\mathbb{P}(E)|\mathbb{P}(E)\times _{X}D}^{i}\,:\,(U\xrightarrow[{}]{\acute{\text{e}}\text{t}}\mathbb{P}(E))$ $\mapsto z^{i}(U|U\times _{X}D,\bullet )$ on $(\mathbb{P}(E)|\mathbb{P}(E)\times _{X}D)_{\text{Nis}}$ and its push-forward $p_{\ast }z_{\mathbb{P}(E)|\mathbb{P}(E)\times _{X}D}^{i}$ to $(X,D)_{\text{Nis}}$ on the one hand.

On the other hand, if we choose an open covering $X=\bigcup _{j}X_{j}$ and trivialization $\unicode[STIX]{x1D719}_{j}\,:\,E_{|X_{j}}\cong {\mathcal{O}}_{X_{j}}^{r}$ , we get a morphism of simplicial schemes from the Čech construction $\unicode[STIX]{x1D719}\,:\,{\check{C}}({\{X_{j}\}}_{j})\rightarrow B\text{GL}_{r}$ hence a simplicial object in $(X,D)_{\text{Nis}}/B\text{GL}_{r}$ (here, we give $X_{j}$ the divisor $D_{j}:=D\times _{X}X_{j}$ ). Therefore we can consider the restriction of $p_{\ast }z_{\mathsf{rel}}^{i}$ to ${\check{C}}(\{{X_{j}\}}_{j})_{\text{Nis}}$ .

One verifies that these two presheaves are canonically isomorphic on ${\check{C}}(\{{X_{j}\}}_{j})_{\text{Nis}}$ .

Remark 1.8. The following nonmodulus version

$$\begin{eqnarray}p_{\ast }z^{i}\,:\,((X,D),\unicode[STIX]{x1D706},f;n,\unicode[STIX]{x1D6FC})\mapsto z^{i}(\mathbb{P}^{r-1}\times X,\bullet ),\end{eqnarray}$$

with the same “presheaf” structure as $p_{\ast }z_{\mathsf{rel}}^{i}$ , plays a minor role later on. Note that since we are not assuming the smoothness of $X$ along $D$ for objects of $\mathsf{MSm}$ , it is difficult to make $p_{\ast }z^{i}$ functorial in a sensible way. However, this does not pose a real problem because $p_{\ast }z^{i}$ is only used as a conceptual aid in some constructions and can be avoided if one is willing to write down all the raw data instead.

1.2 Line bundles and codimension $1$ cycles

We know that codimension $1$ cycles are closely related to line bundles. The following version of this relationship is repeatedly used in this article. Below, we adopt the convention $\Box ^{n}:=(\mathbb{P}^{1}\setminus \{\infty \})^{n}=\text{Spec}(k[t_{1},\ldots ,t_{n}])$ rather than $(\mathbb{P}^{1}\setminus \{1\})^{n}$ used in [Reference Binda and SaitoBS17].

Let $X_{\bullet }$ be a semi-simplicial scheme with flat face maps (so that cycles and nonzero divisors can always be pulled back). Let $L$ be a line bundle on it, that is, the data of line bundles $L_{n}$ on $X_{n}$ for $n\geqslant 0$ and isomorphisms $d_{i}^{\ast }L_{n-1}\cong L_{n}$ for face maps $d_{i}\,:\,X_{n}\rightarrow X_{n-1}$ , compatible with each other in an obvious sense.

Suppose we are given a section $\unicode[STIX]{x1D70E}\in \unicode[STIX]{x1D6E4}(X_{0},L_{0})$ which is everywhere a nonzero divisor. We are going to define sections

$$\begin{eqnarray}F\text{}_{n}^{(\unicode[STIX]{x1D70E})}=F\text{}_{n}^{(\unicode[STIX]{x1D70E})}(t_{1},\ldots ,t_{n})\in \unicode[STIX]{x1D6E4}(X_{n},L_{n})\otimes _{k}k[t_{1},\ldots ,t_{n}]\end{eqnarray}$$

on $X_{n}\times \Box ^{n}$ which are nonzero divisors everywhere.

Let us denote by $v_{k}^{[n]}\,:\,[0]\rightarrow [n]$ the inclusion of the $k$ th vertex. Note that the groups $\unicode[STIX]{x1D6E4}(X_{m},L_{m})\,\otimes _{k}\,k[t_{1},\ldots ,t_{n}]$ are semi-cosimplicial in $m$ and cubical in $n$ . In particular, we have sections $(v_{k}^{[n]})_{\ast }\unicode[STIX]{x1D70E}\in \unicode[STIX]{x1D6E4}(X_{n},L_{n}).$ We define $F\text{}_{n}^{(\unicode[STIX]{x1D70E})}$ by the formula

$$\begin{eqnarray}F\text{}_{n}^{(\unicode[STIX]{x1D70E})}(t_{1},\ldots ,t_{n}):=\mathop{\sum }_{k=0}^{n}\biggl((v_{k}^{[n]})_{\ast }\unicode[STIX]{x1D70E}\otimes t_{k}\mathop{\prod }_{\ell =k+1}^{n}(1-t_{\ell })\biggr)\end{eqnarray}$$

where $t_{0}=1$ by convention. Of course, it is the map corresponding to the composite of a map $\displaystyle \Box ^{n}\rightarrow \unicode[STIX]{x1D6E5}^{n}$ from the $n$ -cube to the (algebraic) $n$ -simplex, followed by the affine map $\unicode[STIX]{x1D6E5}^{n}\rightarrow \unicode[STIX]{x1D6E4}(X_{n},L_{n})$ sending the $k$ th vertex to $(v_{k}^{[n]})_{\ast }\unicode[STIX]{x1D70E}$ . Footnote ¹

Recall that for $1\leqslant j\leqslant n$ and $\unicode[STIX]{x1D716}=0,1$ , the map $\unicode[STIX]{x2202}_{j,\unicode[STIX]{x1D716}}\,:\,\Box ^{n-1}\rightarrow \Box ^{n}$ is the embedding of the face $\{t_{j}=\unicode[STIX]{x1D716}\}$ : $(t_{1},\ldots ,t_{n-1})\mapsto (t_{1},\ldots ,t_{j-1},\unicode[STIX]{x1D716},t_{j},\ldots ,t_{n-1})$ . Degenerate elements of a cubical set mean elements obtained by pull-back along degeneracy maps $\Box ^{n}\rightarrow \Box ^{n-1}$ . Also for $0\leqslant i\leqslant n$ , denote by $d_{i}\,:\,[n-1]\rightarrow [n]$ the $i$ th face of $[n]$ . It is routine to check the following relations.

Lemma 1.9. We have equalities in $\unicode[STIX]{x1D6E4}(X_{n},L_{n})\,\otimes _{k}\,k[t_{1},\ldots ,t_{n-1}]$ :

$$\begin{eqnarray}d_{i}^{\ast }F\text{}_{n-1}^{(\unicode[STIX]{x1D70E})}=\left\{\begin{array}{@{}ll@{}}\unicode[STIX]{x2202}_{1,1}^{\ast }F\text{}_{n}^{(\unicode[STIX]{x1D70E})}\quad & \text{ if }i=0,\\ \unicode[STIX]{x2202}_{i,0}^{\ast }F\text{}_{n}^{(\unicode[STIX]{x1D70E})}\quad & \text{ if }1\leqslant i\leqslant n.\end{array}\right.\end{eqnarray}$$

Also, the functions $\unicode[STIX]{x2202}_{j,1}^{\ast }F\text{}_{n}^{(\unicode[STIX]{x1D70E})}$ are degenerate for $1<j\leqslant n$ .

Let $z^{1}(-,\bullet )$ be Bloch’s codimension $1$ cycle complex, which is a presheaf on the category of schemes and flat morphisms. For a scheme $X$ , let $\mathbb{Z}[X]$ be the additive presheaf generated by the presheaf of sets represented by $X$ . Lemma 1.9 implies that the set of cycles $\{\unicode[STIX]{x1D6E4}\text{}_{n}^{(\unicode[STIX]{x1D70E})}\}_{n}:=\{\text{div}(F\text{}_{n}^{(\unicode[STIX]{x1D70E})})\}_{n}$ determines a map of presheaves of complexes

$$\begin{eqnarray}\unicode[STIX]{x1D6E4}\text{}^{(\unicode[STIX]{x1D70E})}\,:\,\mathbb{Z}[X_{\bullet }]\rightarrow z^{1}(-,\bullet )\end{eqnarray}$$

on the small flat site over the semi-simplicial scheme $X_{\bullet }$ .

Let $\unicode[STIX]{x1D70E}^{\prime }\in \unicode[STIX]{x1D6E4}(X_{0},L_{0})$ be another nowhere zero divisor. We set

(1) $$\begin{eqnarray}\displaystyle F\text{}_{n}^{(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70E}^{\prime })} & := & \displaystyle t_{1}F\text{}_{n}^{(\unicode[STIX]{x1D70E}^{\prime })}(t_{2},\ldots ,t_{n+1})+(1-t_{1})F\text{}_{n}^{(\unicode[STIX]{x1D70E})}(t_{2},\ldots ,t_{n+1})\nonumber\\ \displaystyle & \in & \displaystyle \unicode[STIX]{x1D6E4}(X_{n},L_{n})\,\otimes _{k}\,k[t_{1},\ldots ,t_{n+1}].\end{eqnarray}$$

The cycles $\unicode[STIX]{x1D6E4}\text{}_{n}^{(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70E}^{\prime })}:=\text{div}(F\text{}_{n}^{(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70E}^{\prime })})$ will serve as a homotopy of $\unicode[STIX]{x1D6E4}\text{}^{(\unicode[STIX]{x1D70E})}$ and $\unicode[STIX]{x1D6E4}\text{}^{(\unicode[STIX]{x1D70E}^{\prime })}$ .

1.2.1 Variant

Occasionally our line bundle $L$ will be given as the difference $L=L^{+}\otimes (L^{-})^{\vee }$ of two line bundles, each equipped with a nowhere zero divisor $\unicode[STIX]{x1D70E}^{\pm }\in \unicode[STIX]{x1D6E4}(X_{0},L_{0}^{\pm })$ in degree $0$ . The construction of $\unicode[STIX]{x1D6E4}\text{}^{(\unicode[STIX]{x1D70E})}$ makes sense for the ratio $\unicode[STIX]{x1D70E}=\unicode[STIX]{x1D70E}^{+}/\unicode[STIX]{x1D70E}^{-}$ . In this case $F\text{}_{n}^{(\unicode[STIX]{x1D70E})}$ is the ratio of an element in $\unicode[STIX]{x1D6E4}(X_{n},L_{n}^{+}\otimes (L_{n}^{-})^{\otimes n})\,\otimes _{k}\,k[t_{1},\ldots ,t_{n}]$ and an element in $\unicode[STIX]{x1D6E4}(X_{n},(L_{n}^{-})^{\otimes n+1})$ (for example, $F_{0}^{(\unicode[STIX]{x1D70E})}=\unicode[STIX]{x1D70E}^{+}/\unicode[STIX]{x1D70E}^{-}$ is the ratio of an element in $\unicode[STIX]{x1D6E4}(X_{0},L_{0}^{+})$ and one in $\unicode[STIX]{x1D6E4}(X_{0},L_{0}^{-})$ ).

Also if a second presentation $L=L^{\prime +}\otimes (L^{\prime -})^{\vee }$ and nonzero divisors $\unicode[STIX]{x1D70E}^{\prime \pm }\in \unicode[STIX]{x1D6E4}(X_{0},L_{0}^{\prime \pm })$ are given, the construction of the homotopy $F\text{}^{(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70E}^{\prime })}$ makes sense to give $F\text{}_{n}^{(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70E}^{\prime })}$ as the ratio of an element in $\unicode[STIX]{x1D6E4}(X_{n},L\otimes (L^{-})^{\otimes n+1}\otimes (L^{\prime -})^{\otimes n+1})\otimes _{k}k[t_{1},\ldots ,t_{n+1}]$ and an element in $\unicode[STIX]{x1D6E4}(X_{n},(L^{-})^{\otimes n+1}\otimes (L^{\prime -})^{\otimes n+1})$ .

We will need the next complete intersection criterion. For the proof, the reader is referred to Lemma B.3.

Lemma 1.10. Let $L^{(1)\pm },\ldots ,L^{(i)\pm }$ be $2i$ line bundles on $X_{\bullet }$ equipped with sections $\unicode[STIX]{x1D70E}_{a}^{\pm }$ of $L_{0}^{(a)\pm }$ which are nonzero divisors ( $1\leqslant a\leqslant i$ ). Suppose the sequence of $i$ sections

$$\begin{eqnarray}(v_{k_{1}}^{[n]})_{\ast }\unicode[STIX]{x1D70E}_{1}^{\pm },\ldots ,(v_{k_{i}}^{[n]})_{\ast }\unicode[STIX]{x1D70E}_{i}^{\pm }\end{eqnarray}$$

is a regular sequence for every $n\geqslant 0$ and every choice of $0\leqslant k_{1}\leqslant \ldots \leqslant k_{i}\leqslant n$ and signs $\pm$ . Then the cup product

$$\begin{eqnarray}\unicode[STIX]{x1D6E4}\text{}^{(\unicode[STIX]{x1D70E}_{1})}\,\cdot \,\ldots \,\cdot \,\unicode[STIX]{x1D6E4}\text{}^{(\unicode[STIX]{x1D70E}_{i})}\,:\,\mathbb{Z}[X_{\bullet }]\rightarrow z^{i}(-,\bullet )\end{eqnarray}$$

is well defined.

