Finiteness with Coefficients via a Local Monodromy Theorem

Abstract

The main result of this chapter is that \(\mathscr {E}_K^\dagger \)-valued rigid cohomology \(H^i_\mathrm {rig}(X/\mathscr {E}_K^\dagger ,\mathscr {E})\) is finite dimensional for any smooth scheme \(X/\mathscr {E}_K^\dagger \) and any \(\mathscr {E}\in F\text {-}\mathrm {Isoc}^\dagger (X/\mathscr {E}_K^\dagger )\), and moreover the base change of these vector spaces to \(\mathscr {E}_K\) coincides with classical rigid cohomology. After introducing the appropriate notion of a dagger algebra in our context, the key point is to prove a relative version of the p-adic local monodromy theorem for \((\varphi ,\nabla )\)-modules over Rbba rings attached to these dagger algebras, which we do by descending the corresponding result from affinoid algebras over \(\mathscr {E}_K\). Once certain other properties of \(\mathscr {E}_K^\dagger \)-valued rigid cohomology have been established, such as excision, a Gysin isomorphism&c. have been established, the eventual proof of finite dimensionality for smooth varieties proceeds in the usual way. Base change is proved simultaneously, and this then allows us to deduce results such as a Kunneth formula from their counterparts over \(\mathscr {E}_K\).