Let $k$ be an algebraically closed field of characteristic $p$, and let ${\mathcal{O}}$ be either $k$ or its ring of Witt vectors $W(k)$. Let $G$ be a finite group and $B$ a block of ${\mathcal{O}} G$ with normal abelian defect group and abelian $p^{\prime}$ inertial quotient $L$. We show that $B$ is isomorphic to its second Frobenius twist. This is motivated by the fact that bounding Frobenius numbers is one of the key steps towards Donovan’s conjecture. For ${\mathcal{O}}=k$, we give an explicit description of the basic algebra of $B$ as a quiver with relations. It is a quantized version of the group algebra of the semidirect product $P\rtimes L$.
The Cohomological Brauer Group of a Torsion ${\mathbb{G}}_{m}$-Gerbe
Let $S$ be a scheme and let $\pi : \mathcal{G} \to S$ be a ${\mathbb{G}}_{m,S}$-gerbe corresponding to a torsion class $[\mathcal{G}]$ in the cohomological Brauer group ${\operatorname{Br}}^{\prime}(S)$ of $S$. We show that the cohomological Brauer group ${\operatorname{Br}}^{\prime}(\mathcal{G})$ of $\mathcal{G}$ is isomorphic to the quotient of ${\operatorname{Br}}^{\prime}(S)$ by the subgroup generated by the class $[\mathcal{G}]$. This is analogous to a theorem proved by Gabber for Brauer–Severi schemes.
more » « less- Award ID(s):
- 1646385
- NSF-PAR ID:
- 10123040
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- ISSN:
- 1073-7928
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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