Here, the cup product is defined by the concrete formulas in Appendix B.2. It is said to be well defined if all the intersection products appearing in the expression are well defined.

1.3 The projective bundle formula

Now we can state and prove the main result of this section:

Theorem 1.11. (Projective bundle formula)

For every $i\geqslant 0$ , we have a canonical isomorphism in the Nisnevich-local derived category $D(\mathsf{MSm}^{\ast }/B\text{GL}_{r})$ :

$$\begin{eqnarray}\bigoplus _{j=0}^{r-1}z_{\mathsf{rel}}^{i-j}\xrightarrow[{{\sim}}]{p^{\ast }(-)\,\cdot \,\unicode[STIX]{x1D709}^{j}}p_{\ast }z_{\mathsf{rel}}^{i}.\end{eqnarray}$$

First, we have to construct the maps. The following is a consequence of the Friedlander–Lawson moving lemma [Reference Friedlander and LawsonFL98, Theorem 3.1]. In the lemma and onwards, the superscript $(-)^{\circ }$ will be used to indicate that some moving procedure is involved.

Let $H\text{}^{(1)\circ }:=\text{div}(\unicode[STIX]{x1D70E}_{1})\subset \mathbb{P}^{r-1}$ be any hyperplane (i.e., $\unicode[STIX]{x1D70E}_{1}\in {\mathcal{O}}(1)$ ). Applying Lemma with $e=1$ gives a codimension $1$ cycle $H\text{}^{(2)\circ }$ on $\mathbb{P}^{r-1}$ which intersect every linear translate of $H\text{}^{(1)\circ }$ properly. $H\text{}^{(2)\circ }$ is written as the difference of the divisor of a section $\unicode[STIX]{x1D70E}_{2}^{+}\in {\mathcal{O}}(d+1)$ of some degree $d>0$ and that of $\unicode[STIX]{x1D70E}_{2}^{-}\in {\mathcal{O}}(d)$ . (Of course, any sections with $d\geqslant 2$ work for this first step without appealing to Lemma.) Next, apply Lemma 1.12 with $e=d+1$ and find sections $\unicode[STIX]{x1D70E}_{3}^{+}\in {\mathcal{O}}(d^{\prime }+1)$ , $\unicode[STIX]{x1D70E}_{3}^{-}\in {\mathcal{O}}(d^{\prime })$ of some large degrees so that the cycle $H\text{}^{(3)\circ }:=\text{div}(\unicode[STIX]{x1D70E}_{3}^{+}/\unicode[STIX]{x1D70E}_{3}^{-})$ satisfies the condition that for every $(\unicode[STIX]{x1D6FC}_{1},\unicode[STIX]{x1D6FC}_{2})\in \text{GL}_{r}(\bar{k})^{2}$ the following set has codimension $3$ in $\mathbb{P}_{\bar{k}}^{r-1}$ (the symbol $|-|$ denotes the support of a cycle):

$$\begin{eqnarray}[\unicode[STIX]{x1D6FC}_{1}]^{\ast }|H\text{}^{(1)\circ }|~\cap ~[\unicode[STIX]{x1D6FC}_{2}]^{\ast }|H\text{}^{(2)\circ }|\cap |H\text{}^{(3)\circ }|.\end{eqnarray}$$

This works because the intersection of the first two factors (with the reduced structure) has been assured to have codimension 2 and has degree ${\leqslant}1\cdot (d+1)$ . Next apply Lemma 1.12 with $e=(d+1)(d^{\prime }+1)$ to get $H\text{}^{(4)\circ }=\text{div}(\unicode[STIX]{x1D70E}_{4}^{+}/\unicode[STIX]{x1D70E}_{4}^{-})$ and so on.

In the end, we get codimension $1$ cycles $H\text{}^{(a)\circ }=\text{div}(\unicode[STIX]{x1D70E}_{a}^{+}/\unicode[STIX]{x1D70E}_{a}^{-})$ on $\mathbb{P}^{r-1}$ ( $1\leqslant a\leqslant r-1$ ) with the property that for every $(\unicode[STIX]{x1D6FC}_{1},\ldots ,\unicode[STIX]{x1D6FC}_{r-1})\in \text{GL}_{r}(\bar{k})^{r-1}$ the intersection

(2) $$\begin{eqnarray}[\unicode[STIX]{x1D6FC}_{1}]^{\ast }|H\text{}^{(1)\circ }|\cap \cdots \cap [\unicode[STIX]{x1D6FC}_{r-1}]^{\ast }|H\text{}^{(r-1)\circ }|\quad \text{in }\mathbb{P}_{\bar{k}}^{r-1}\end{eqnarray}$$

is zero-dimensional. In other words, the sequence $[\unicode[STIX]{x1D6FC}_{1}]^{\ast }\unicode[STIX]{x1D70E}_{1}^{\pm },\ldots ,[\unicode[STIX]{x1D6FC}_{r-1}]^{\ast }\unicode[STIX]{x1D70E}_{r-1}^{\pm }$ is a regular sequence for every possible choice of signs $\pm$ . Applying the procedure in Section 1.2 to these data, we get cycles which we denote by $\unicode[STIX]{x1D6E4}\text{}_{n}^{(a)\circ }$ ( $1\leqslant a\leqslant r-1$ ):

$$\begin{eqnarray}\unicode[STIX]{x1D6E4}\text{}_{n}^{(a)\circ }:=\unicode[STIX]{x1D6E4}_{n}^{(\unicode[STIX]{x1D70E}_{a})}\in z^{1}(\mathbb{P}(E_{n}\text{GL}_{r}),n).\end{eqnarray}$$

Now, let us note that giving a section $((X,D),\unicode[STIX]{x1D706},f)\xrightarrow[{}]{\unicode[STIX]{x1D6FC}}B_{n}\text{GL}_{r}$ of $B\text{GL}_{r}$ in degree $n$ is equivalent to giving a map $((X,D),\unicode[STIX]{x1D706},f)\times \unicode[STIX]{x1D6E5}^{n}\rightarrow B\text{GL}_{r}$ of simplicial presheaves on $\mathsf{MSm}^{\ast }$ (here, $\unicode[STIX]{x1D6E5}^{n}$ is a simplicial set, not a scheme). This motivates the following:

Definition 1.13. Let $\unicode[STIX]{x1D6E5}$ be the simplicial presheaf on $\mathsf{MSm}^{\ast }/B\text{GL}_{r}$ given by

$$\begin{eqnarray}((X,D),\unicode[STIX]{x1D706},f;n,\unicode[STIX]{x1D6FC})\mapsto \unicode[STIX]{x1D6E5}^{n}\end{eqnarray}$$

on objects and $(g,\unicode[STIX]{x1D711},\unicode[STIX]{x1D703})\mapsto \unicode[STIX]{x1D703}$ on morphisms.

The projection $\unicode[STIX]{x1D6E5}\rightarrow \ast$ to the singleton is a sectionwise weak equivalence of simplicial presheaves on $\mathsf{MSm}^{\ast }/B\text{GL}_{r}$ because $\unicode[STIX]{x1D6E5}^{n}$ is contractible.

Definition 1.14. For every object $\mathfrak{X}:=((X,D),\unicode[STIX]{x1D706},f;n,\unicode[STIX]{x1D6FC})\in \mathsf{MSm}^{\ast }/B\text{GL}_{r}$ , a simplex $\unicode[STIX]{x1D703}\in \unicode[STIX]{x1D6E5}_{m}^{n}$ and an index $a\in \{1,\ldots ,r-1\}$ , denote by

$$\begin{eqnarray}\unicode[STIX]{x1D6E4}\text{}_{m}^{(a)\circ }(\mathfrak{X},\unicode[STIX]{x1D703})\in (p_{\ast }z^{1})(\mathfrak{X},m)=z^{1}(\mathbb{P}^{r-1}\times X,m)\end{eqnarray}$$

the pull-back of $\unicode[STIX]{x1D6E4}\text{}_{m}^{(a)\circ }$ by the map $\{\mathbb{P}(E\text{GL}_{r})(\unicode[STIX]{x1D703})\times \text{id}_{\Box ^{m}}\}\circ \{\text{id}_{\mathbb{P}^{r-1}}\times \unicode[STIX]{x1D6FC}\times \text{id}_{\Box ^{m}}\}$ :

$$\begin{eqnarray}\begin{array}{@{}rll@{}}\mathbb{P}^{r-1}\times X\times \Box ^{m}\xrightarrow[{}]{\unicode[STIX]{x1D6FC}} & \mathbb{P}^{r-1}\times (\text{GL}_{r})^{n}\times \Box ^{m}\\ & =\mathbb{P}(E_{n}\text{GL}_{r})\times \Box ^{m}\xrightarrow[{\unicode[STIX]{x1D703}}]{} & \mathbb{P}(E_{m}\text{GL}_{r})\times \Box ^{m}.\end{array}\end{eqnarray}$$

Using them, we define a map of complexes

$$\begin{eqnarray}\unicode[STIX]{x1D6E4}\text{}^{(a)\circ }\,:\,\mathbb{Z}\otimes \unicode[STIX]{x1D6E5}\rightarrow p_{\ast }z^{1}\quad \text{ on }\mathsf{MSm}^{\ast }/B\text{GL}_{r}\end{eqnarray}$$

as follows: On an object $\mathfrak{X}$ as above and in degree $m$ , we must give a map of presheaves $\mathbb{Z}\otimes \unicode[STIX]{x1D6E5}_{m}^{n}\rightarrow p_{\ast }z^{1}(\mathfrak{X},m)=z^{1}(\mathbb{P}^{r-1}\times X,m)$ . We do this by mapping $\unicode[STIX]{x1D703}\in \unicode[STIX]{x1D6E5}_{m}^{n}$ to $\unicode[STIX]{x1D6E4}\text{}_{m}^{(a)\circ }(\mathfrak{X},\unicode[STIX]{x1D703})$ .

Of course, we could have pulled back the function $F\text{}_{m}^{(\unicode[STIX]{x1D70E}_{a}^{+}/\unicode[STIX]{x1D70E}_{a}^{-})}$ to define a function $F\text{}_{m}^{(a)\circ }(\mathfrak{X},\unicode[STIX]{x1D703})$ (a ratio of nowhere zero divisors by direct inspection) and set $\unicode[STIX]{x1D6E4}\text{}_{m}^{(a)\circ }(\mathfrak{X},\unicode[STIX]{x1D703})$ as its divisor. Both give the same cycle.

The cup product below is well defined for every $1\leqslant j\leqslant r-1$ :

$$\begin{eqnarray}C^{j\circ }:=\unicode[STIX]{x1D6E4}\text{}^{(1)\circ }\,\cdot \,\ldots \,\cdot \,\unicode[STIX]{x1D6E4}\text{}^{(j)\circ }\,:\,\mathbb{Z}\otimes \unicode[STIX]{x1D6E5}\rightarrow p_{\ast }z^{j}\end{eqnarray}$$

thanks to proper intersection (2) and Lemma 1.10 applied on each object of $\mathsf{MSm}^{\ast }/B\text{GL}_{r}$ . We tensor both sides with $z_{\mathsf{rel}}^{i-j}$ and apply the intersection product in the Nisnevich-local derived category (Appendix B.1):

$$\begin{eqnarray}z_{\mathsf{rel}}^{i-j}\otimes \unicode[STIX]{x1D6E5}\xrightarrow[{}]{\text{id}\otimes C^{j\circ }}z_{\mathsf{rel}}^{i-j}\otimes p_{\ast }z^{j}\xrightarrow[{}]{p^{\ast }(-)\,\cdot \,(-)}p_{\ast }z_{\mathsf{rel}}^{i}.\end{eqnarray}$$

Composed with the inverse of the quasi-isomorphism $z_{\mathsf{rel}}^{i-j}\otimes \unicode[STIX]{x1D6E5}\xrightarrow[{}]{{\sim}}z_{\mathsf{rel}}^{i-j}$ , it gives us the maps which we call $p^{\ast }(-)\,\cdot \,\unicode[STIX]{x1D709}^{j}$ :

$$\begin{eqnarray}p^{\ast }(-)\,\cdot \,\unicode[STIX]{x1D709}^{j}\,:\,\quad z_{\mathsf{rel}}^{i-j}\rightarrow p_{\ast }z_{\mathsf{rel}}^{i}\quad \text{ in }D(\mathsf{MSm}^{\ast }/B\text{GL}_{r}).\end{eqnarray}$$

2.1 Hyperplanes

We often consider data $(\mathfrak{X},\unicode[STIX]{x1D703})$ of an object $\mathfrak{X}=((X,D),\unicode[STIX]{x1D706},f;n,\unicode[STIX]{x1D6FC})$ $\in \mathsf{MSm}^{\ast }/B\text{GL}_{r}$ and an ordered map $\unicode[STIX]{x1D703}\,:\,[m]\rightarrow [n]$ .

