Indecomposable direct summands of cohomologies of curves

TL;DR

This paper investigates the indecomposable summands of cohomologies of smooth projective curves with group actions, revealing that infinitely many such summands can occur, especially in the context of certain covers.

Contribution

It provides a detailed analysis of the building blocks of cohomologies of curves with group actions, demonstrating the abundance of possible indecomposable summands.

Findings

Infinitely many indecomposable summands can occur in cohomologies of curves with group actions.

Complete description of cohomologies for a family of bp imes bp-covers.

Highlights the complexity of modular representations arising from algebraic varieties with group actions.

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.