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Motivic weight complexes for arithmetic varieties. (English) Zbl 1186.14020

The authors extend the results of their paper [J. Reine Angew. Math. 478, 127–176 (1996; Zbl 0863.19002)] to arithmetic varieties. They associate weight complexes of homological motives to arithmetic varieties and Deligne-Mumford stacks. In the mentioned paper, a similar result was proved for varieties over a field of characteristic zero and in terms of Chow motives with integral coefficients. In the current paper because of lack of resolution of singularities, the authors apply de Jong’s results and use \(K_{0}\)-motives with rational coefficients. Let \(S\) be fixed base scheme which is regular, excellent and finite-dimensional. In addition assume that the following condition is fulfilled:
Condition (C) For every finite morphism \(\pi : T\rightarrow S\), and finite group \(G\) acting on \(T\) over \(S\), the pair \((T,G)\) satisfies condition 5.12.1 of [A. J. de Jong, Ann. Inst. Fourier 47, No. 2, 599–621 (1997; Zbl 0868.14012)].
The main theorem of the paper is the following:
{Theorem 0.1.} There is a covariant functor \(h : { Stack}_{S}\rightarrow Ho (C_{*}(K(S))\) from the category of (separated) Deligne-Mumford stacks of finite type over \(S\), to the category of homotopy classes of maps of bounded complexes of (homological) \(K_{0}\)-motives over \(S\) with rational coefficients, having the following properties:
\(\bullet\)
If \(X\) is a regular scheme, projective over \(S\), then \(h(X)\) is the usual motive of \(X\).
\(\bullet\)
If \(X\) is a regular scheme, projective over \(S\), and \(G\) is a finite group acting on \(X\), then \(h({[X/G]})=h(X)^{G}.\) Here \([X/G]\) is the quotient stack associated to the action.
\(\bullet\)
If \({\mathfrak D}\subset {\mathfrak X}\) is a closed substack with complement \(\mathfrak U\), then we have a triangle \[ h({\mathfrak D})\rightarrow h({\mathfrak X})\rightarrow h({\mathfrak U})\rightarrow h({\mathfrak D})[+1] . \]
As a consequence of the theorem one obtains existence of the Euler characteristic in the Grothendieck group \(K_{0}(\mathbf{KM}_S)\)
{Corollary 0.2.} One can associate to any reduced separated Deligne-Mumford stack \(\mathfrak X\) of finite type over \(S\) an element \({\chi}_{c}(\mathfrak X)\) in the Grothendieck group \(K_{0}(\mathbf{KM}_S)\) of the category \(\mathbf{KM}_S\) of \(K_{0}\)-motives over \(S\), with the following properties:
(i)
If \(X\) is a regular projective scheme over \(S\) equipped with an action by a finite group \(G\) then \({\chi}_{c}([ X/G])\) is the class of \[ \Bigl( X, \frac{1}{\# (G)}{\sum}_{g\in G}[{\mathcal O}_{\Gamma (g_{*})}]\Bigr) ; \] here \([X/G]\) is the quotient stack associated to the action of \(G\) on \(X\) and \({\Gamma (g_{*})}\) is the graph of the action \(g_{*}: X\rightarrow X\) of an element \(g\in G .\)
(ii)
If \({\mathfrak D}\subset {\mathfrak X}\) is a closed substack, with complement \(\mathfrak X \backslash \mathfrak D\), then \[ {\chi}_{c}(\mathfrak X) = {\chi}_{c}(\mathfrak D) + {\chi}_{c}(\mathfrak X \backslash \mathfrak D ). \]

MSC:

14F42 Motivic cohomology; motivic homotopy theory
19A99 Grothendieck groups and \(K_0\)
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References:

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