Relative Brauer relations of abelian p-groups

Non-abelian reciprocity laws and higher Brauer-Manin obstructions

We reinterpret Kim's non-abelian reciprocity maps for algebraic varieties as obstruction towers of mapping spaces of etale homotopy types, removing technical hypotheses such as global basepoints and cohomological constraints. We then extend the theory by considering alternative natural series of extensions, one of which gives an obstruction tower whose first stage is the Brauer--Manin obstruction, allowing us to determine when Kim's maps recover the Brauer-Manin locus. A tower based on relative completions yields non-trivial reciprocity maps even for Shimura varieties; for the stacky modular c…

On saturated fusion systems and Brauer indecomposability of Scott modules

Let $p$ be a prime number, $G$ a finite group, $P$ a $p$-subgroup of $G$ and $k$ an algebraically closed field of characteristic $p$. We study the relationship between the category $\Ff_P(G)$ and the behavior of $p$-permutation $kG$-modules with vertex $P$ under the Brauer construction. We give a sufficient condition for $\Ff_P(G)$ to be a saturated fusion system. We prove that for Scott modules with abelian vertex, our condition is also necessary. In order to obtain our results, we prove a criterion for the categories arising from the data of $(b, G)$-Brauer pairs in the sense of Alperin-Brou…

K(1)-局部谱的相对 Brauer 群

The relative Brauer group of $K(1)$-local spectra

本文利用有限 Galois 下降计算了 K(1)-局部范畴相对于 Morava E-理论的 Brauer 群，给出奇素数及素数 2 处的生成元描述，并研究其在 KO 的 Brauer 群中的像。

有墙 Brauer 范畴与 $\mathrm{IA}_n$ 的稳定上同调

The walled Brauer category and stable cohomology of $\mathrm{IA}_n$

本文借助有墙 Brauer 范畴改造 Kupers–Randal-Williams 方法，在稳定有限维假设下刻画 IA-自同构群稳定有理上同调的代数部分至次数 Q+1，并应用于稳定 Albanese（共）同调。

A Dixmier-Douady Theory for strongly self-absorbing C*-algebras II: the Brauer group

We have previously shown that the isomorphism classes of orientable locally trivial fields of $C^*$-algebras over a compact metrizable space $X$ with fiber $D\otimes \mathbb{K}$, where $D$ is a strongly self-absorbing $C^*$-algebra, form an abelian group under the operation of tensor product. Moreover this group is isomorphic to the first group $\bar{E}^1_D(X)$ of the (reduced) generalized cohomology theory associated to the unit spectrum of topological K-theory with coefficients in $D$. Here we show that all the torsion elements of the group $\bar{E}^1_D(X)$ arise from locally trivial fields …

The Relative Burnside Kernel - The Elementary Abelian Case

We give a conjectural description for the kernel of the map assigning to each finite $\mathbb Z_p$-free $G\times\mathbb Z_p$-set its rational permutation module where G is a finite p-group. We prove that this conjecture is true when G is an elementary abelian p-group or a cyclic p-group.