Groups of order \(p^3\) are mixed Tate (Q6618797)

The study of finite \(p\)-groups is important for understanding the structure of finite groups, particularly through the application of Sylow's theorems. These groups, as building blocks of all finite groups, help reveal important properties of group actions, cohomology and representations. Investigating \(p\)-groups provides key insights into the behavior of more general finite groups and their associated algebraic structures.\N\NThe author observed from many research works spanning decades of efforts, the following results:\N\begin{itemize}\N\item There exist groups of order \(p^7\) and \(p^9\) which are not \textit{stably rational}.\N\item Every \(p\)-group \(G\) of order \(\leq p^4\) or \(2\)-group of order \(\leq 2^5\) and every \(G\)-representation \(V\), the quotient \(V/\! /G\) is \textit{rational}. And this is stronger than saying that \(BG\) is \textit{stably rational}.\N\item Every \(p\)-group \(G\) of order \(\leq p^4\) or \(2\)-group of order \(\leq 2^5\) has the \textit{weak Chow-Künneth property}.\N\item Every \(p\)-group \(G\) of order \(\leq p^4\) or \(2\)-group of order \(\leq 2^5\) has \textit{trivial unramified cohomology}.\N\end{itemize}\NMost recently, \textit{B. Totaro} [Geom. Topol. 20, No. 4, 2079--2133 (2016; Zbl 1375.14027)]\Nproved the following result: ``If \(G\) is mixed Tate, then \(BG\) is stably rational, satisfies the weak Chow-Künneth property and has trivial unramified cohomology.'' This raises the question: Are any of these properties of finite groups equivalent?\N\NIn addition to the result shown by Totaro [loc. cit.] which states that every abelian \(p\)-group is mixed Tate, the author recognized the importance of further studying these \(p\)-groups and stated: ``It's worth investigating whether all \(p\)-groups of order \(\leq p^4\) or all 2-groups of order \(\leq 2^5\) are actually mixed Tate.''\N\NAs a proposition, the main result of the paper (Theorem 1.1) aims to prove that every \(p\)-group \(G\) of order \(p^3\) is mixed Tate by showing \(M^c(BG)\) is mixed Tate and thus groups of order \(p^3\) are mixed Tate. However, all of this reduces to a problem of showing a linear scheme for the quotient \(V/\! /G\), where \(G\) is a non-abelian \(p\)-group of order \(p^3\) (with \(p\) an odd prime) and \(V\) is a faithful irreducible representation of \(G\) (Theorem 3.3).\N\NThe author finally shows that \(V/\! /G\) is a linear scheme, specifically for the only two non-abelian \(p\)-groups of order \(p^3\), where he begins by working in a more general framework. He assumes that \(G\) is a non-abelian \(p\)-group of order \(\leq p^4\) and has a faithful irreducible representation \((\rho, V)\) of dimension (degree) \(p\), from which the group structure is deduced: there exists an abelian \(p\)-group \(N\) of order \(\leq p^3\), such that \(N \unlhd G\), \(Z(G) \subset N\) and \(G\) can be written as a semi-direct product \(N \rtimes M\), with \(M \cong \mathbb{Z}/p\). Furthermore, \(Z(G)\) is cyclic. Consequently, it follows that any group of the form \(\mathbb{Z}/p^s \rtimes \mathbb{Z}/p\) is mixed Tate, which includes some \(p\)-groups of order \(p^4\).\N\NIn proving that \(M(BG)\) is mixed Tate if and only if \(M^c(BG)\) is mixed Tate for a finite group \(G\) over a field of characteristic zero, the author follows the same strategy as with the first result, reducing the problem and aiming to prove that \(M(BG)\) is mixed Tate if and only if \(M(\mathrm{GL}(n)/G)\) is mixed Tate for a faithful representation \(G \rightarrow \mathrm{GL}(n)\), thanks to a result by \textit{B. Totaro} [Geom. Topol. 20, No. 4, 2079--2133 (2016; Zbl 1375.14027)]. The author then works with a more general case that includes a principal \N\(\mathrm{GL}(n)\)-bundle and further extends the results to \(\mathbb{G}_m\)-bundles.