Related papers: Bounds on Gorenstein Dimensions and Exceptional Co…

Let (R,m,k) be a local ring. We establish a totally reflexive analogue of the New Intersection Theorem, provided for every totally reflexive R-module M, there is a big Cohen-Macaulay R-module B_M such that the socle of B_M\otimes_RM is…

Let $(A,\mathfrak{m})$ be an excellent Gorenstein local ring of dimension $d \geq 2$ which is an isolated singularity. Let $\widehat{A}$ denote the completion of $A$. If $G(A)$ is the Grothendieck group of $A$ then by $G(A)_\mathbb{Q}$ we…

In this paper we characterize the relative Gorenstein weak global dimension of the generalized Gorenstein $\mathrm{FP}_n$-flat $R$-modules and Projective Coresolved $\mathrm{FP}_n$-flat $R$-modules recently studied by S. Estrada, A. Iacob,…

Let $(R, \frak m)$ denote a local Cohen-Macaulay ring and $I$ a non-nilpotent ideal of $R$. The purpose of this article is to investigate Faltings' finiteness dimension $f_I(R)$ and equidimensionalness of certain homomorphic image of $R$.…

Given a bounded complex of finitely generated modules $M$ over a commutative noetherian local ring $R$, one assigns to it a variety, $\mathcal V_R(M)$, called the cohomological support variety of $M$ over $R$. The variety $\mathcal V_R(M)$…

Distinctive characteristics of Iwanaga--Gorenstein rings are typically understood through their intrinsic symmetry. We show that several of those that pertain to the Gorenstein global dimensions carry over to the one-sided situation, even…

Let $R$ be a commutative Noetherian ring. In this paper, we study those finitely generated $R$-modules whose Cousin complexes provide Gorenstein injective resolutions. We call such a module a G-Gorenstein module. Characterizations of…

Let $R$ be a commutative Noetherian local ring of prime characteristic $p$ and $f:R\to R$ the Frobenius ring homomorphism. For $e\ge 1$ let $R^{(e)}$ denote the ring $R$ viewed as an $R$-module via $f^e$. Results of Peskine, Szpiro, and…

Let R be a local Noetherian domain of positive characteristic. A theorem of Hochster and Huneke (1992) states that if R is excellent, then the absolute integral closure of R is a big Cohen-Macaulay algebra. We prove that if R is the…

Starting from the notion of totally reflexive modules, we survey the theory of Gorenstein homological dimensions for modules over commutative rings. The account includes the theory's connections with relative homological algebra and with…

Let $(R,\mathfrak{m})$ be a commutative Noetherian local ring, $\mathfrak{a}$ be a proper ideal of $R$ and $M$ be an $R$-complex in $\mathrm{D}(R)$. We prove that if $M\in\mathrm{D}^f_\sqsubset(R)$ (respectively,…

We prove that if M, N are finite modules over a Gorenstein local ring R of codimension at most 4, then the vanishing of Ext^n_R(M,N) for n\gg 0 is equivalent to the vanishing of Ext^n_R(N,M) for n\gg 0. Furthermore, if the completion of $R$…

Let R be a commutative ring and S be an R-algebra. It is well-known that if N is an injective R-module, then Hom(S,N) is an injective S-module. The converse is not true, not even if R is a commutative noetherian local ring and S is its…

Let $R$ be a commutative noetherian ring, and let $C$ be a semidualizing $R$-module. In this paper, we study levels of bounded complexes of finitely generated $R$-modules with respect to the full subcategory $\mathsf{G}_{C}(R)$ consisting…

Let $S$ be a deeply embedded, equicharacteristic, Artinian Gorenstein local ring. We prove that if $R$ is a non-Gorenstein quotient of $S$ of small colength, then every totally reflexive $R$-module is free. Indeed, the second syzygy of the…

Let T be a commutative Noetherian local ring of dimension at least two and R=T[x_1,...,x_n] a polynomial ring in n variables over T. Consider R as a graded ring with deg T = 0 and deg x_i = 1 for all i. Let I=R_+ and f a homogeneous…

Let (R,m) -> (S,n) be a flat local homomorphism of excellent local rings. We investigate the conditions under which the weak or strong F-regularity of R passes to S. We show that is suffices that the closed fiber S/mS be Gorenstein and…

Let $A$ be a virtually Gorenstein algebra of finite CM-type. We establish a duality between the subcategory of compact objects in the homotopy category of Gorenstein projective left $A$-modules and the bounded Gorenstein derived category of…

New homotopy invariant finiteness conditions on modules over commutative rings are introduced, and their properties are studied systematically. A number of finiteness results for classical homological invariants like flat dimension,…

Let $R$ be a commutative Noetherian local ring. We prove that the finiteness of the injective dimension of a finitely generated $R$-module $C$ is determined by the existence of a Cohen--Macaulay module $M$ that satisfies an inequality…