Groups of order $p^3$ are mixed Tate

TL;DR

This paper proves that for groups of order p^3 over certain fields, the motive of their classifying space is a mixed Tate motive, linking the properties of motives M(BG) and M^c(BG).

Contribution

It establishes that all groups of order p^3 have mixed Tate motives for their classifying spaces over specific fields, extending the understanding of motives in algebraic geometry.

Findings

M(BG) is a mixed Tate motive for groups of order p^3 over certain fields.

M(BG) is mixed Tate if and only if M^c(BG) is mixed Tate over characteristic zero fields.

The result applies to fields containing primitive p^2-th roots of unity and characteristic not p.

Abstract

A natural place to study the Chow ring of the classifying space $BG$, for $G$ a linear algebraic group, is Voevodsky's triangulated category of motives, inside which Morel and Voevodsky, and Totaro have defined motives $M(BG)$ and $M^c(BG)$, respectively. We show that, for any group $G$ of order $p^3$ over a field of characteristic not $p$ which contains a primitive $p^2$-th root of unity, the motive $M(BG)$ is a mixed Tate motive. We also show that, for a finite group $G$ over a field of characteristic zero, $M(BG)$ is a mixed Tate motive if and only $M^c(BG)$ is a mixed Tate motive.