In joint work with Gebhard Böckle, we systematically study modules with a right action of Frobenius.
Up to nilpotent actions, the resulting category of Cartier Crystals satisfies some surprising and very strong finiteness conditions. The theory has connections to questions in commutative algebra (finiteness of local cohomology), birational geometry (test ideals) and number theory (constructible p-torsion sheaves, L-functions). In my talk I will explain the key features of the theory via simple examples, elaborate its connection with Grothendieck-Serre duality and discuss some of the aforementioned applications.