2.1.1 The standard hyperplanes

Let $T_{1},\ldots ,T_{r}\in \unicode[STIX]{x1D6E4}(\mathbb{P}^{r-1},{\mathcal{O}}(1))$ be the homogeneous coordinates on $\mathbb{P}^{r-1}$ . We apply Section 1.2 to the line bundle ${\mathcal{O}}(1)$ on $\mathbb{P}(E\text{GL}_{r})$ and sections $T_{a}$ for $a\in \{1,\ldots ,r\}$ to get functions

$$\begin{eqnarray}F\text{}_{n}^{(a)\ast }:=F\text{}_{n}^{(T_{a})}\in \unicode[STIX]{x1D6E4}(\mathbb{P}(E_{n}\text{GL}_{r}),{\mathcal{O}}(1))\,\otimes _{k}\,k[t_{1},\ldots ,t_{n}]\end{eqnarray}$$

and their divisors $\unicode[STIX]{x1D6E4}\text{}_{n}^{(a)\ast }:=\text{div}(F\text{}_{n}^{(a)\ast })\in z^{1}(\mathbb{P}(E_{n}\text{GL}_{r}),n)$ . Here we use superscripts $(-)^{\ast }$ because they will be useful only over certain open subsets which will be indicated by the same superscripts.

Repeat the construction of Definition 1.14 on these data to define cycles $\unicode[STIX]{x1D6E4}\text{}_{m}^{(a)\ast }(\mathfrak{X},\unicode[STIX]{x1D703})\in z^{1}(\mathbb{P}^{r-1}\times X,m)$ . They determine a map of complexes

(3) $$\begin{eqnarray}\unicode[STIX]{x1D6E4}\text{}_{\mathsf{MSm}}^{(a)\ast }\,:\,\mathbb{Z}\otimes \unicode[STIX]{x1D6E5}\rightarrow p_{\ast }z^{1}\quad \text{ on }\mathsf{MSm}^{\ast }/B\text{GL}_{r}.\end{eqnarray}$$

Remark 2.1. One may want to construct $\unicode[STIX]{x1D709}_{\mathsf{rel}}^{r}$ as the cup product $\unicode[STIX]{x1D6E4}\text{}_{\mathsf{MSm}}^{(1)\ast }\,\cdot \,\ldots \,\cdot \,\unicode[STIX]{x1D6E4}\text{}_{\mathsf{MSm}}^{(r)\ast }$ : $\mathbb{Z}\otimes \unicode[STIX]{x1D6E5}\rightarrow p_{\ast }z^{r}$ . But the cup product is not well defined due to the failure of proper intersection. The easiest such example would be, taking $\unicode[STIX]{x1D6EC}=\{\ast \}$ : $r=2$ , $\mathfrak{X}=((X,\emptyset );n=1,\unicode[STIX]{x1D6FC}=(\!\begin{smallmatrix}0 & 1\\ 1 & 0\end{smallmatrix}\!)\in (\boldsymbol{X}_{2}^{\mathsf{rel}})_{1})$ , $\unicode[STIX]{x1D703}=\text{id}_{[1]}$ . In this case the cycle that is supposed to represent the $(\mathfrak{X},\unicode[STIX]{x1D703})$ -component of $\unicode[STIX]{x1D6E4}\text{}_{\mathsf{MSm}}^{(1)\ast }\,\cdot \,\unicode[STIX]{x1D6E4}\text{}_{\mathsf{MSm}}^{(2)\ast }$ always does not satisfy the face condition. This is why we need the following constructions.

2.1.2 The generic hyperplanes

Let $\{{x_{ab}\}}_{1\leqslant a,b\leqslant r}$ be the coordinates of $\text{GL}_{r}$ , so that its function field is $k(\text{GL}_{r})=k({\{x_{ab}\}}_{a,b})$ . For each $a\in \{1,\ldots ,r\}$ , let us consider the “generic translation” of the coordinates: $T_{a}^{\circ }:=\sum _{b=1}^{r}T_{b}x_{ba}\in \unicode[STIX]{x1D6E4}(\mathbb{P}_{k(\text{GL}_{r})}^{r-1},{\mathcal{O}}(1))$ . As in Section 1.3, we are using the superscript $(-)^{\circ }$ to indicate the involvement of moving procedure. Applying the construction in Section 1.2 and Definition 1.14 on $T_{a}^{\circ }$ , we define cycles:

$$\begin{eqnarray}\unicode[STIX]{x1D6E4}\text{}_{m,k(\text{GL}_{r})}^{(a)\circ }(\mathfrak{X},\unicode[STIX]{x1D703})\in z^{1}((\mathbb{P}^{r-1}\times X)_{k(\text{GL}_{r})},m).\end{eqnarray}$$

These objects are different from those denoted by similar symbols in Definition 1.14, but since the older ones are not going to be used again in this section, there is no risk of confusion.

2.1.3 Homotopy of the hyperplanes

The homotopy in equation (1) gives us cycles

$$\begin{eqnarray}\unicode[STIX]{x1D6E4}\text{}_{n,k(\text{GL}_{r})}^{(a)\circ \ast }:=\unicode[STIX]{x1D6E4}\text{}_{n}^{(T_{a}^{\circ },T_{a})}\in z^{1}(\mathbb{P}(E_{n}\text{GL}_{r})_{k(\text{GL}_{r})},n+1)\end{eqnarray}$$

and by pull-back as in Definition 1.14, we define

$$\begin{eqnarray}\unicode[STIX]{x1D6E4}\text{}_{m,k(\text{GL}_{r})}^{(a)\circ \ast }(\mathfrak{X},\unicode[STIX]{x1D703})\in z^{1}(\mathbb{P}^{r-1}\times X_{k(\text{GL}_{r})},m+1).\end{eqnarray}$$

Definition 2.2. For a field extension $L/k$ , denote by $p_{\ast }z_{L}^{i}$ (the nonmodulus cycle complex with scalar extension) the association of complexes $\mathfrak{X}\mapsto p_{\ast }z(\mathfrak{X}\otimes _{k}L)$ for objects of $\mathsf{MSm}^{\ast }/B\text{GL}_{r}$ . There is an obvious scalar extension map $p_{\ast }z^{i}\rightarrow p_{\ast }z_{L}^{i}$ .

As in Remark 1.8, this is not a presheaf on $\mathsf{MSm}^{\ast }/B\text{GL}_{r}$ unless one restricts to some nice subcategory. The set of cycles $\{\unicode[STIX]{x1D6E4}\text{}_{m,k(\text{GL}_{r})}^{(a)\circ }(\mathfrak{X},\unicode[STIX]{x1D703})\}_{m,\mathfrak{X},\unicode[STIX]{x1D703}}$ determines a map of complexes

$$\begin{eqnarray}\unicode[STIX]{x1D6E4}\text{}_{\mathsf{MSm},k(\text{GL}_{r})}^{(a)\circ }\,:\,\mathbb{Z}\otimes \unicode[STIX]{x1D6E5}\rightarrow p_{\ast }z_{k(\text{GL}_{r})}^{1}.\end{eqnarray}$$

2.2 The open covering and the complex ${\mathcal{Z}}$

Definition 2.3. Let $S\subset [n]$ be a subset consisting of $m+1$ elements. We denote the unique injection $[m]{\hookrightarrow}[n]$ into $S$ also by the same letter. Let us denote by $\unicode[STIX]{x1D6E4}\text{}^{(a)\ast }(S)$ the pull-back of $\unicode[STIX]{x1D6E4}\text{}_{m}^{(a)\ast }$ by the map

$$\begin{eqnarray}\mathbb{P}(E\text{GL}_{r})(S)\times \text{id}_{\Box ^{m}}:\quad \mathbb{P}(E_{n}\text{GL}_{r})\times \Box ^{m}\rightarrow \mathbb{P}(E_{m}\text{GL}_{r})\times \Box ^{m}.\end{eqnarray}$$

Of course, it is the divisor of the function $F\text{}^{(a)\ast }(S)$ similarly defined.

Definition 2.4. Let $B_{n}\text{GL}_{r}^{\ast }$ be the following open subset of $B_{n}\text{GL}_{r}$ :

$$\begin{eqnarray}B_{n}\text{GL}_{r}^{\ast }:=B_{n}\text{GL}_{r}\setminus p\biggl(\mathop{\bigcup }_{0\leqslant k_{1}\leqslant \ldots \leqslant k_{r}\leqslant n}\unicode[STIX]{x1D6E4}\text{}^{(1)\ast }(v_{k_{1}}^{[n]})\cap \cdots \cap \unicode[STIX]{x1D6E4}\text{}^{(r)\ast }(v_{k_{r}}^{[n]})\biggr)\end{eqnarray}$$

where $p\,:\,\mathbb{P}(E_{n}\text{GL}_{r})=\mathbb{P}^{r-1}\times B_{n}\text{GL}_{r}\rightarrow B_{n}\text{GL}_{r}$ is the second projection.

It follows that a point $\unicode[STIX]{x1D6FC}=(\unicode[STIX]{x1D6FC}_{1},\ldots ,\unicode[STIX]{x1D6FC}_{n})\in B_{n}\text{GL}_{r}(k(\unicode[STIX]{x1D6FC}))$ is in $B_{n}\text{GL}_{r}^{\ast }$ if and only if for every choice of $0\leqslant k_{1}\leqslant \ldots \leqslant k_{r}\leqslant n$ , the intersection in $\mathbb{P}_{k(\unicode[STIX]{x1D6FC})}^{r-1}$

$$\begin{eqnarray}[\unicode[STIX]{x1D6FC}_{1}\unicode[STIX]{x1D6FC}_{2}\cdots \unicode[STIX]{x1D6FC}_{k_{1}}]^{\ast }\{T_{1}=0\}\cap \cdots \cap [\unicode[STIX]{x1D6FC}_{1}\unicode[STIX]{x1D6FC}_{2}\cdots \unicode[STIX]{x1D6FC}_{k_{r}}]^{\ast }\{T_{r}=0\}\end{eqnarray}$$

is empty. Note that whenever $\unicode[STIX]{x1D6FC}_{1},\ldots ,\unicode[STIX]{x1D6FC}_{n}$ are all upper triangular, the sequence $(\unicode[STIX]{x1D6FC}_{1},\ldots ,\unicode[STIX]{x1D6FC}_{n})$ belongs to $B_{n}\text{GL}_{r}^{\ast }$ . The simplicial structure of $B\text{GL}_{r}$ restricts to the schemes ${\{B_{n}\text{GL}_{r}^{\ast }\}}_{n}$ .

Definition 2.6. Define the simplicial subpresheaves $\unicode[STIX]{x1D6E5}^{\ast }$ and $\unicode[STIX]{x1D6E5}^{\circ }$ of $\unicode[STIX]{x1D6E5}$ on $\mathsf{MSm}^{\ast }/B\text{GL}_{r}$ by (where $\mathfrak{X}=((X,D),\unicode[STIX]{x1D706},f;n,\unicode[STIX]{x1D6FC})$ ):

$$\begin{eqnarray}\unicode[STIX]{x1D6E5}^{\ast }(\mathfrak{X})_{m}=\{\unicode[STIX]{x1D703}\in \unicode[STIX]{x1D6E5}_{m}^{n}\mid X_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D703}}^{\ast }=X\},\qquad \unicode[STIX]{x1D6E5}^{\circ }(\mathfrak{X})=\left\{\begin{array}{@{}ll@{}}\unicode[STIX]{x1D6E5}^{n}\quad & \text{ if }D=\emptyset ,\\ \emptyset \quad & \text{ if }D\neq \emptyset .\end{array}\right.\end{eqnarray}$$

Also, set $\unicode[STIX]{x1D6E5}\text{}^{\circ \ast }:=\unicode[STIX]{x1D6E5}^{\circ }\cap \unicode[STIX]{x1D6E5}^{\ast }$ .

Let us say that a morphism of the form $(f,\text{id},\text{id})\,:\,\mathfrak{X}^{\prime }\rightarrow \mathfrak{X}$ in $\mathsf{MSm}^{\ast }/B\text{GL}_{r}$ is an open immersion if the underlying morphism $f\,:\,X^{\prime }\rightarrow X$ is an open immersion and if $D^{\prime }=X^{\prime }\times _{X}D$ . Then $\unicode[STIX]{x1D6E5}^{\circ }$ and $\unicode[STIX]{x1D6E5}^{\ast }$ are open subpresheaves of $\unicode[STIX]{x1D6E5}$ . The map $\unicode[STIX]{x1D6E5}^{\circ }\sqcup \unicode[STIX]{x1D6E5}^{\ast }\rightarrow \unicode[STIX]{x1D6E5}$ is not a surjection of Zariski sheaves, but becomes so on the smaller category $\mathsf{MSm}^{\ast }/\boldsymbol{X}_{r}^{\mathsf{rel}}$ introduced in Section 2.3. Consequently, the complex ${\mathcal{Z}}$ below becomes quasi-isomorphic to $\mathbb{Z}$ on that category.

