We prove that Vopěnka's Principle implies for every class X of modules over any ring, the X-Gorenstein Projective (X-GP) is a precovering class. In particular, it not possible to (unless inconsistent) there ring which Ding Projectives (DP) or Gorenstein (GP) do form (Šaroch previously obtained this result GP, using different methods). The key innovation new “top-down” characterization deconstru...

Let Λ and Γ be artin algebras and ΛUΓ a faithfully balanced selforthogonal bimodule. In this paper, we first introduce the notion of k-Gorenstein modules with respect to ΛUΓ and then characterize it in terms of the U -resolution dimension of some special injective modules and the property of the functors Ext(Ext(−, U), U) preserving monomorphisms, which develops a classical result of Auslander....

Let Λ and Γ be artin algebras and ΛUΓ a faithfully balanced selforthogonal bimodule. In this paper, we first introduce the notion of k-Gorenstein modules with respect to ΛUΓ and then characterize it in terms of the U -resolution dimension of some special injective modules and the property of the functors Exti(Exti(−, U), U) preserving monomorphisms, which develops a classical result of Auslande...

In the present note we investigate projective varieties which are geometric models of binary symmetric phylogenetic 3-valent trees. We prove that these varieties have Gorenstein terminal singularities (with small resolution) and they are Fano varieties of index 4. Moreover any two such varieties associated to trees with the same number of leaves are deformation equivalent, that is, they are in ...

In [Pas15], we described the Minimal Model Program in the family of Q-Gorenstein projective horospherical varieties, by studying a family of polytopes defined from the moment polytope of an ample Q-Cartier Q-divisor of the variety we begin with. Here, we summarize the results of [Pas15] and we explain how to generalize them in order to describe the Log Minimal Model Program for pairs (X,∆) wher...