Some results in local rings on ramification in low codimension

Abstract

One way to study the behavior of the algebraic variety Spec A, where A is a normal domain, is to obtain finite maps f: Spec A -+ Spec R, where R is regular, and to investigate properties of Spec A relative to those of Spec R through the mapf. In case A is finitely generated over a field k one can obtain such a map f via Noether normalization (see [19, p. 911). In case A is local, complete, and contains a coefficient field k, one obtains an S by choosing (any) system of parameters xi, . . . . x, for A and then setting R=k[[x,, . . . . x,]] (see [19, pp. 210-2121; the case of unequal characteristic is slightly more delicate). Having constructed f: Spec A -+ Spec R by some means, the map f itself exerts a great deal of influence on the situation. In particular, the ramification which occurs in each fiber plays a central role in the interplay between properties of A and those of R. For example, should the map f be unramilied one concludes that A must be regular and the map f must be &tale. Moreover, the question of whether or not A/R is a ramified extension can be answered by computing ramification indices in codimension one. This is the essence of the theorem on “purity of branch loci” which can be found in [20, Sect. 411 or in [2]. When the regularity assumption on R is removed, then whether or not A/R is ramified cannot simply be detected in codimension one (e.g., see [lo, Example 16.5, p. 851). Nevertheless, one might expect to find “good” properties of R reflected in those of A in such a setting. For example, if R is a complete intersection (or perhaps just Gorenstein) and if A/R is unramified in codimension one, must A at least be Cohen-Macaulay? In Section 1 of this article we obtain sufficient conditions in Theorem 1.4 (and resulting properties) for a complete local normal domain A to be a