In this paper we shall apply modules of generalized fractions to extend the notion of the grade of an ideal, and to obtain characterizations of Gorenstein rings.

Let P and Q be finite partially ordered sets with |P | = |Q| = d, and C(P ) ⊂ R and C(Q) ⊂ R their chain polytopes. The twinned chain polytope of P and Q is the normal Gorenstein Fano polytope Γ(C(P ),−C(Q)) ⊂ R which is the convex hull of C(P )∪(−C(Q)). In this paper, we study combinatorial properties of twinned chain polytopes. First, we will give the formula of the volume of twinned chain po...

Given a homomorphism of commutative noetherian rings R → S and an S–module N , it is proved that the Gorenstein flat dimension of N over R, when finite, may be computed locally over S. When, in addition, the homomorphism is local and N is finitely generated over S, the Gorenstein flat dimension equals sup {m ∈ Z | Torm(E,N) 6= 0}, where E is the injective hull of the residue field of R. This re...

In this note we establish a (non-trivial) lower bound on the degree two entry h2 of a Gorenstein h-vector of any given socle degree e and any codimension r. In particular, when e = 4, that is for Gorenstein h-vectors of the form h = (1, r, h2, r, 1), our lower bound allows us to prove a conjecture of Stanley on the order of magnitude of the minimum value, say f(r), that h2 may assume. In fact, ...

Contravariantly finite resolving subcategories of the category of finitely generated modules have been playing an important role in the representation theory of algebras. In this paper we study contravariantly finite resolving subcategories over commutative rings. The main purpose of this paper is to classify contravariantly finite resolving subcategories over a henselian Gorenstein local ring;...

R ring (always commutative and Noetherian) (R,m,k) local ring with maximal ideal m and k = R/m L,M,N, . . . R-modules (always finitely generated) M HomR(M,R), the dual of M D(M) the Auslander dual of M (Definition 2) σM : M wM∗∗ the natural evaluation map; KM = Ker(σM ), CM = Coker(σM ) G-dimR(M),G-dim(M) Gorenstein dimension of M (Definition 16) G-dim(M) <loc ∞ M has locally finite Gorenstein ...

Let [Formula: see text] be an extriangulated category with a proper class of text]-triangles. We study complete cohomology objects in by applying text]-projective resolutions and text]-injective coresolutions constructed text]. Vanishing detects finite dimension dimension. As consequence, we obtain some criteria for the validity Wakamatsu tilting conjecture give necessary sufficient condition v...

We use Quillen model structures to show a systematic method lift recollements of hereditary abelian categories their associated homotopy categories. To that end, we the notion adjoint triples and investigate transfers along pairs. Applications include liftings module derived counterpart, provide models for stable Gorenstein projective injective modules n-morphism over Iwanaga-Gorenstein rings.

We show that any quasi-Gorenstein deformation of a 3-dimensional Buchsbaum local ring with I-invariant 1 admits maximal Cohen-Macaulay module, provided it is quotient Gorenstein ring. Such class rings includes two instances unique factorization domains constructed by Marcel-Schenzel and Imtiaz-Schenzel, respectively. Apart from this result, motivated the small conjecture in prime characteristic...

We find new classes of non noetherian rings which have the same homological behavior that Gorenstein rings.