Definition 2.7. Define the complex ${\mathcal{Z}}$ on $\mathsf{MSm}^{\ast }/B\text{GL}_{r}$ by:

The maps $\unicode[STIX]{x1D6E4}\text{}_{\mathsf{MSm},k(\text{GL}_{r})}^{(a)\circ }$ on $\mathbb{Z}\otimes \unicode[STIX]{x1D6E5}^{\circ }$ , $\unicode[STIX]{x1D6E4}\text{}_{\mathsf{MSm}}^{(a)\ast }$ on $\mathbb{Z}\otimes \unicode[STIX]{x1D6E5}^{\ast }$ and the homotopy $\{\unicode[STIX]{x1D6E4}\text{}_{m,k(\text{GL}_{r})}^{(a)\circ \ast }\}_{m}$ on $\mathbb{Z}\otimes \unicode[STIX]{x1D6E5}\text{}^{\circ \ast }$ determine a map of complexes

(4) $$\begin{eqnarray}\unicode[STIX]{x1D6E4}\text{}_{\mathsf{MSm}}^{(a)}\,:\,{\mathcal{Z}}\rightarrow p_{\ast }z_{k(\text{GL}_{r})}^{1}.\end{eqnarray}$$

The next lemma follows from the definition of $B\text{GL}_{r}^{\ast }$ and the algebraic independence of ${\{x_{ab}\}}_{a,b}$ .

By Lemma 2.8, we can apply Lemmas 1.10 and B.4 to conclude that the cup product

(5) $$\begin{eqnarray}\unicode[STIX]{x1D6E4}\text{}_{\mathsf{MSm}}^{(1)}\,\cdot \,\ldots \,\cdot \,\unicode[STIX]{x1D6E4}\text{}_{\mathsf{MSm}}^{(r)}\,:\,{\mathcal{Z}}\rightarrow p_{\ast }z_{k(\text{GL}_{r})}^{r}\end{eqnarray}$$

is well defined. We now introduce a modulus version $p_{\ast }z_{\mathsf{rel},k(\text{GL}_{r})}^{i}\subset p_{\ast }z_{k(\text{GL}_{r})}^{i}$ of the target. We shall show that the map (5) factors through it when restricted to the category $\mathsf{MSm}^{\ast }/\boldsymbol{X}_{r}^{\mathsf{rel}}$ introduced in Section 2.3.

Definition 2.9. For a field extension $L/k$ , define the presheaf of complexes $p_{\ast }z_{\mathsf{rel},L}^{i}$ (cycle complex with modulus with scalar extension) on $\mathsf{MSm}^{\ast }/B\text{GL}_{r}$ by the rule

$$\begin{eqnarray}\mathfrak{X}=((X,D),\unicode[STIX]{x1D706},f)\mapsto \left\{\begin{array}{@{}ll@{}}p_{\ast }z_{\mathsf{rel}}^{i}(\mathfrak{X}_{L})\quad & \text{ if }D=\emptyset ,\\ p_{\ast }z_{\mathsf{rel}}^{i}(\mathfrak{X})\quad & \text{ if }D\neq \emptyset ,\end{array}\right.\end{eqnarray}$$

with the same presheaf structure as $p_{\ast }z_{\mathsf{rel}}^{i}$ . There is a scalar extension map $p_{\ast }z_{\mathsf{rel}}^{i}\rightarrow p_{\ast }z_{\mathsf{rel},L}^{i}$ .

This definition may appear strange at first glance. It is motivated by the fact that Bloch’s specialization map (Section 2.4) is available (and necessary) only when $D=\emptyset$ .

2.3 The relative Volodin space

Now we introduce the relative Volodin space presheaf. Its significance lies in its relation to the relative $K$ -theory; see Theorem 2.24.

Definition 2.10. Let $(X,D)\in \mathsf{MSm}$ . Denote by $I=I_{D}\subset {\mathcal{O}}_{X}$ the ideal sheaf defining $D$ . Let $r\geqslant 0$ be a non-negative integer and $\unicode[STIX]{x1D70E}$ a partial order on the set $\{1,\ldots ,r\}$ . Then the subgroup $T^{\unicode[STIX]{x1D70E}}(X,D)\subset \text{GL}_{r}(X)$ is defined to be the set of matrices $(x_{ab})_{1\leqslant a,b\leqslant r}$ such that $x_{ab}\equiv \unicode[STIX]{x1D6FF}_{ab}\hspace{0.6em}{\rm mod}\hspace{0.2em}I_{D}$ (Kronecker’s  $\unicode[STIX]{x1D6FF}$ ) unless $a\overset{\unicode[STIX]{x1D70E}}{{<}}b$ . For example, if $\unicode[STIX]{x1D70E}$ is the usual total order on $\{1,2,3\}$ , then elements of $T^{\unicode[STIX]{x1D70E}}(X,D)$ look like:

$$\begin{eqnarray}\left(\begin{array}{@{}ccc@{}}1+I & {\mathcal{O}}_{X} & {\mathcal{O}}_{X}\\ I & 1+I & {\mathcal{O}}_{X}\\ I & I & 1+I\end{array}\right)\!.\end{eqnarray}$$

If another order $\unicode[STIX]{x1D70E}^{\prime }$ extends $\unicode[STIX]{x1D70E}$ (i.e., if $a\overset{\unicode[STIX]{x1D70E}}{{<}}b$ implies $a\overset{\unicode[STIX]{x1D70E}^{\prime }}{{<}}b$ ), we have $T^{\unicode[STIX]{x1D70E}}(X,D)\subset T^{\unicode[STIX]{x1D70E}^{\prime }}(X,D)$ . The Volodin space $\boldsymbol{X}_{r}(X,D)$ is the simplicial subset of $B\text{GL}_{r}(X)$ defined by

$$\begin{eqnarray}\boldsymbol{X}_{r}(X,D)=\mathop{\bigcup }_{\unicode[STIX]{x1D70E}}BT^{\unicode[STIX]{x1D70E}}(X,D)\subset B\text{GL}_{r}(X).\end{eqnarray}$$

We set $\boldsymbol{X}(X,D)=\mathop{\varinjlim }\nolimits_{r}\boldsymbol{X}_{r}(X,D)\subset B\text{GL}(X)$ . Define a Nisnevich sheaf $\boldsymbol{X}_{r}^{\mathsf{rel}}$ on $\mathsf{MSm}$ by $(X,D)\mapsto \boldsymbol{X}_{r}(X,D)$ .

We also denote by $\boldsymbol{X}_{r}^{\mathsf{rel}}$ the presheaf induced by the forgetful functor $\mathsf{MSm}^{\ast }\rightarrow \mathsf{MSm}$ . In particular, we have the site $\mathsf{MSm}^{\ast }/\boldsymbol{X}_{r}^{\mathsf{rel}}$ fibered over it. The inclusion $\boldsymbol{X}_{r}^{\mathsf{rel}}{\hookrightarrow}B\text{GL}_{r}$ induces a functor $\mathsf{MSm}^{\ast }/\boldsymbol{X}_{r}^{\mathsf{rel}}\rightarrow \mathsf{MSm}^{\ast }/B\text{GL}_{r}$ . Every presheaf and map of presheaves on the latter restrict to the former. It follows for example that the projective bundle formula in Section 1 holds on $\mathsf{MSm}^{\ast }/\boldsymbol{X}_{r}^{\mathsf{rel}}$ in the same form.

By Lemma 2.11, the complex ${\mathcal{Z}}$ is Zariski locally quasi-isomorphic to $\mathbb{Z}\otimes \unicode[STIX]{x1D6E5}\simeq \mathbb{Z}$ by the map $(\mathbb{Z}\otimes \unicode[STIX]{x1D6E5}^{\circ })\oplus (\mathbb{Z}\otimes \unicode[STIX]{x1D6E5}^{\ast })$ $\xrightarrow[{\text{incl.}\sqcup (-\text{incl.})}]{}$ $\mathbb{Z}\otimes \unicode[STIX]{x1D6E5}$ when restricted to $\mathsf{MSm}^{\ast }/\boldsymbol{X}_{r}^{\mathsf{rel}}$ .

The rest of this subsection is devoted to the proof of the following:

Note that the assertion only concerns the part $\unicode[STIX]{x1D6E4}\text{}_{\mathsf{MSm}}^{(1)\ast }\,\cdot \,\ldots \,\cdot \,\unicode[STIX]{x1D6E4}\text{}_{\mathsf{MSm}}^{(r)\ast }\,:\,\mathbb{Z}\otimes \unicode[STIX]{x1D6E5}^{\ast }\rightarrow p_{\ast }z^{r}$ . The following criterion for the modulus condition will be useful.

Definition 2.13. [Reference Binda and SaitoBS17, Section 4]

Let $A$ be a commutative ring with unit and $I$ be an ideal. A polynomial

$$\begin{eqnarray}f=\mathop{\sum }_{\unicode[STIX]{x1D706}_{1},\ldots ,\unicode[STIX]{x1D706}_{n}}a_{\unicode[STIX]{x1D706}_{1},\ldots ,\unicode[STIX]{x1D706}_{n}}t^{\unicode[STIX]{x1D706}_{1}}\cdots t^{\unicode[STIX]{x1D706}_{n}}\in A[t_{1},\ldots ,t_{n}]\end{eqnarray}$$

is said to be admissible if $a_{\unicode[STIX]{x1D706}_{1},\ldots ,\unicode[STIX]{x1D706}_{n}}\in I^{\max _{i}\{\unicode[STIX]{x1D706}_{i}\}}$ and if $a_{0,\ldots ,0}$ maps into $(A/I)^{\ast }$ .

When $X$ is a $k$ -scheme of finite type equipped with an ideal sheaf $I$ , let us say that an ideal sheaf $J$ on $X\times \Box ^{n}$ is admissible if there exists an affine open covering ${\{U_{\unicode[STIX]{x1D6FC}}\}}_{\unicode[STIX]{x1D6FC}}$ of $X$ such that $J$ restricted to each $U_{\unicode[STIX]{x1D6FC}}\times \Box ^{n}$ contains an admissible polynomial with respect to $I(U_{\unicode[STIX]{x1D6FC}})$ . Note that if $J$ is admissible and $f\,:\,X^{\prime }\rightarrow X$ is a morphism from another scheme, the ideal sheaf $(f\times \text{id}_{\Box ^{n}})^{\ast }J$ on $X^{\prime }\times \Box ^{n}$ is admissible with respect to $f^{\ast }I$ , because elements in a power $I^{\unicode[STIX]{x1D706}}$ pull back into the power $(f^{\ast }I)^{\unicode[STIX]{x1D706}}$ .

Under this notation, a section $\unicode[STIX]{x1D6FC}\in (\boldsymbol{X}_{r}^{\mathsf{rel}})_{n}(X,D)$ is the same as a morphism of schemes $X\rightarrow B_{n}\text{GL}_{r}$ which maps the subscheme $D$ into the closed subscheme $V(I^{\unicode[STIX]{x1D70E}})$ for some $\unicode[STIX]{x1D70E}$ . For subsets $S,T\subset [n]$ , let us write $S\leqslant T$ to mean $s\leqslant t$ for all $s\in S$ and $t\in T$ . Also, recall the symbol $F\text{}^{(a)\ast }(S)$ from Definition 2.3.

Lemma 2.16. Let $n\geqslant 0$ and $r\geqslant 1$ be integers and $\unicode[STIX]{x1D70E}$ an order on $\{1,\ldots ,r\}$ . Let $S_{1}\leqslant \ldots \leqslant S_{r}$ be nonempty subsets of $[n]$ . Then the ideal sheaf on $\mathbb{P}^{r-1}\times B_{n}\text{GL}_{r}\times \Box ^{n}$ associated to the homogeneous ideal generated by:

$$\begin{eqnarray}F\text{}^{(a)\ast }(S_{a})\quad 1\leqslant a\leqslant r\end{eqnarray}$$

is admissible with respect to the ideal sheaf ${\mathcal{O}}_{\mathbb{P}^{r-1}}\otimes _{k}I^{\unicode[STIX]{x1D70E}}$ on $\mathbb{P}^{r-1}\times B_{n}\text{GL}_{r}$ .

Lemma 2.16 follows from a more precise claim below. Note that we may obviously assume that $\unicode[STIX]{x1D70E}$ is a total order and, by symmetry, that $\unicode[STIX]{x1D70E}$ is the usual order $\unicode[STIX]{x1D70E}=\{1<\cdots <r\}$ . Let us write $I:=I^{\unicode[STIX]{x1D70E}}$ for this $\unicode[STIX]{x1D70E}$ .

For $S\subset \{1,\ldots ,n\}$ , let us denote by $[t_{i}\mid i\in S]\subset k[t_{1},\ldots ,t_{n}]$ the $2^{|S|}$ -dimensional $k$ -vector space spanned by monomials $\prod _{i\in S}t_{i}^{\unicode[STIX]{x1D716}_{i}}$ where $\unicode[STIX]{x1D716}_{i}\in \{0,1\}$ . To ease the notation, we shall use the phrase:

to mean a sum of polynomials of the form $T_{c}\cdot x\cdot f$ with $x\in I\subset k[x_{ab}^{i}]$ and $f\in [t_{i}\mid i\in S]$ . For a nonempty subset $S$ of $[n]$ , we write $S^{\prime }$ for the set $S\setminus \{$ the minimum element of $S\}$ .

