Finite homological dimension and primes associated to integrally closed ideals, II

Let R be a Noetherian local ring with the maximal ideal m. Assume that R contains ideals I and J satisfying the conditions (1) I ⊆ J, (2) I : m ⊈ J, and (3) J is m-full, that is mJ : x = J for some x ∈ m. Then the theorem says that R is a regular local ring, if the projective dimension pd_RI of I is finite. Let q = (a₁, a₂,..., a_t)R be an ideal in a Noetherian local ring R generated by a maximal R-regular sequence a₁, a₂,..., a_t and let q̄ denote the integral closure of q. Then, thanks to the theorem applied to the ideals I = q : m and J = q̄, it follows that I² = qI, unless A is a regular local ring. Consequences are discussed.