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 $\mathcal{M}o{t}_{\mathrm{dg}}^{\mathrm{nilp}}$ 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 $\mathrm{Ho}(\mathcal{M}o{t}_{\mathrm{dg}}^{\mathrm{nilp}})$, namely $M_{\mathrm{nilp}}(X)$ and $M_{\mathrm{nilp}}^{\prime }(X)$. In general $M_{\mathrm{nilp}}(X)$ is not isomorphic to $M_{\mathrm{nilp}}^{\prime }(X_{\mathrm{red}})$. But there exists a canonical isomorphism $M_{\mathrm{nilp}}^{\prime }(X)\simeq M_{\mathrm{nilp}}^{\prime }(X_{\mathrm{red}})$ and if $X$ is regular noetherian separated, $M(X)$ is canonically isomorphic to $M^{\prime }(X)$.

References in corpus (1)

• Non-connective K-theory of relative exact categories