Claim 2.17. For any $a\in \{1,\ldots ,r\}$ , the ideal of the polynomial ring

$$\begin{eqnarray}k[T_{1},\ldots ,T_{r}][x_{bc}^{i}\mid _{b,c\in \{1,\ldots ,r\}}^{1\leqslant i\leqslant n}][t_{1},\ldots ,t_{n}]\end{eqnarray}$$

generated by $\{F\text{}^{(b)\ast }(S_{b})\}_{a\leqslant b\leqslant r}$ contains a polynomial of the form

(6a ) $$\begin{eqnarray}T_{a}+\mathop{\sum }_{c=1}^{r}\big(T_{c}\cdot I\cdot \big[t_{i}\mid i\in S_{a}^{\prime }\cup S_{a+1}^{\prime }\cup \cdots \cup S_{r}^{\prime }\big]\big).\end{eqnarray}$$

Claim 2.17 implies Lemma 2.16 because formula (6a ) divided by $T_{a}$ gives an admissible polynomial over the affine open set $\{T_{a}\neq 0\}$ .

Proof of Claim. We proceed by descending induction on the index $a$ starting with $a=r$ . Let us write down the definition of $F\text{}^{(a)\ast }(S)$ , where $a\in \{1,\ldots ,r\}$ and $S\subset [n]$ is a nonempty subset with $s+1$ elements:

$$\begin{eqnarray}\displaystyle F\text{}^{(a)\ast }(S) & = & \displaystyle \mathop{\sum }_{i=1}^{s}\biggl((S_{\ast }(v_{i}^{[s]})_{\ast }T_{a})\cdot t_{S(i)}\mathop{\prod }_{j=i+1}^{s}(1-t_{S(j)})\biggr)\nonumber\\ \displaystyle & & \displaystyle +\,(S_{\ast }(v_{0}^{[s]})_{\ast }T_{a})\cdot \mathop{\prod }_{j=1}^{s}(1-t_{S(j)}).\nonumber\end{eqnarray}$$

Since we are in the group $T(X,D)$ of upper triangular matrices modulo $I$ , for any map $v\,:\,[0]\rightarrow [n]$ (such as $S\circ v_{i}^{[s]}$ ) the function $v_{\ast }T_{a}$ has the form

$$\begin{eqnarray}\biggl(\mathop{\sum }_{b=1}^{a-1}T_{b}\cdot I\biggr)+T_{a}\cdot (1+I)+\biggl(\mathop{\sum }_{b=a+1}^{r}T_{b}\cdot {\mathcal{O}}\biggr),\end{eqnarray}$$

where ${\mathcal{O}}:={\mathcal{O}}_{B_{n}\text{GL}_{r}}$ . By these two formulas and the identity $t_{S(s)}+t_{S(s-1)}(1-t_{S(s)})+\cdots +(1-t_{S(1)})\cdots (1-t_{S(s)})=1$ , we get:

$$\begin{eqnarray}F\text{}^{(a)\ast }(S)=T_{a}+\mathop{\sum }_{b=1}^{a}(T_{b}\cdot I\cdot [t_{i}\mid i\in S_{a}^{\prime }])+\mathop{\sum }_{b=a+1}^{r}(T_{b}\cdot {\mathcal{O}}\cdot [t_{i}\mid i\in S_{a}^{\prime }]).\end{eqnarray}$$

This already proves the assertion for $a=r$ .

Now suppose $a<r$ . By descending induction, we know that the ideal in question contains polynomials of the form ( $6b$ ) for $b=a+1,\ldots ,r$ . In particular, we get:

$$\begin{eqnarray}\mathop{\sum }_{b=a+1}^{r}(T_{b}\cdot {\mathcal{O}}\cdot [t_{i}\mid i\in S_{a}^{\prime }])\equiv \mathop{\sum }_{c=1}^{r}(T_{c}\cdot I\cdot [t_{i}\mid i\in S_{a}^{\prime }\cup S_{a+1}^{\prime }\cup \cdots \cup S_{r}^{\prime }])\end{eqnarray}$$

modulo the ideal in question. Here we used the fact that the product of an element in $[t_{i}\mid i\in S]$ and one in $[t_{i}\mid i\in T]$ with $S\cap T=\emptyset$ belongs to $[t_{i}\mid i\in S\cup T]$ . The last two formulas give a formula of the form (6a ). This completes the proof of Claim 2.17, hence also of Theorem 2.12.

Hence we have obtained a map

(7) $$\begin{eqnarray}\unicode[STIX]{x1D6E4}\text{}_{\mathsf{MSm}}^{(1)}\cdot \ldots \cdot \unicode[STIX]{x1D6E4}\text{}_{\mathsf{MSm}}^{(r)}\,:\,{\mathcal{Z}}\rightarrow p_{\ast }z_{\mathsf{rel},k(\text{GL}_{r})}^{r}\end{eqnarray}$$

in $D(\mathsf{MSm}^{\ast }/\boldsymbol{X}_{r}^{\mathsf{rel}})$ .

2.4 Specialization map, and end of construction of $\unicode[STIX]{x1D709}_{\mathsf{rel}}^{r}$

Bloch defined a specialization map $z^{i}(X_{L},\bullet )\rightarrow z^{i}(X,\bullet )$ in the derived category when $L/k$ is a purely transcendental extension of finite degree equipped with a transcendence basis and $X$ is an equidimensional $k$ -scheme [Reference BlochBl86, pages 291, 292]. Likewise, we can define a specialization map

$$\begin{eqnarray}\text{sp}_{L/k}\,:\,p_{\ast }z_{\mathsf{rel},L}^{i}\rightarrow p_{\ast }z_{\mathsf{rel}}^{i}\end{eqnarray}$$

in $D(\mathsf{MSm}^{\ast }/\boldsymbol{X}_{r}^{\mathsf{rel}})$ by using his map when $D=\emptyset$ and setting it to be the identity when $D\neq \emptyset$ , roughly speaking. See Appendix B.4 for a careful definition.

This applies in particular to the field $L=k(\text{GL}_{r})$ . Since the specialization map depends on the transcendental basis and the order thereof, we fix a total order on the set $\mathbb{N}\times \mathbb{N}$ once and for all, and use the induced order on the variables $\{{x_{ab}\}}_{(a,b)\in \{1,\ldots ,r\}^{2}}$ .

Definition-Lemma 2.18. We define the map $\unicode[STIX]{x1D709}_{\mathsf{rel}}^{r}$ in $D(\mathsf{MSm}^{\ast }/\boldsymbol{X}_{r}^{\mathsf{rel}})$ by:

(8)

It does not depend on the choice of the ordering.

2.5 Chern classes on the relative Volodin space

Let $r>0$ . In Theorem 1.11, we have proved an isomorphism

$$\begin{eqnarray}p^{\ast }(-)\cdot \unicode[STIX]{x1D709}^{j}\,:\,\bigoplus _{j=0}^{r-1}z_{\mathsf{rel}}^{r-j}\xrightarrow[{}]{\simeq }p_{\ast }z_{\mathsf{rel}}^{r}\end{eqnarray}$$

in $D(\mathsf{MSm}^{\ast }/B\text{GL}_{r})$ , and thus in $D(\mathsf{MSm}^{\ast }/\boldsymbol{X}_{r}^{\mathsf{rel}})$ . In Definition-Lemma 2.18, we have constructed a map

$$\begin{eqnarray}\unicode[STIX]{x1D709}_{\mathsf{rel}}^{r}\,:\,\mathbb{Z}\rightarrow p_{\ast }z_{\mathsf{rel}}^{r}\end{eqnarray}$$

in $D(\mathsf{MSm}^{\ast }/\boldsymbol{X}_{r}^{\mathsf{rel}})$ . It follows that there are unique morphisms $c_{i}\,:\,\mathbb{Z}\rightarrow z_{\mathsf{rel}}^{i}$ in $D(\mathsf{MSm}^{\ast }/\boldsymbol{X}_{r}^{\mathsf{rel}})$ for $1\leqslant i\leqslant r$ which satisfy the equality of maps $\mathbb{Z}\rightrightarrows p_{\ast }z_{\mathsf{rel}}^{r}$ :

(9) $$\begin{eqnarray}\unicode[STIX]{x1D709}_{\mathsf{rel}}^{r}+(p^{\ast }c_{1})\cdot \unicode[STIX]{x1D709}^{r-1}+\cdots +p^{\ast }c_{r}=0.\end{eqnarray}$$

It is convenient to define $c_{i}:=0$ for $i>r$ . By Theorem A.5 applied to ${\mathcal{C}}=\mathsf{MSm}^{\ast }$ , we have an isomorphism

$$\begin{eqnarray}\operatorname{Hom}_{\mathsf{Ho}(s\mathsf{PSh}(\{))}(\boldsymbol{X}_{r}^{\mathsf{rel}},K(z_{\mathsf{rel}}^{i},0))\cong \operatorname{Hom}_{D(\mathsf{MSm}^{\ast }/\boldsymbol{X}_{r}^{\mathsf{rel}})}(\mathbb{Z},z_{\mathsf{rel}}^{i}).\end{eqnarray}$$

Denote again by $c_{i}$ the corresponding map $c_{i}\,:\,\boldsymbol{X}_{r}^{\mathsf{rel}}\rightarrow K(z_{\mathsf{rel}}^{i},0)$ in the Nisnevich-local homotopy category of simplicial presheaves $\mathsf{Ho}(s\mathsf{PSh}(\mathsf{MSm}^{\ast }))$ .

Definition 2.20. The above-defined maps:

$$\begin{eqnarray}c_{i}\,:\,\mathbb{Z}\rightarrow z_{\mathsf{rel}}^{i}\quad \text{or }\quad c_{i}\,:\,\boldsymbol{X}_{r}^{\mathsf{rel}}\rightarrow z_{\mathsf{rel}}^{i}\end{eqnarray}$$

are called the Chern classes (of rank $r$ ).

Note that for every $r,i\geqslant 1$ the composite in $\mathsf{Ho}(s\mathsf{PSh}(\mathsf{MSm}^{\ast }))$ :

$$\begin{eqnarray}\ast =(\text{the identity matrix}){\hookrightarrow}\boldsymbol{X}_{r}^{\mathsf{rel}}\xrightarrow[{}]{c_{i}}K(z_{\mathsf{rel}}^{i},0)\end{eqnarray}$$

equals the constant map to the base point because the map $\unicode[STIX]{x1D709}_{\mathsf{rel}}^{r}$ is represented by the empty cycle when restricted to $\mathsf{MSm}^{\ast }/\{\text{id}\}$ . By this fact and a somewhat standard result below, it follows that $c_{i}$ come from unique maps $\boldsymbol{X}_{r}^{\mathsf{rel}}\rightarrow K(z_{\mathsf{rel}}^{i},0)$ in $\mathsf{Ho}(s\mathsf{PSh}_{\ast }(\mathsf{MSm}^{\ast }))$ , the homotopy category of pointed simplicial presheaves.

Lemma 2.21. cf. [Reference Asakura and SatoAS13, Proposition 5.2]

Let $(X,x)$ be a pointed object in $s\mathsf{PSh}(\mathsf{MSm}^{\ast })$ and $(K,e_{K})$ be a group object in the same category. Then the square of sets below is cartesian:

where the left vertical arrow is the “forget the base point” map and the right vertical arrow maps the point to the constant map at $e_{K}$ .

Next, associated with the embedding $\unicode[STIX]{x1D704}\,:\,\text{GL}_{r}{\hookrightarrow}\text{GL}_{r+1};~\unicode[STIX]{x1D6FC}\mapsto \left(\begin{array}{@{}cc@{}}\unicode[STIX]{x1D6FC} & 0\\ 0 & 1\end{array}\right)$ , we have embeddings $\boldsymbol{X}_{r}^{\mathsf{rel}}{\hookrightarrow}\boldsymbol{X}_{r+1}^{\mathsf{rel}}$ and $\mathbb{P}^{r-1}{\hookrightarrow}\mathbb{P}^{r};(T_{1}:\ldots :T_{r})\mapsto (T_{1}:\ldots :T_{r}:0)$ . The following is a special case of Proposition 3.4.

Lemma 2.22. The following diagram in $\mathsf{Ho}(s\mathsf{PSh}_{\ast }(\mathsf{MSm}^{\ast }))$ :

commutes for $i\geqslant 1$ .

2.6 Chern classes on the relative K-theory

We let $\text{K}$ be a functorial model of Thomason–Trobaugh’s $K$ -theory [Reference Thomason and TrobaughTT90, 3.1], that is, it is a presheaf of spectra $\text{K}$ such that for every quasicompact quasiseparated scheme $X$ , $\text{K}(X)$ is the $K$ -theory spectrum of the Waldhausen category of perfect complexes on $X$ .

We use Bousfield–Kan’s $\mathbb{Z}$ -completion in [Reference Bousfield and KanBK72] as a functorial model of Quillen’s plus construction. The $\mathbb{Z}$ -completion is an endofunctor $\mathbb{Z}_{\infty }\,:\,{\mathcal{S}}\rightarrow {\mathcal{S}}$ of the category of spaces (= simplicial sets) with a natural transformation $\text{Id}_{{\mathcal{S}}}\rightarrow \mathbb{Z}_{\infty }$ . We will apply $\mathbb{Z}_{\infty }$ sectionwise to simplicial presheaves.

The following is a relative and functorial version of Quillen’s “ $+=Q$ ” theorem.

