Differential Modules and Ramification

Abstract

Differential modules and ramification is the subject of this chapter. With regard to the principal properties of differential modules, we refer to chapter 16 of Eisenbud’s book [63], and add some more results in section 1. After dealing with norms and traces in section 2, we introduce the notion of formally unramified and unramified extensions of rings R → S in section 3 and study, in particular, ramification of pairs of local rings in subsection 3.3, ramification of pairs of quasilocal rings in section 5. Under additional hypotheses for a ring extension R → S, the set of prime ideals of R which are unramified in S can be described as the closure in Spec(R) of an ideal, the Noether discriminant ideal, which is generated by discriminants; this is done in section 4. In the last section 6 we prove a theorem of Chevalley and a theorem of Zariski: Let k be a perfect field, and let A be a local k-algebra essentially of finite type. Then A is analytically irreducible and analytically normal These two results shall play an important role when proving the uniformization theorem VIII(6.9): in the proof of it we use namely VIII(6.4) and VIII(6.3) which rely on Chevalley’s result.