Nilpotent invariant motives I

Abstract

The purpose of this article is to clarify the question what makes motives $\mathbb{A}^1$-homotopy invariance. we give construction of the stable model category of nilpotent invariant motives $\mathcal{M}ot_{\operatorname{dg}}^{\operatorname{nilp}}$ and define the nilpotent invriant motives associated with schemes and relative exact categories. For a noetherian scheme $X$, there are two kind of motives associated with $X$ in the homotopy category $\operatorname{Ho}(\mathcal{M}ot^{\operatorname{nilp}}_{\operatorname{dg}})$, namely $M_{\operatorname{nilp}}(X)$ and $M_{\operatorname{nilp}}'(X)$. In general $M_{\operatorname{nilp}}(X)$ is not isomorphic to $M'_{\operatorname{nilp}}(X_{\operatorname{red}})$. But there exists a canonical isomorphism $M_{\operatorname{nilp}}'(X)\simeq M_{\operatorname{nilp}}'(X_{\operatorname{red}})$ and if $X$ is regular noetherian separated, $M(X)$ is canonically isomorphic to $M'(X)$.