Arithmetic of normal Rees algebras

The stable set of associated prime ideals of a polymatroidal ideal

“…Proof. It is a well-known fact that R(I) is a normal ring if and only if I is a normal ideal (see [15,Proposition 2.1.2]). By definition, I is normal if all powers of I are integrally closed.…”

Serre’s condition 𝑅_{𝑘} for associated graded rings

“…also [6], [4], [5]). Recall that a Noetherian ring S is said to be quasi-Gorenstein if for every maximal ideal m of S, S m is the canonical module of S m .…”

“…Another immediate consequence of our main theorem is a rather surprising result by Huneke, Simis, and Vasconcelos to the effect that if the associated graded ring of a prime ideal of finite projective dimension is reduced, then it is already a domain ( [8]) (Corollary 3.1). We use this fact in turn to investigate the connection between the reducedness and the quasi-Gorenstein property of associated graded rings (Theorem 3.2), a theme that goes back to earlier work by Hochster and by Herzog, Simis, and Vasconcelos ( [6], [4], [5]). …”

Canonical modules of Rees algebras

“…Its proof relies on the fact that the formation of the canonical module commutes with subintersections in important cases. As an application we treat the classical determinantal ideals and the corresponding algebras of minors.A considerable part of Wolmer Vasconcelos' work has been devoted to Rees algebras, in particular to their divisorial structure and the computation of the canonical module (see [16][17][18]23,28]). In this paper we give a generalization of the formula of Herzog and Vasconcelos [18] who have computed the canonical module under more special assumptions.…”

“…A considerable part of Wolmer Vasconcelos' work has been devoted to Rees algebras, in particular to their divisorial structure and the computation of the canonical module (see [HSV1,HSV2,HV,MV,Va]). In this paper we give a generalization of the formula of Herzog and Vasconcelos [HV] who have computed the canonical module under more special assumptions.…”