In this part of the book, we bring together the streams of differential algebra (from Part III) and difference algebra (from Part IV), realizing Dwork's fundamental insight that the study of differential modules on discs and annuli is greatly enhanced by the introduction of Frobenius structures.

This chapter sets the foundations for this study. First, we introduce the notion of a Frobenius structure on a differential module, with some examples. Then we consider the effect of Frobenius structures on the generic radius of convergence and obtain the fact that a differential module on a disc has a full basis of horizontal sections (“Dwork's trick”). We also show that the existence of a Frobenius structure does not depend on the particular choice of Frobenius lift; this independence plays an important role in rigid cohomology (Chapter 23).

Throughout Part V, Hypothesis 14.0.1 remains in force unless explicitly contravened. In particular, K will by default be a discretely valued complete nonarchimedean field.

Frobenius structures

We start with the basic compatibility between differential and difference structures.

Definition 17.1.1. Let R be a ring as in Definition 15.2.1. For M a finite free differential module over R, a Frobenius structure on M with respect to a Frobenius lift ϕ on R is an isomorphism Φ : ϕ*M ≅ M of differential modules.