Generalized Artin and Brauer induction for compact Lie groups

实-全局等变Segal-Becker分裂，显式Brauer诱导，和全局Adams操作

A Real-global equivariant Segal--Becker splitting, explicit Brauer induction, and global Adams operations

本文证明了全局等变同调论中一个分裂结果，同时细化了Segal-Becker分裂及其Real和等变推广，以及Boltje和Symonds的显式Brauer诱导。

A Topological Proof for a Version of Artin's Induction Theorem

We define a Euler characteristic $χ(X,G)$ for a finite cell complex $X$ with a finite group $G$ acting cellularly on it. Then, each $K_{i}(X)$ (a complex vector space with basis the $i$-cells of $X$) is a representation of $G$, and we define $χ(X,G)$ to be the alternating sum of the representations $K_{i}(X)$, as elements of the representation ring $R(G)$ of $G$. By adapting the ordinary proof that the alternating sum of the dimensions of the chain complexes is equal to the alternating sum of the dimensions of the homology groups, we prove that there is another definition of $χ(X,G)$ with the …

有限约化群的Brauer对的单纯复形

The simplicial complex of Brauer pairs of a finite reductive group

本文研究有限约化群的Brauer对偏序集诱导的单纯复形，明确其在定义特征下与Tits building同伦型一致，在非定义特征下与广义Harish-Chandra理论确定的复形同伦等价，拓展了Brown复形相关研究。

Brauer groups and étale cohomology in derived algebraic geometry

In this paper, we study Azumaya algebras and Brauer groups in derived algebraic geometry. We establish various fundamental facts about Brauer groups in this setting, and we provide a computational tool, which we use to compute the Brauer group in several examples. In particular, we show that the Brauer group of the sphere spectrum vanishes, and we use this to prove two uniqueness theorems for the stable homotopy category. Our key technical results include the local geometricity, in the sense of Artin n-stacks, of the moduli space of perfect modules over a smooth and proper algebra, the étale l…

A remark on the higher torsion invariants for flat vector bundles with finite holonomy

We show that the Igusa-Klein topological torsion and the Bismut-Lott analytic torsion are equivalent for any flat vector bundle whose holonomy is a finite subgroup of $\mathrm{GL}_n(\mathbb{Q})$. Our proof uses Artin's induction theorem in representation theory to reduce the problem to the special case of trivial flat line bundles, which is a recent result of Puchol, Zhu and the second author. The idea of using Artin's induction theorem appeared in a paper of Ohrt on the same topic, of which our present work is an improvement.

Derived induction and restriction theory

Let $G$ be a finite group. To any family $\mathscr{F}$ of subgroups of $G$, we associate a thick $\otimes$-ideal $\mathscr{F}^{\mathrm{Nil}}$ of the category of $G$-spectra with the property that every $G$-spectrum in $\mathscr{F}^{\mathrm{Nil}}$ (which we call $\mathscr{F}$-nilpotent) can be reconstructed from its underlying $H$-spectra as $H$ varies over $\mathscr{F}$. A similar result holds for calculating $G$-equivariant homotopy classes of maps into such spectra via an appropriate homotopy limit spectral sequence. In general, the condition $E\in \mathscr{F}^{\mathrm{Nil}}$ implies strong …