On injective dimension of the conormal module

Abstract

Let $Q$ be a Gorenstein local ring, let $I$ be a proper ideal of $Q$, and set $R=Q/I$. We investigate when finite injective dimension of the conormal module $I/I^2$ forces $I$ to be generated by a regular sequence. We prove this implication for every ideal in the linkage class of a complete intersection. When $Q$ is regular, we also establish it under several numerical conditions in terms of the number of generators of $I$ and the type of $R$. For normal domain quotients, we show that finite injective dimension of $I/I^2$ forces $(\mathrm{ht}(I)+1)[\omega_R]=0$ in $\operatorname{Cl}(R)$, yielding the desired conclusion when the divisor class group is torsionfree. In the graded setting, we further prove the implication for all monomial ideals in polynomial rings. We also obtain a converse to Kunz's description of the first Koszul homology of an almost complete intersection.