Regular sequences, projective dimension and criteria for regularity of local rings

Abstract

Foxby proves the following proposition (Math. Z. 132 (1973)). Let ( R , m ) (R,\mathfrak {m}) be a noetherian local ring and M M any finitely generated R R -module such that the projective dimension of M / a M M/\mathfrak {a}M is finite for all ideals a \mathfrak {a} of finite projective dimension. Then either M M is free or R R is regular local. In this article we prove that the conclusion holds if we restrict only to ideals generated by regular sequences, with the empty sequence being interpreted as the zero ideal.