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Stacks of group representations

Paul Balmer — 2015

Journal of the European Mathematical Society

We start with a small paradigm shift about group representations, namely the observation that restriction to a subgroup can be understood as an extension-of-scalars. We deduce that, given a group $G$ , the derived and the stable categories of representations of a subgroup $H$ can be constructed out of the corresponding category for $G$ by a purely triangulated-categorical construction, analogous to étale extension in algebraic geometry. In the case of finite groups, we then use descent methods to investigate...

Modular representations of finite groups with trivial restriction to Sylow subgroups

Paul Balmer — 2013

Journal of the European Mathematical Society

Let $k$ be a field of characteristic $p$ . Let $G$ be a finite group of order divisible by $p$ and $P$ a $p$ -Sylow subgroup of $G$ . We describe the kernel of the restriction homomorphism $T (G) \to T (P)$ , for $T (-)$ the group of endotrivial representations. Our description involves functions $G \to k^{\times}$ that we call weak $P$ -homomorphisms. These are generalizations to possibly non-normal $P \leq G$ of the classical homomorphisms $G / P \to k^{\times}$ appearing in the normal case.

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Stacks of group representations

Modular representations of finite groups with trivial restriction to Sylow subgroups

A Gersten–Witt spectral sequence for regular schemes

Geometric description of the connecting homomorphism for Witt groups.

The Gersten conjecture for Witt groups in the equicharacteristic case.