Indecomposable Direct Summands of Cohomologies of Curves

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. Investigating those indecomposable modules could determine whether describing modular representations arising from cohomologies of curves is a “wild” problem. We show that usually there are infinitely many such possible summands.