Nilpotent invariant motives I

TL;DR

This paper constructs a new stable model category for nilpotent invariant motives, clarifies their relation to schemes, and explores conditions under which different motives are isomorphic, advancing understanding of $ extit{A}^1$-homotopy invariance.

Contribution

It introduces the stable model category of nilpotent invariant motives and analyzes their properties and relations to schemes, especially regarding $ extit{A}^1$-homotopy invariance.

Findings

Construction of the stable model category of nilpotent invariant motives.

Existence of two types of motives associated with schemes and their relations.

Isomorphism conditions for motives of regular noetherian schemes.

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 $M'(X)$.