The chow weight structure for geometric motives of quotient stacks

TL;DR

The work constructs a Chow weight structure on the derived category of geometric motives over quotient stacks $[X/G]$ with coefficients in a ring $\Lambda$, under $k$ of characteristic $0$ and affine $G$, and identifies its heart with the classical category of Chow motives $\operatorname{CHM}([X/G], \Lambda)$. It develops an extension of motivic categories $\operatorname{DM}$ to Nis-loc stacks, establishes descent properties, and proves that $\operatorname{DM}_{\operatorname{gm}}(\mathcal{X}, \Lambda)$ is generated by Chow motives, enabling a stable weight structure with a heart equivalent to $\textbf{Chow}_{\infty}(\mathcal{X}, \Lambda)$. The mapping spectra between Chow motives are shown to be connective and relate to equivariant Chow groups via a Totaro gadget, while the equivariant motive theory is linked to Laterveer and Corti–Hanamura frameworks. Collectively, the paper extends motivic weight structures to stacks, connects geometric motives to equivariant Chow theory, and lays groundwork for further study of motivic measures and perverse-type motivic objects on stacks.

Abstract

We construct the Chow weight structure on the derived category of geometric motives with arbitrary coefficients for X a finite type scheme over a field characteristic 0 and G an affine algebraic group. In particular we also show that the heart of this weight structure recovers the category of Chow motives on [X/G].

Theorems & Definitions (110)

• Theorem 1
• Theorem 2
• Definition 2.1
• Remark 2.2
• Definition 2.4
• Example 2.5
• Example 2.6
• Theorem 2.7
• proof
• Definition 2.8