Let (R,m) denote an n-dimensional Gorenstein ring. For an ideal I ⊂ R of height c we are interested in the endomorphism ring B = HomR(H c I (R),H c I (R)). It turns out that B is a commutative ring. In the case of (R, m) a regular local ring containing a field B is a Cohen-Macaulay ring. Its properties are related to the highest Lyubeznik number l = dimk Ext d R(k, H c I (R)). In particular R ≃...

We develop a theory of G-dimension over local homomorphisms which encompasses the classical theory of G-dimension for finitely generated modules over local rings. As an application, we prove that a local ring R of characteristic p is Gorenstein if and only if it possesses a nonzero finitely generated module of finite projective dimension that has finite G-dimension when considered as an R-modul...

Let R be a commutative noetherian local ring and consider the set of isomorphism classes of indecomposable totally reflexive R-modules. We prove that if this set is finite, then either it has exactly one element, represented by the rank 1 free module, or R is Gorenstein and an isolated singularity (if R is complete, then it is even a simple hypersurface singularity). The crux of our proof is to...

Let R be a left and right Noetherian ring. We introduce the notion of the torsionfree dimension of finitely generated R-modules. For any n 0, we prove that R is a Gorenstein ring with self-injective dimension at most n if and only if every finitely generated left R-module and every finitely generated right R-module have torsionfree dimension at most n, if and only if every finitely generated le...

We introduce and investigate the notion of GC -projective modules over (possibly non-noetherian) commutative rings, where C is a semidualizing module. This extends Holm and Jørgensen’s notion of C-Gorenstein projective modules to the non-noetherian setting and generalizes projective and Gorenstein projective modules within this setting. We then study the resulting modules of finite GC-projectiv...

In this paper, we explore the relation between index of reducibility and Hilbert coefficients in local rings. Consequently, main result study provides a characterization sequentially Cohen-Macaulay ring terms its non-parameter ideals. As corollaries to theorem, obtain characterizations Gorenstein/Cohen-Macaulay Chern

Let R be a general ring. Duality pairs of R-modules were introduced by Holm-Jørgensen. Most examples satisfy further properties making them what we call semi-complete duality in this paper. We attach relative theory Gorenstein homological algebra to any given pair