An artin algebra is called CM-free provided that all its finitely generated Gorenstein projective modules are projective. We show that a connected artin algebra with radical square zero is either self-injective or CM-free. As a consequence, we prove that a connected artin algebra with radical square zero is Gorenstein if and only if its valued quiver is either an oriented cycle with the trivial...

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...

Let R = k[x 1 ,. .. , xn] and let I be the ideal of n + 1 generically chosen forms of degrees d 1 ≤ · · · ≤ d n+1. We give the precise graded Betti numbers of R/I in the following cases: • n = 3; • n = 4 and 5 i=1 d i is even; • n = 4, 5 i=1 d i is odd and d 2 + d 3 + d 4 < d 1 + d 5 + 4; • n is even and all generators have the same degree, a, which is even; • (n+1 i=1 d i) − n is even and d 2 ...

Foreman [For13] proved the Duality Theorem, which gives an algebraic characterization of certain ideal quotients in generic extensions. As an application he proved that generic supercompactness of ω1 is preserved by any proper forcing. We generalize portions of Foreman’s Duality Theorem to the context of generic extender embeddings and ideal extenders (as introduced by Claverie [Cla10]). As an ...

We prove the following statement (theorem 2.1). Let V/P be a birationally rigid Gorenstein Mori fibration on del Pezzo surfaces. If a Mori fibration W is birational to V , then either they are isomorphic, or W is non-Gorenstein. The result was obtained during my stay at the Max-Planck-Institut für Mathematik in Bonn. I would like to use this opportunity and thank the Directors and the staff of ...

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...

In this paper we describe the facets cone associated to transversal polymatroid presented by A = {{1, 2}, {2, 3}, . . . , {n − 1, n}, {n, 1}}. Using the Danilov-Stanley theorem to characterize the canonicale module, we deduce that the base ring associated to this polymatroid is Gorenstein ring. Also, starting from this polymatroid we describe the transversal polymatroids with Gorenstein base ri...