2 - Tubes

Summary

Up to the next to last chapter, I will mainly follow Berthelot's preprint on rigid cohomology ([13] and [8]).

Some rigid geometry

Ultrametric fields

We assume that the reader is familiar with rigid analytic geometry such as in [21] or [45]. We fix a complete ultrametric field K (complete for a non trivial non archimedean absolute value) with valuation ring V, maximal ideal m and residue field k. We will denote by π a non zero, non invertible element of V. Finally, when we pick up a positive real number, it is always assumed to live in |K*| ⊗ Q ⊂ R_>0.

We will be mainly interested in the discrete valuation case. Then, unless otherwise specified, we will implicitly assume that π is a uniformizer. In the equicharacteristic situation, we have K ≃ k((t)). Actually, we will be mainly concerned with the mixed characteristic case, namely CharK = 0 and Chark = p > 0.

Note that, starting with a field k of characteristic p, there always exists a Cohen ring for k. This is a minimal discrete valuation ring V of mixed characteristic with uniformizer p and residue field k. Actually, any complete discrete valuation ring with residue field k contains a Cohen ring of k.