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arXiv · 2410.03319

Indecomposable direct summands of cohomologies of curves

Jędrzej Garnek
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Abstract

Groups with a non-cyclic Sylow $p$-subgroup have too many representations over a field of characteristic~$p$ to describe them fully. A~natural question arises, whether the world of representations coming from algebraic varieties with a group action is as vast as the realm of all modular representations. In this article, we explore the possible ``building blocks'' (the indecomposable direct summands) of cohomologies of smooth projective curves with a group action. We show that usually there are infinitely many such possible summands. To prove this, we study a family of $\mathbb Z/p \times \mathbb Z/p$-covers and describe the cohomologies of the members of this family completely.

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Jędrzej Garnek. 2024-10-04. Indecomposable direct summands of cohomologies of curves. https://arxiv.org/abs/2410.03319

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