Theorem 2.24. There exists an isomorphism

$$\begin{eqnarray}\unicode[STIX]{x1D6FA}^{\infty }\text{K}^{\mathsf{rel}}\simeq \mathbb{Z}^{\mathsf{rel}}\times \mathbb{Z}_{\infty }\boldsymbol{X}^{\mathsf{rel}}\end{eqnarray}$$

in $\mathsf{Ho}(s\mathsf{PSh}_{\ast }(\mathsf{MSm}))$ .Footnote ² Under this isomorphism, the multiplication of $\unicode[STIX]{x1D6FA}^{\infty }\text{K}^{\mathsf{rel}}$ coming from loop composition is compatible with the one of $\mathbb{Z}^{\mathsf{rel}}\times \mathbb{Z}_{\infty }\boldsymbol{X}^{\mathsf{rel}}$ coming from the group law of $\mathbb{Z}$ and the diagonal sum of matrices.

Proof. As in [Reference LodayLo98, 11.3.6], for any ring $A$ with an ideal $I$ , there exists an isomorphism

$$\begin{eqnarray}\unicode[STIX]{x1D6FA}^{\infty }\text{K}(A,I)\simeq \text{K}_{0}(A,I)\times \mathbb{Z}_{\infty }\boldsymbol{X}(A,I).\end{eqnarray}$$

The construction of the isomorphism can be functorial for the connected components (cf. [Reference GilletGi81, Proposition 2.15] for the case $I=0$ ), and thus the desired isomorphism follows. We can verify easily that each step of the construction of the isomorphism is compatible with the multiplications.

Theorem 2.25. For $r\geqslant 2l+2$ , the canonical map

$$\begin{eqnarray}\mathbb{Z}_{\infty }\boldsymbol{X}_{r}^{\mathsf{rel}}\rightarrow \mathbb{Z}_{\infty }\boldsymbol{X}^{\mathsf{rel}}\end{eqnarray}$$

is a Zariski-local $l$ -equivalence of simplicial presheaves on $\mathsf{MSm}$ , that is, a Zariski-local weak equivalence after taking the $l$ th Postnikov filtration.

Proof. By Suslin’s stability as formulated in [Reference BeilinsonBe14, Section 5], for any local ring $A$ with an ideal $I$ , the canonical map $\boldsymbol{X}_{r}(A,I)\rightarrow \boldsymbol{X}(A,I)$ induces homology isomorphisms in degree less or equal to $(r-1)/2$ . Then it follows from [Reference Bousfield and KanBK72, Chapter I 6.2] that the morphism

$$\begin{eqnarray}\unicode[STIX]{x1D70B}_{l}\mathbb{Z}_{\infty }\boldsymbol{X}_{r}(A,I)\rightarrow \unicode[STIX]{x1D70B}_{l}\mathbb{Z}_{\infty }\boldsymbol{X}(A,I)\end{eqnarray}$$

is an isomorphism for $l\leqslant (r-2)/2$ . This proves the theorem.

In Definition 2.20, we have constructed maps

$$\begin{eqnarray}c_{i}\,:\,\boldsymbol{X}_{r}^{\mathsf{rel}}\rightarrow K(z_{\mathsf{rel}}^{i},0)\end{eqnarray}$$

for $1\leqslant i\leqslant r$ in $\mathsf{Ho}(s\mathsf{PSh}_{\ast }(\mathsf{MSm}^{\ast }))$ . Let $l>0$ . For $r\gg l$ , we have the following sequence of morphisms in $\mathsf{Ho}(s\mathsf{PSh}_{\ast }(\mathsf{MSm}^{\ast }))$ :

where $P_{l}$ is the $l$ th Postnikov filtration and $\unicode[STIX]{x1D70F}_{{\leqslant}l}$ is the $l$ th canonical filtration. According to Lemma 2.22, the composite $\unicode[STIX]{x1D70F}_{{\leqslant}l}\mathsf{C}_{i}$ is independent of the choice of $r$ . Also, the diagram

commutes, where the vertical map is the obvious one.

Theorem 2.26. Let $i>0$ . There exists a morphism

in $\text{pro}-\mathsf{Ho}(s\mathsf{PSh}_{\ast }(\mathsf{MSm}^{\ast }))$ . For $n\geqslant 0$ , its $(-n)$ th hypercohomology on a modulus pair $(X,D)$ yields a map

$$\begin{eqnarray}\mathsf{C}_{n,i}\,:\,\text{K}_{n}(X,D)\rightarrow H_{\text{Nis}}^{-n}((X,D),z_{\mathsf{rel}}^{i}),\end{eqnarray}$$

which is functorial in $(X,D)\in \mathsf{MSm}$ and is a group homomorphism for $n>0$ . This map coincides with Bloch’s Chern class [Reference BlochBl86, Section 7] when $D=\emptyset$ .

Proof. We define . Since the Nisnevich cohomological dimension of $X$ is finite, by taking the $(-n)$ th hypercohomology of $\mathsf{C}_{i}$ , we obtain

$$\begin{eqnarray}\displaystyle \mathsf{C}_{n,i}\,:\,\text{K}_{n}(X,D) & \xrightarrow[{}]{\simeq } & \displaystyle H_{\text{Nis}}^{-n}((X,D),\text{K}^{\mathsf{rel}})\nonumber\\ \displaystyle & \rightarrow & \displaystyle H_{\text{Nis}}^{-n}((X,D),\unicode[STIX]{x1D70F}_{{\leqslant}l}z_{\mathsf{rel}}^{i})\simeq H_{\text{Nis}}^{-n}((X,D),z_{\mathsf{rel}}^{i}).\nonumber\end{eqnarray}$$

The first map is an isomorphism by Thomason–Trobaugh’s Nisnevich descent. Recall that $\mathsf{MSm}^{\ast }$ could be the category over any finite diagram in $\mathsf{MSm}$ , which ensures the functoriality. The map $\mathsf{C}_{n,i}$ is a group homomorphism for $n>0$ since it is defined by taking the $n$ th homotopy groups. Compatibility with Bloch’s Chern class is immediate from the construction.

3.1 Algebraic join

Let $X$ be a scheme. Consider the projective spaces over $X$ : $\mathbb{P}_{X}^{r-1}=\mathbf{Proj}({\mathcal{O}}_{X}[T_{1},\ldots ,T_{r}])$ , $\mathbb{P}_{X}^{s-1}=\mathbf{Proj}({\mathcal{O}}_{X}[T_{r+1},\ldots ,T_{r+s}])$ and

(10) $$\begin{eqnarray}\mathbb{P}_{X}^{r+s-1}=\mathbf{Proj}({\mathcal{O}}_{X}[T_{1},\ldots ,T_{r+s}]).\end{eqnarray}$$

The schemes $\mathbb{P}_{X}^{r-1}$ and $\mathbb{P}_{X}^{s-1}$ are naturally closed subschemes of $\mathbb{P}_{X}^{r+s-1}$ . We consider the rational maps $q_{1}\,:\,\mathbb{P}_{X}^{r+s-1}{\dashrightarrow}\mathbb{P}_{X}^{r-1}$ and $q_{2}\,:\,\mathbb{P}_{X}^{r+s-1}{\dashrightarrow}\mathbb{P}_{X}^{s-1}$ defined by $(T_{1},\ldots ,T_{r+s})\mapsto (T_{1},\ldots ,T_{r})$ and $\mapsto (T_{r+1},\ldots ,T_{r+s})$ .

Denote by $\unicode[STIX]{x1D70B}_{1}\,:\,P_{1}\rightarrow \mathbb{P}_{X}^{r+s-1}$ the blow-up along the ill-defined locus $\mathbb{P}_{X}^{s-1}$ of $q_{1}$ . Then $q_{1}$ induces a morphism $q_{1}^{\prime }\,:\,P_{1}\rightarrow \mathbb{P}_{X}^{r-1}$ which is a $\mathbb{P}^{s}$ -bundle. Denote by $q_{1}^{\ast }$ the operation on cycles defined as flat pull-back $q_{1}^{\prime \ast }$ followed by proper push-forward $\unicode[STIX]{x1D70B}_{1\ast }$ . Similarly, if $\unicode[STIX]{x1D70B}_{2}\,:\,P_{2}\rightarrow \mathbb{P}_{X}^{s-1}$ is the blow-up along $\mathbb{P}_{X}^{r-1}$ , the rational map $q_{2}$ induces a morphism $q_{2}^{\prime }\,:\,P_{2}\rightarrow \mathbb{P}_{X}^{s-1}$ which is a $\mathbb{P}^{r}$ -bundle. Denote by $q_{2}^{\ast }$ the flat pull-back $q_{2}^{\prime \ast }$ followed by push-forward $\unicode[STIX]{x1D70B}_{2\ast }$ .

Observe the obvious fact that the cycle in $\mathbb{P}_{X}^{r-1}$ given by a set of homogeneous equations $\{f_{\unicode[STIX]{x1D6FC}}(T_{1},\ldots ,T_{r})\}_{\unicode[STIX]{x1D6FC}}$ is mapped by $q_{1}^{\ast }$ to the cycle in $\mathbb{P}_{X}^{r+s-1}$ defined by the same equations.

Lemma 3.1. Let $\unicode[STIX]{x1D6FC}$ be an element in $z^{i}(\mathbb{P}_{X}^{r-1},m)$ and $\unicode[STIX]{x1D6FD}$ be an element in $z^{j}(\mathbb{P}_{X}^{r-1},n)$ . Suppose that the intersection product $\unicode[STIX]{x1D6FC}\,\cdot \,\unicode[STIX]{x1D6FD}\in z^{i+j}(\mathbb{P}_{X}^{r-1},m+n)$ is defined. Then the same holds for cycles $q_{1}^{\ast }\unicode[STIX]{x1D6FC}\in z^{i}(\mathbb{P}_{X}^{r+s-1},m)$ and $q_{1}^{\ast }\unicode[STIX]{x1D6FD}\in z^{j}(\mathbb{P}_{X}^{r+s-1},n)$ and we have an equality in $z^{i+j}(\mathbb{P}_{X}^{r+s-1},m+n)$ :

$$\begin{eqnarray}q_{1}^{\ast }(\unicode[STIX]{x1D6FC}\,\cdot \,\unicode[STIX]{x1D6FD})=(q_{1}^{\ast }\unicode[STIX]{x1D6FC})\,\cdot \,(q_{1}^{\ast }\unicode[STIX]{x1D6FD}).\end{eqnarray}$$

The same is true for the operation $q_{2}^{\ast }$ .

The operation $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})\mapsto \unicode[STIX]{x1D6FC}\#\unicode[STIX]{x1D6FD}$ is called algebraic join. It has been systematically used by authors like Friedlander, Lawson, Michelsohn and Walker. The authors of the present article learned this technique mainly in [Reference Friedlander and LawsonFL98].

3.2 The equalities

Let $r,s$ be non-negative integers. We keep the coordinate convention (10) when considering projective bundles $\mathbb{P}(E\text{GL}_{r})$ and $\mathbb{P}(E\text{GL}_{s})$ . On the category $\mathsf{MSm}^{\ast }/\boldsymbol{X}_{r}^{\mathsf{rel}}\times \boldsymbol{X}_{s}^{\mathsf{rel}}$ we consider presheaves:

• ∙ $p_{1\ast }z_{\mathsf{rel}}^{i}$ , which is induced from $p_{\ast }z_{\mathsf{rel}}^{i}$ on $\mathsf{MSm}^{\ast }/\boldsymbol{X}_{r}^{\mathsf{rel}}$ by the first projection $\boldsymbol{X}_{r}^{\mathsf{rel}}\times \boldsymbol{X}_{s}^{\mathsf{rel}}\rightarrow \boldsymbol{X}_{r}^{\mathsf{rel}}$ ;

• ∙ $p_{2\ast }z_{\mathsf{rel}}^{i}$ , induced by the second projection $\boldsymbol{X}_{r}^{\mathsf{rel}}\times \boldsymbol{X}_{s}^{\mathsf{rel}}\rightarrow \boldsymbol{X}_{s}^{\mathsf{rel}}$ ;

• ∙ $p_{\ast }z_{\mathsf{rel}}^{i}$ , induced by the inclusion $\boldsymbol{X}_{r}^{\mathsf{rel}}\times \boldsymbol{X}_{s}^{\mathsf{rel}}{\hookrightarrow}\boldsymbol{X}_{r+s}^{\,\mathsf{rel}}$ ;

• ∙ their nonmodulus counterparts $p_{1\ast }z^{i}$ , $p_{2\ast }z^{i}$ and $p_{\ast }z^{i}$ .

Let us distinguish the pull-back maps by writing $p_{1}^{\ast }\,:\,z_{\mathsf{rel}}^{i}\rightarrow p_{1\ast }z_{\mathsf{rel}}^{i}$ , $p_{2}^{\ast }\,:\,z_{\mathsf{rel}}^{i}\rightarrow p_{2\ast }z_{\mathsf{rel}}^{i}$ and $p^{\ast }\,:\,z_{\mathsf{rel}}^{i}\rightarrow p_{\ast }z_{\mathsf{rel}}^{i}$ . Similarly, we can consider three different versions of ${\mathcal{Z}}$ ’s, denoted by ${\mathcal{Z}}_{1}$ , ${\mathcal{Z}}_{2}$ and ${\mathcal{Z}}$ . There are obvious maps ${\mathcal{Z}}\rightarrow {\mathcal{Z}}_{1}$ and ${\mathcal{Z}}\rightarrow {\mathcal{Z}}_{2}$ .

