arXiv subjects

Shawn Baland

Publications and source records attributed to Shawn Baland.

3 recordsLinked to original sources

arXiv2017

The prime spectra of relative stable module categories

For a finite group $G$ and an arbitrary commutative ring $R$, Broué has placed a Frobenius exact structure on the category of finitely generated $RG$-modules by taking the exact sequences to be those that split upon restriction to the trivial subgroup. The corresponding stable category is then tensor triangulated. In this paper we examine the case $R=S/t^n$, where $S$ is a discrete valuation ring having uniformising parameter $t$. We prove that the prime ideal spectrum (in the sense of Balmer) of this `relative' version of the stable module category of $RG$ is a disjoint union of $n$ copies of that for $kG$, where $k$ is the residue field of $S$.

math.RT↗

arXiv2016

Comparisons between singularity categories and relative stable categories of finite groups

We consider the relationship between the relative stable category of Benson, Iyengar, and Krause and the usual singularity category for group algebras with coefficients in a commutative noetherian ring. When the coefficient ring is self-injective we show that these categories share a common, relatively large, Verdier quotient. At the other extreme, when the coefficient ring has finite global dimension, there is a semi-orthogonal decomposition, due to Poulton, relating the two categories. We prove that this decomposition is partially compatible with the monoidal structure and study the morphism it induces on spectra.

math.RT↗

arXiv2015

Modules of constant Jordan type, pullbacks of bundles and generic kernel filtrations

Let $kE$ denote the group algebra of an elementary abelian $p$-group of rank $r$ over an algebraically closed field of characteristic $p$. We investigate the functors $\mathcal{F}_i$ from $kE$-modules of constant Jordan type to vector bundles on $\mathbb{P}^{r-1}(k)$, constructed by Benson and Pevtsova. For a $kE$-module $M$ of constant Jordan type, we show that restricting the sheaf $\mathcal{F}_i(M)$ to a dimension $s-1$ linear subvariety of $\mathbb{P}^{r-1}(k)$ is equivalent to restricting $M$ along a corresponding rank $s$ shifted subgroup of $kE$ and then applying $\mathcal{F}_i$. In the case $r=2$, we examine the generic kernel filtration of $M$ in order to show that $\mathcal{F}_i(M)$ may be computed on certain subquotients of $M$ whose Loewy lengths are bounded in terms of $i$. More precise information is obtained by applying similar techniques to the $n$th power generic kernel filtration of $M$. The latter approach also allows us to generalise our results to higher ranks $r$.

math.RT↗