Overview of Rigid Analytic Geometry

The idea is simple: we want to develop a theory of analytic manifolds and spaces over fields equipped with an arbitrary complete valuation. Of course, it is a standard fact that such a field must be either R, C, or a field with a nonarchimedean valuation, so what we really mean is that we want to develop a theory of nonarchimedean analytic spaces. Doing this näıvely (i.e., defining manifolds in the same way as one does over R or C) leads to serious difficulties, mostly due to the fact that the resulting topologies are totally disconnected, so we cannot use connectedness arguments and it is unclear how to develop useful cohomology theories. There are many ways of getting around this, and in this talk I’ll discuss the (chronologically) first method, due to Tate and generally known as rigid analytic geometry (other methods include the theories of Berkovich spaces and Huber’s adic spaces). Once and for all, fix a field k that is complete for a nonarchimedean valuation | · | (by definition, a valuation here is multiplicative and nontrivial). We can then consider the ring of integers