We may consider the partially defined join operator $\#\,:\,p_{1\ast }z_{\mathsf{rel}}^{i}\otimes p_{2\ast }z_{\mathsf{rel}}^{j}{\dashrightarrow}p_{\ast }z_{\mathsf{rel}}^{i+j}$ and its nonmodulus version. The modulus version is a well-defined map in $D(\mathsf{MSm}^{\ast }/\boldsymbol{X}_{r}^{\mathsf{rel}}\times \boldsymbol{X}_{s}^{\mathsf{rel}})$ (Section B.1.2). Given two maps $\unicode[STIX]{x1D6FC}\,:\,{\mathcal{Z}}\rightarrow p_{1\ast }z^{i}$ and $\unicode[STIX]{x1D6FD}\,:\,{\mathcal{Z}}\rightarrow p_{2\ast }z^{j}$ , we consider their cup product followed by algebraic join to get a map $\unicode[STIX]{x1D6FC}\#\unicode[STIX]{x1D6FD}\,:\,{\mathcal{Z}}\rightarrow p_{\ast }z^{i+j}$ whenever it is well defined.

In Remark 2.19 (ii), we defined maps $C_{\mathsf{MSm}^{\ast }/\boldsymbol{X}_{r}^{\mathsf{rel}}}^{i}$ of presheaves on $\mathsf{MSm}^{\ast }/\boldsymbol{X}_{r}^{\mathsf{rel}}$ . Via ${\mathcal{Z}}\rightarrow {\mathcal{Z}}_{1}$ , it induces a map

$$\begin{eqnarray}C_{\mathsf{MSm}^{\ast }/\boldsymbol{X}_{r}^{\mathsf{rel}}}^{i}\,:\,{\mathcal{Z}}\rightarrow p_{1\ast }z_{k(\text{GL}_{r})}^{i}\quad \text{ on }\mathsf{MSm}^{\ast }/\boldsymbol{X}_{r}^{\mathsf{rel}}\times \boldsymbol{X}_{s}^{\mathsf{rel}}\end{eqnarray}$$

which we denote by the same symbol. Similarly for $C_{\mathsf{MSm}^{\ast }/\boldsymbol{X}_{s}^{\mathsf{rel}}}^{j}$ . Recall that they depend on the choice of $i$ indices out of $\{1,\ldots ,r\}$ and $j$ out of $\{r+1,\ldots ,r+s\}$ although it is not explicit in the notation.

Also, thanks to Remark 2.19 (i)(ii), we have yet other maps

$$\begin{eqnarray}C_{\mathsf{MSm}^{\ast }/\boldsymbol{X}_{r+s}^{\mathsf{rel}}}^{k}\,:\,{\mathcal{Z}}\rightarrow p_{\ast }z_{\mathsf{rel},k(\text{GL}_{r}\times \text{GL}_{s})}^{k}\end{eqnarray}$$

for integers $1\leqslant k\leqslant r+s$ .

Proposition 3.3. For any $0\leqslant i\leqslant r$ and $0\leqslant j\leqslant s$ , the two maps

$$\begin{eqnarray}C_{\mathsf{MSm}^{\ast }/\boldsymbol{X}_{r}^{\mathsf{rel}}}^{i}\#C_{\mathsf{MSm}^{\ast }/\boldsymbol{X}_{s}^{\mathsf{rel}}}^{j}\text{ and }C_{\mathsf{MSm}^{\ast }/\boldsymbol{X}_{r+s}^{\mathsf{rel}}}^{i+j}\,:\,\quad {\mathcal{Z}}\rightrightarrows p_{\ast }z_{k(\text{GL}_{r}\times \text{GL}_{s})}^{i+j}\end{eqnarray}$$

are equal as maps of presheaves on $\mathsf{MSm}^{\ast }/\boldsymbol{X}_{r}^{\mathsf{rel}}\times \boldsymbol{X}_{s}^{\mathsf{rel}}$ under appropriate choices of indices (specified in the proof). In particular we have $\unicode[STIX]{x1D709}_{\mathsf{rel}}^{r}\#\unicode[STIX]{x1D709}_{\mathsf{rel}}^{s}=\unicode[STIX]{x1D709}_{\mathsf{rel}}^{r+s}$ as maps $\mathbb{Z}\rightrightarrows p_{\ast }z_{\mathsf{rel}}^{r+s}$ in $D(\mathsf{MSm}^{\ast }/\boldsymbol{X}_{r}^{\mathsf{rel}}\times \boldsymbol{X}_{s}^{\mathsf{rel}})$ .

Proof. It suffices to prove the corresponding equality of cycles on the simplicial scheme $\mathbb{P}(E\text{GL}_{r+s})_{|B\text{GL}_{r}\times B\text{GL}_{s}}$ . By the definition of algebraic join of maps, the problem is to prove the equality of maps:

$$\begin{eqnarray}\displaystyle & & \displaystyle q_{1}^{\ast }(\unicode[STIX]{x1D6E4}\text{}_{B\text{GL}_{r}}^{(1)}\,\cdot \,\ldots \,\cdot \,\unicode[STIX]{x1D6E4}\text{}_{B\text{GL}_{r}}^{(i)})~\,\cdot \,~q_{2}^{\ast }(\unicode[STIX]{x1D6E4}\text{}_{B\text{GL}_{s}}^{(r+1)}\,\cdot \,\ldots \,\cdot \,\unicode[STIX]{x1D6E4}\text{}_{B\text{GL}_{s}}^{(r+j)})\nonumber\\ \displaystyle & & \displaystyle \qquad =\unicode[STIX]{x1D6E4}\text{}_{B\text{GL}_{r+s}}^{(1)}\,\cdot \,\ldots \,\cdot \,\unicode[STIX]{x1D6E4}\text{}_{B\text{GL}_{r+s}}^{(i)}\,\cdot \,\unicode[STIX]{x1D6E4}\text{}_{B\text{GL}_{r+s}}^{(r+1)}\,\cdot \,\unicode[STIX]{x1D6E4}\text{}_{B\text{GL}_{r+s}}^{(r+j)}\nonumber\end{eqnarray}$$

from the cone of

$$\begin{eqnarray}\mathbb{Z}[B_{\bullet }\text{GL}_{r}^{\ast }\times B_{\bullet }\text{GL}_{s}^{\ast }]\rightarrow \mathbb{Z}[B_{\bullet }(\text{GL}_{r}\times \text{GL}_{s})]\oplus \mathbb{Z}[B_{\bullet }\text{GL}_{r}^{\ast }\times B_{\bullet }\text{GL}_{s}^{\ast }]\end{eqnarray}$$

to $z^{r+s}(\mathbb{P}_{k(\text{GL}_{r}\times \text{GL}_{s})}^{r+s-1}\times -,\bullet )$ . By Lemma 3.1, it is reduced to the equalities of cycles $q_{1}^{\ast }\unicode[STIX]{x1D6E4}\text{}_{B\text{GL}_{r}}^{(a)\ast }(S)=\unicode[STIX]{x1D6E4}\text{}_{B\text{GL}_{r+s}}^{(a)\ast }(S)$ on $\mathbb{P}(E_{n}\text{GL}_{r+s})_{|B\text{GL}_{r}^{\ast }\times B\text{GL}_{s}^{\ast }}$ for all $1\leqslant a\leqslant r$ and nonempty subsets $S\subset [n]$ , and its variants involving $\circ$ , $\circ \ast$ and $q_{2}^{\ast }$ . In view of the fact observed before Lemma 3.1, this last equality clearly holds. This completes the proof of the proposition.

Now, consider the following diagram in $D(\mathsf{MSm}^{\ast }/\boldsymbol{X}_{r}^{\mathsf{rel}}\times \boldsymbol{X}_{s}^{\mathsf{rel}})$ (see Sections B.1.1, B.1.2 for the tensor products $\overset{\text{:D}}{\otimes }$ ):

(11)

where the vertical map “intersection” sends an element $(\unicode[STIX]{x1D6FC}_{0},(\unicode[STIX]{x1D6FC}_{i})_{i})\otimes (\unicode[STIX]{x1D6FD}_{0},(\unicode[STIX]{x1D6FD}_{j})_{j})$ to the tuple of cycles $(\sum _{k=i+j}\unicode[STIX]{x1D6FC}_{i}\,\cdot \,\unicode[STIX]{x1D6FD}_{j})_{0\leqslant k\leqslant r+s}$ . The horizontal maps $\unicode[STIX]{x1D70E}$ are defined by $\unicode[STIX]{x1D70E}(r)\,:\,(\unicode[STIX]{x1D6FC}_{0},(\unicode[STIX]{x1D6FC}_{i})_{i=1}^{r})\mapsto \unicode[STIX]{x1D6FC}_{0}\unicode[STIX]{x1D709}_{\mathsf{rel}}^{r}+\sum _{i=1}^{r}p^{\ast }(\unicode[STIX]{x1D6FC}_{i})\,\cdot \,\unicode[STIX]{x1D709}^{r-i}.$ Applying Lemma 3.1, Proposition 3.3 and the commutativity of intersection product in the derived category, one checks that the diagram (11) commutes.

The rank $r$ Chern classes $c_{i}$ are characterized by the property that the composite map $\mathbb{Z}\xrightarrow[{}]{(1,c_{1},\ldots ,c_{r})}\mathbb{Z}\oplus \bigoplus _{i=1}^{r}z_{\mathsf{rel}}^{i}\xrightarrow[{}]{\unicode[STIX]{x1D70E}}p_{\ast }z_{\mathsf{rel}}^{r}$ is zero. Of course the same holds for the rank $s$ case. It follows by the commutativity of (11) that the composite $\mathbb{Z}\xrightarrow[{}]{(\sum _{i+j=k}c_{i}\,\cdot \,c_{j})_{k\geqslant 0}}\mathbb{Z}\oplus \bigoplus _{k=1}^{r+s}z_{\mathsf{rel}}^{k}\xrightarrow[{}]{\unicode[STIX]{x1D70E}}p_{\ast }z_{\mathsf{rel}}^{r+s}$ is zero. From the characterization of Chern classes we get:

Equivalently, it is an equality of maps $\boldsymbol{X}_{r}^{\mathsf{rel}}\times \boldsymbol{X}_{s}^{\mathsf{rel}}\rightrightarrows K(z_{\mathsf{rel}}^{k},0)$ in the homotopy category of pointed simplicial presheaves $\mathsf{Ho}(s\mathsf{PSh}_{\ast }(\mathsf{MSm}^{\ast }))$ .

Corollary 3.5. The diagram in $\mathsf{Ho}(s\mathsf{PSh}_{\ast }(\mathsf{MSm}^{\ast }))$

is commutative, where the left vertical map is defined by the diagonal sum of matrices and the right one is defined by the intersection product.

3.3 Whitney sum formula on the relative $K$ -theory

We set $\tilde{z}_{\mathsf{rel}}^{\ast }:=\mathbb{Z}\oplus (\bigoplus _{i\geqslant 1}z_{\mathsf{rel}}^{i})$ . We define the total Chern class map

(12)

by the product of the canonical map $\mathbb{Z}^{\mathsf{rel}}\rightarrow \mathbb{Z}$ and $(1,\mathsf{C}_{1},\mathsf{C}_{2},\ldots \,\!)$ . We have seen that the diagonal sum of $\boldsymbol{X}^{\mathsf{rel}}$ and the group law of $\mathbb{Z}^{\mathsf{rel}}$ is compatible with the loop composition of $\unicode[STIX]{x1D6FA}^{\infty }\text{K}^{\mathsf{rel}}$ (Theorem 2.24). It follows from Corollary 3.5 that the diagram in $\text{pro}-\mathsf{Ho}(s\mathsf{PSh}_{\ast }(\mathsf{MSm}^{\ast }))$

is commutative. By taking the $0$ th hypercohomology of $\mathsf{C}_{\mathsf{tot}}$ on a modulus pair $(X,D)$ , we obtain a map

(13) $$\begin{eqnarray}\text{K}_{0}(X,D)\rightarrow \mathbb{Z}\times \{1\}\times \bigoplus _{i\geqslant 1}H_{\text{Nis}}^{0}((X,D),z_{\mathsf{rel}}^{i}).\end{eqnarray}$$

We regard the target as a group by

$$\begin{eqnarray}\biggl(n,1+\mathop{\sum }_{i\geqslant 1}\unicode[STIX]{x1D6FC}_{i}\biggr)\cdot \biggl(m,1+\mathop{\sum }_{j\geqslant 1}\unicode[STIX]{x1D6FD}_{j}\biggr)=\biggl(n+m,(1+\mathop{\sum }_{i\geqslant 1}\unicode[STIX]{x1D6FC}_{i})(1+\mathop{\sum }_{j\geqslant 1}\unicode[STIX]{x1D6FD}_{j})\biggr).\end{eqnarray}$$

It follows from the above commutative diagram that:

Theorem 3.6. The map (13) is a group homomorphism. In other words, we have

$$\begin{eqnarray}\mathsf{C}_{0,i}(\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD})=\mathop{\sum }_{j+k=i,\,j,k\geqslant 0}\mathsf{C}_{0,j}(\unicode[STIX]{x1D6FC})\mathsf{C}_{0,k}(\unicode[STIX]{x1D6FD})\end{eqnarray}$$

for $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}\in \text{K}_{0}(X,D)$ with the convention $\mathsf{C}_{0,0}=1$ .

