Strict local rings

Abstract:

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