Related papers: Free resolutions over short Gorenstein local rings

Hochster established the existence of a commutative noetherian ring $\tilde C$ and a universal resolution $U$ of the form $0\to \tilde C^{e}\to \tilde C^{f}\to \tilde C^{g}\to 0$ such that for any commutative noetherian ring $S$ and any…

Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring with canonical module that is generically Gorenstein. In this paper, I prove isomorphisms relating the minimal MCM approximations and minimal FID hulls of modules constructed from a…

We give counterexamples to the following conjecture of Auslander: given a finitely generated module $M$ over an Artin algebra $\Lambda$, there exists a positive integer $n_M$ such that for all finitely generated $\Lambda$-modules $N$, if…

A sparse generic matrix is a matrix whose entries are distinct variables and zeros. Such matrices were studied by Giusti and Merle who computed some invariants of their ideals of maximal minors. In this paper we extend these results by…

In 1978, Stanley constructed an example of an Artinian Gorenstein (AG) ring $A$ with non-unimodal $H$-vector $(1,13,12,13,1)$. Migliore-Zanello later showed that for regularity $r=4$, Stanley's example has the smallest possible codimension…

Let I be a finitely supported complete m-primary ideal of a regular local ring (R, m). A theorem of Lipman implies that I has a unique factorization as a *-product of special *-simple complete ideals with possibly negative exponents for…

In this paper we study minimal free resolutions of some classes of monomial ideals. we first give a sufficient condition to check the minimality of the resolution obtained by the mapping cone. Using it, we obtain the Betti numbers of…

Quasi-socle ideals, that is the ideals $I$ of the form $I= Q : \mathfrak{m}^q$ in a Noetherian local ring $(A, \mathfrak{m})$ with the Gorenstein tangent cone $\mathrm{G}(\mathfrak{m}) = \bigoplus_{n \geq…

Let \fa be an ideal of a commutative Noetherian ring R and M and N two finitely generated R-modules. Let \cd_{\fa}(M,N) denote the supremum of the i's such that H^i_{\fa}(M,N)\neq 0. First, by using the theory of Gorenstein homological…

Let $R$ be a commutative Noetherian ring, $M$ a finitely generated $R$-module and $n$ be a non-negative integer. In this article, it is shown that there is a finitely generated submodule $N_i$ of $H_{\frak a}^i(M)$ such that $\dim{\rm Supp…

Let X be a zero-dimensional scheme in P1 \times P1. Then X has a minimal free resolution of length 2 if and only if X is ACM. In this paper we determine a class of reduced schemes whose resolutions, similarly to the ACM case, can be…

The main achievement of this paper is to provide a structure theorem for Artinian, Gorenstein local rings with the property that the square of the maximal ideal is generated by two elements. The moduli problem for this class of local…

In this paper, we study Gorenstein injective, projective, and flat modules over a Noetherian ring $R$. For an $R$-module $M$, we denote by ${\rm Gpd}_RM$ and ${\rm Gfd}_R M$ the Gorenstein projective and flat dimensions of $M$,…

Let $(R,\mathfrak{m},\Bbbk)$ be a regular local ring of dimension 3. Let $I$ be a Gorenstein ideal of $R$ of grade 3. It follows from a result of Buchsbaum and Eisenbud that there is a skew-symmetric matrix of odd size such that $I$ is…

Let $ R $ be a Cohen-Macaulay local ring. We prove that the $ n $th syzygy module of a maximal Cohen-Macaulay $ R $-module cannot have a semidualizing direct summand for every $ n \ge 1 $. In particular, it follows that $ R $ is Gorenstein…

We investigate homological and depth-theoretic properties of finitely generated modules of finite projective dimension over Noetherian local rings. A central theme is the study of criteria for freeness and reflexivity derived from the…

Let $M$ be a finitely generated graded module over a Noetherian homogeneous ring $R = \bigoplus_{n \in \mathbb{N}_0}R_n$. For each $i \in \mathbb{N}_0$ let $H^i_{R_{+}}(M)$ denote the $i$-th local cohomology module of $M$ with respect to…

Let $k$ be a field with characteristic zero, $R$ be the ring $k[x_1, \cdots, x_n]$ and $I$ be a monomial ideal of $R$. We study the Artinian local algebra $R/I$ when considered as an $R$-module $M$. We show that the largest reduced…

Let R be a commutative Noetherian ring, I and J ideals of R and M a finitely generated R-module. Let F be a covariant R-linear functor from the category of finitely generated R-modules to itself. We first show that if F is coherent, then…

We say that a local ring $R$ is good, in the sense of Roos, if all finitely generated $R$-modules have rational Poincar\'e series that share a common denominator; otherwise, $R$ is said to be bad. An important class of good rings is the…