4.1 Chern character

We set

where the Hom group is taken in the category $\text{pro}-\mathsf{Ho}(s\mathsf{PSh}_{\ast }(\mathsf{MSm}^{\ast }))$ . Under this convention, the total Chern class (12) is in the set $A^{0}\times \{1\}\times \prod _{i\geqslant 1}A^{i}$ . We define a map

$$\begin{eqnarray}\text{ch}\,:\,A^{0}\times \{1\}\times \bigoplus _{i\geqslant 1}A^{i}\rightarrow A_{\mathbb{ Q}}^{\ast }:=\mathop{\prod }_{i\geqslant 0}A^{i}\otimes \mathbb{Q}\end{eqnarray}$$

as in [Reference Berthelot, Grothendieck and IllusieSGA6, Exposé 0, Appendix 1.26], that is,

$$\begin{eqnarray}\text{ch}\Bigl(\Bigl(n,1+\mathop{\sum }_{i\geqslant 1}x^{i}\Bigr)\Bigr)=n+\unicode[STIX]{x1D702}\Bigl(\log \Bigl(1+\mathop{\sum }_{i\geqslant 1}x^{i}\Bigr)\Bigr),\end{eqnarray}$$

where $\unicode[STIX]{x1D702}$ is an endomorphism of $A_{\mathbb{Q}}^{\ast }$ defined by $\unicode[STIX]{x1D702}(x^{i})=(-1)^{i-1}x^{i}/(i-1)!$ .

The image of $\mathsf{C}_{\mathsf{tot}}$ by $\text{ch}$ gives a map

(14)

in $\text{pro}-\mathsf{Ho}(s\mathsf{PSh}_{\ast }(\mathsf{MSm}^{\ast }))$ . According to Theorems 2.26 and 3.6, we obtain the following result.

Theorem 4.1. Let $(X,D)\in \mathsf{MSm}$ and $n\geqslant 0$ . The $(-n)$ th hypercohomology of (14) yields a group homomorphism

$$\begin{eqnarray}\mathsf{ch}_{n}\,:\,\text{K}_{n}(X,D)\rightarrow H_{\text{Nis}}^{-n}((X,D),(\tilde{z}_{\mathsf{rel}}^{\ast })_{\mathbb{ Q}}),\end{eqnarray}$$

which is functorial in $(X,D)$ and coincides with Bloch’s Chern character when $D=\emptyset$ .

For an additive category ${\mathcal{A}}$ , let ${\mathcal{A}}_{\mathbb{Q}}$ be the category up to isogeny, which has the same objects as ${\mathcal{A}}$ and $\operatorname{Hom}_{{\mathcal{A}}_{\mathbb{Q}}}(M,N)=\operatorname{Hom}_{{\mathcal{A}}}(M,N)\otimes \mathbb{Q}$ . We denote the image of $M\in {\mathcal{A}}$ in ${\mathcal{A}}_{\mathbb{Q}}$ by $M_{\mathbb{Q}}$ .

For a presheaf $F$ on $\mathsf{MSm}$ , we define a pro-system of presheaves $\hat{F}$ by

$$\begin{eqnarray}\hat{F}((X,D))=\{F(X,mD)\}_{m\geqslant 1}.\end{eqnarray}$$

The above argument can be modified to obtain a map

(15)

in $\text{pro}-\mathsf{Ho}(\text{pro}-s\mathsf{PSh}_{\ast }(\mathsf{MSm}^{\ast }))_{\mathbb{Q}}$ . Here is a variant of Theorem 4.1.

Theorem 4.2. Let $(X,D)\in \mathsf{MSm}$ and $n\geqslant 0$ . The $(-n)$ th hypercohomology of (15) yields a morphism

$$\begin{eqnarray}\mathsf{ch}_{n}\,:\,\{\text{K}_{n}(X,mD)\}_{m,\mathbb{Q}}\rightarrow \{H_{\text{Nis}}^{-n}((X,mD),\tilde{z}_{\mathsf{rel}}^{\ast })\}_{m,\mathbb{Q}}\end{eqnarray}$$

in the category of pro-abelian groups $(\text{pro}-\mathsf{Ab})_{\mathbb{Q}}$ up to isogeny. This is functorial in $(X,D)$ and coincides with Bloch’s Chern character when $D=\emptyset$ .

4.2 Relative motivic cohomology of henselian dvr

Let $k$ be a field of characteristic zero. Let $A$ be a henselian dvr over $k$ and $\unicode[STIX]{x1D70B}$ its uniformizer. Set $X=\text{Spec}A$ and $D=\text{Spec}A/\unicode[STIX]{x1D70B}$ . In this section, we prove the following.

Theorem 4.3. For every $n\geqslant 0$ , there is a natural isomorphism

$$\begin{eqnarray}\displaystyle & & \displaystyle \{\text{CH}^{\ast }(X|mD,n)\}_{m,\mathbb{Q}}\nonumber\\ \displaystyle & & \displaystyle \qquad \simeq \{\text{K}_{n}(X,mD)\oplus \ker (\text{CH}^{\ast }(X|mD,n)\rightarrow \text{CH}^{\ast }(X,n))\}_{m,\mathbb{Q}}\nonumber\end{eqnarray}$$

in the category $(\text{pro}-\mathsf{Ab})_{\mathbb{Q}}$ of pro-abelian groups up to isogeny.

We expect that $\{\ker (\text{CH}^{i}(X|mD,n)\rightarrow \text{CH}^{i}(X,n))\}_{m,\mathbb{Q}}$ vanishes.

Lemma 4.4. The canonical map

$$\begin{eqnarray}\text{K}_{n}(A)\rightarrow \{\text{K}_{n}(A/\unicode[STIX]{x1D70B}^{m})\}\text{}_{m}\end{eqnarray}$$

is a pro-epimorphism.

Proof. By Artin’s approximation theorem [Reference ArtinAr69], we may replace $A$ by its completion $\hat{A}\simeq F[[t]]$ , that is, enough to show that

$$\begin{eqnarray}\text{K}_{n}(F[[t]])\rightarrow {\{\text{K}_{n}(F[t]/t^{m})\}}_{m}\end{eqnarray}$$

is a pro-epimorphism.

Since $\text{K}_{n}(F[[t]])\rightarrow \text{K}_{n}(F)$ is a split surjection, it suffices to show that

$$\begin{eqnarray}\text{K}_{n}(F[[t]],(t))\rightarrow {\{\text{K}_{n}(F[t]/t^{m},(t))\}}_{m}\end{eqnarray}$$

is a pro-epimorphism. By Goodwillie’s theorem [Reference GoodwillieGo86] and the $\operatorname{HC}$ version of pro HKR theorem [Reference MorrowMo15, Theorem 3.23], we have pro-isomorphisms

$$\begin{eqnarray}\displaystyle \{\text{K}_{n}(F[t]/t^{m},(t))\}_{m} & \simeq & \displaystyle \{\operatorname{HC}_{n-1}(F[t]/t^{m},(t))\}_{m}\nonumber\\ \displaystyle & \simeq & \displaystyle \bigg\{\bigoplus _{p=0}^{n-1}H^{2p-(n-1)}(\unicode[STIX]{x1D6FA}_{F[t]/t^{m},(t)}^{{\leqslant}p})\bigg\}_{m},\nonumber\end{eqnarray}$$

where $\unicode[STIX]{x1D6FA}_{A,I}^{j}:=\ker (\unicode[STIX]{x1D6FA}_{A}^{j}\rightarrow \unicode[STIX]{x1D6FA}_{A/I}^{j})$ . By the Poincaré lemma [Reference WeibelWei94, Corollary 9.9.3], we have

$$\begin{eqnarray}H^{j}(\unicode[STIX]{x1D6FA}_{F[t]/t^{m},(t)}^{\bullet })=0\end{eqnarray}$$

for every $m,j\geqslant 0$ . Hence, it follows that

$$\begin{eqnarray}{\{\text{K}_{n}(F[t]/t^{m},(t))\}}_{m}\simeq {\{\unicode[STIX]{x1D6FA}_{F[t]/t^{m},(t)}^{n-1}/d\unicode[STIX]{x1D6FA}_{F[t]/t^{m},(t)}^{n-2}\}}_{m}.\end{eqnarray}$$

Consider the commutative diagram

where the rows are exact and the vertical maps are split surjections. Again by the Poincaré lemma, the left vertical map is an isomorphism, and thus we have an isomorphism

Given an element $f\otimes d\log y_{1}\wedge \cdots \wedge d\log y_{n-1}\in tF[t]/t^{m}\otimes _{F}\unicode[STIX]{x1D6FA}_{F}^{n-1}$ , the element $\{\exp (f),y_{1},\ldots ,y_{n-1}\}\in \text{K}_{n}^{M}(F[[t]])$ lifts it via

Therefore, the composite

$$\begin{eqnarray}\text{K}_{n}(F[[t]],t)\rightarrow {\{\text{K}_{n}(F[t]/t^{m},(t))\}}_{m}\simeq \{tF[t]/t^{m}\otimes _{F}\unicode[STIX]{x1D6FA}_{F}^{n-1}\}\end{eqnarray}$$

is isomorphic to a levelwise epimorphism, and thus the first map is a pro-epimorphism. This proves the lemma.

Proof of Theorem 4.3.

The Chern character $\hat{\mathsf{ch}}_{n}$ in Theorem 4.2 fits into the commutative diagram

in $(\text{pro}-\mathsf{An})_{\mathbb{Q}}$ . By Lemma 4.4, the upper sequence is exact. By Bloch’s comparison theorem in [Reference BlochBl86], the middle vertical map $\mathsf{ch}_{n}$ is an isomorphism. Hence, it follows that the left vertical map $\hat{\mathsf{ch}}_{n}$ is a pro-monomorphism.

We shall show that the composite

$$\begin{eqnarray}\unicode[STIX]{x1D6E9}:=\unicode[STIX]{x1D6FD}\circ \mathsf{ch}^{-1}\circ \unicode[STIX]{x1D6FC}\,:\,\{\text{CH}^{\ast }(X|mD,n)\}_{m,\mathbb{Q}}\rightarrow \{\text{K}_{n}(A/\unicode[STIX]{x1D70B}^{m})\}_{m,\mathbb{Q}}\end{eqnarray}$$

is the zero map.

Binda–Saito [Reference Binda and SaitoBS17] has constructed the cycle map

$$\begin{eqnarray}\text{CH}^{i}(X|mD,n)\rightarrow H^{2i-n}(\unicode[STIX]{x1D6FA}_{X|mD}^{{\geqslant}i}),\end{eqnarray}$$

where $\unicode[STIX]{x1D6FA}_{X|mD}^{j}=\unicode[STIX]{x1D6FA}_{A}^{j}(\log D)\otimes A\unicode[STIX]{x1D70B}^{m}$ . Note that we have a pro-isomorphism ${\{\unicode[STIX]{x1D6FA}_{X|mD}^{j}\}}_{m}\simeq {\{\unicode[STIX]{x1D6FA}_{A}^{j}\otimes A\unicode[STIX]{x1D70B}^{m}\}}_{m}$ . Hence, we have a commutative diagram

and the bottom composite is zero. Here, the second vertical map is the usual cycle map to the de Rham cohomology, and the composite

coincides with Goodwillie’s Chern character by [Reference WeibelWei93]. Therefore, the composite

equals zero, where $c$ is Goodwillie’s Chern character and the last isomorphism is by the pro HKR theorem again. (The pro HKR theorem may not yield a pro-isomorphism for $\operatorname{HN}$ in general, but now the relative part $\operatorname{HN}_{n}(A/\unicode[STIX]{x1D70B}^{m},(\unicode[STIX]{x1D70B}))$ is equal to $\operatorname{HC}_{n-1}(A/\unicode[STIX]{x1D70B}^{m},(\unicode[STIX]{x1D70B}))$ for which we can apply the pro HKR theorem, and we obtain the above pro-isomorphism by the five lemma.)

Consider the commutative diagram

where $c$ and $c_{1}$ are Goodwillie’s Chern characters and $\unicode[STIX]{x1D6FE}$ is the canonical map. We have seen that $c\circ \unicode[STIX]{x1D6E9}=0$ , and it is clear that $\unicode[STIX]{x1D6FE}\circ \unicode[STIX]{x1D6E9}=0$ . We claim that the kernels of $c$ and $c_{1}$ are isomorphic, which implies that $\unicode[STIX]{x1D6E9}=0$ . Indeed, the above square fits into the diagram

with exact rows. Here, the first vertical map is surjective and the second vertical map is an isomorphism by Goodwillie’s theorem [Reference GoodwillieGo86]. Hence, $\ker c\simeq \ker c_{1}$ follows.

Consequently, we obtain a morphism

$$\begin{eqnarray}\unicode[STIX]{x1D719}\,:\,\{\text{CH}^{\ast }(X|mD,n)\}_{m,\mathbb{Q}}\rightarrow \{\text{K}_{n}(A,(\unicode[STIX]{x1D70B})^{m})\}_{m,\mathbb{Q}}.\end{eqnarray}$$

It is clear that $\unicode[STIX]{x1D719}\circ \hat{\mathsf{ch}}_{n}=\text{id}$ and that $\unicode[STIX]{x1D6FC}\circ \hat{\mathsf{ch}}_{n}\circ \unicode[STIX]{x1D719}=\unicode[STIX]{x1D6FC}$ . This completes the proof of Theorem 4.3.