Stability of annihilators of cohomology and closed subsets defined by Jacobian ideals

Let $R$ be a commutative Noetherian ring of dimension $d$. In this paper, we first show that some power of the cohomology annihilator annihilates the $(d+1)$-th Ext modules for all finitely generated modules when either $R$ admits a dualizing complex or $R$ is local. Next, we study the Jacobian ideal of affine algebras over a field and equicharacteristic complete local rings, and characterize the equidimensionality of the ring in terms of the singular locus and the closed subsets defined by the cohomology annihilator and the Jacobian ideal.