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arXiv · 2608.00982

GL-algebras in positive characteristic III: the divided power algebra

Karthik Ganapathy
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Abstract

In this paper, we study GL-equivariant modules over the infinite-variable divided power algebra $D = \text{Div}(k^{\infty})$ with $k$ an algebraically closed field of characteristic $p > 0$. Unlike previously analyzed GL-algebras, the divided power algebra is not noetherian or even finitely generated. We show that $D$ is GL-coherent and prove a ``shift theorem'' for finitely presented $D$-modules. Using this, we obtain a (semi-infinite) semi-orthogonal decomposition of its bounded derived category with one piece corresponding to each Frobenius twist $D^{(r)}$ of $D$. Crucial to our approach is the fact that $D$ is a flat colimit of subalgebras which are GL-noetherian.

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Karthik Ganapathy. 2026-08-02. GL-algebras in positive characteristic III: the divided power algebra. https://arxiv.org/abs/2608.00982

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