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

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.

Disclosure

“is GL-coherent) All tensor products ⊗ are over k unless denoted otherwise. Acknowledgements. The author thanks Steven Sam for helpful discussions and Kent Vashaw for helpful references. Disclosure of LLM use. The author used Claude Opus 4.8 and Fable 5 to revise the paper; these models identified gaps in the proofs of some results and errors in some statements that substantially changed the mathematical meaning; all such issues were corrected by the author.”

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2026-08-23T19:17:08.404566Z

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