A finite module M over a noetherian local ring R is said to be Gorenstein if Ext(k, M) = 0 for all i 6= dimR. A endomorphism φ : R → R of rings is called contracting if φ(m) ⊆ m for some i ≥ 1. Letting R denote the R-module R with action induced by φ, we prove: A finite R-module M is Gorenstein if and only if HomR( R,M) ∼= M and ExtiR( R,M) = 0 for 1 ≤ i ≤ depthR.

Let Λ and Γ be left and right noetherian rings and ΛU a Wakamatsu tilting module with Γ = End(ΛT ). We introduce a new definition of U -dominant dimensions and show that the U -dominant dimensions of ΛU and UΓ are identical. We characterize k-Gorenstein modules in terms of homological dimensions and the property of double homological functors preserving monomorphisms. We also study a generaliza...

Abstract We show that finitely generated Cox rings are Gorenstein. This leads to a refined characterization of varieties Fano type: they exactly those projective with Gorenstein canonical quasicone ring. then for type and Kawamata log terminal quasicones X , iteration is finite factorial master In particular, even if the class group has torsion, we can express such as quotients by solvable redu...