Strict local rings

Abstract

In this paper we introduce the notion of a strict local ring. A local Cohen-Macaulay ring ( B , m ) (B,m) is called strict if, whenever a local ring ( A , n ) (A,n) specializes by a regular sequence to B B , then the associated graded ring g r n ( A ) {\text {g}}{{\text {r}}_n}(A) is Cohen-Macaulay. We show that an artinian graded algebra B B is strict if for the graded cotangent module we have T 1 ( B / k , B ) r = 0 for ν > − 1 {T^1}{(B/k,B)_r} = 0{\text {for }}\nu > - 1 . Various examples are considered where this condition holds. In particular, with this method we reprove a result of J. Sally [6].