Related papers: On certain DG-algebra resolutions

We generalize a theorem of Ding relating the generalized Loewy length $\text{g}\ell\ell(R)$ and index of a one-dimensional Cohen-Macaulay local ring $(R,\mathfrak{m},k)$. Ding proved that if $R$ is Gorenstein, the associated graded ring is…

Let R be a (not necessarily local) Noetherian ring and M a finitely generated R-module of finite dimension d. Let \fa be an ideal of R and \fM denote the intersection of all prime ideals \fp in Supp_RH^d_{\fa}(M). It is shown that…

This paper investigates when local cohomology modules have an annihilator that does not depend on the choice of an ideal. Takahashi classified the dominant resolving subcategories of the category of finitely generated modules over a…

In this paper, we continue the study of cominimaxness modules with respect to an ideal of a commutative Noetherian ring (cf. \cite{ANV}), and Bass numbers of local cohomology modules. Let $R$ denote a commutative Noetherian local ring and…

Let $(A,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d \geq 2$ with infinite residue field and let $I$ be an $\mathfrak{m}$-primary ideal. For $0 \leq i \leq d$ let $I_i$ be the $i^{th}$-coefficient ideal of $I$. Also let…

Let $\A$ be a $\k$-algebra where $\k$ a field of arbitrary characteristic, and let $\mathscr{A}_\k$ be a full subcategory of $\A$-Mod, the abelian category of left $\A$-modules.Following M. Kleiner and I. Reiten, $\mathscr{A}_\k$ is {\it…

Let $(A,\mathfrak{m})$ be a regular local ring of dimension $d \geq 1$, $I$ an $\mathfrak{m}$-primary ideal. Let $N$ be a non-zero finitely generated $A$-module. Consider the functions \[ t^I(N, n) = \sum_{i = 0}^{ d}\ell(\text{Tor}^A_i(N,…

In Commutative Algebra structure results on minimal free resolutions of Gorenstein modules are of classical interest. We define Gorenstein modules of finite length over the weighted polynomial ring via symmetric matrices in divided powers.…

Let $R$ be a noetherian ring, $\fa$ an ideal of $R$, and $M$ an $R$--module. We prove that for a finite module $M$, if $\LC^{i}_{\fa}(M)$ is minimax for all $i\geq r\geq 1$, then $\LC^{i}_{\fa}(M)$ is artinian for $i\geq r$. A Local-global…

Let $R$ be a commutative Noetherian ring, $I$ an ideal of $R$ and $M$ a finitely generated $R$-module with $\dim_R(M)=d$. Denote by $\depth_R(I,M)$ the depth of $M$ in $I$. In \cite{HT}, C. Huneke and V. Trivedi proved that if $R$ is a…

We prove that if M is a finitely-generated module of dimension d with finite local cohomologies over a Noetherian local ring, and if the ith local cohomology module of M is zero unless i = d, i = 0, and i = r for some r strictly between 0…

Let $R$ be a local ring with residue field $k$ and $M$, $N$ be finitely generated modules over $R$. It is well known that $Tor^R_i(M, N) = 0$ for $i \gg 0$ if $pd_R(M) < \infty$ or $pd_R(N) < \infty$. The ring $R$ is said to satisfy the…

In this paper, an algebraic theory for local rings of finite embedding dimension is developed. Several extensions of (Krull) dimension are proposed, which are then used to generalize singularity notions from commutative algebra. Finally,…

A classical result of Micali asserts that a Noetherian local ring is regular if and only if the Rees algebra of its maximal ideal is defined by an ideal of linear forms. In this case, this defining ideal may be realized as a determinantal…

Let $A$ be an Iwanaga-Gorenstein ring. Enomoto conjectured that a self-orthogonal $A$-module has finite projective dimension. We prove this conjecture for $A$ having the property that every indecomposable non-projective maximal…

Let $\frak a$ be an ideal of a commutative noetherian ring $R$ with unity and $M$ an $R$-module supported at $\V(\fa)$. Let $n$ be the supermum of the integers $i$ for which $H^{\fa}_i(M)\neq 0$. We show that $M$ is $\fa$-cofinite if and…

This is the author's diploma thesis. In the first part of the thesis the algebra structure on the Ext-spaces Ext^k(M(x), M(y)) of Verma modules M(x) and M(y) in the parabolic category O for the case of the parabolic subalgebras gl(n) x…

Avramov and Buchweitz proved that for finitely generated modules $M$ and $N$ over a complete intersection local ring $R$, $\Ext^i_R(M,N)=0$ for all $i\gg 0$ implies $\Ext^i_R(N,M)=0$ for all $i\gg 0$. In this note we give some…

Let $\fa$ be an ideal of a local ring $(R,\fm)$ and $M$ a finitely generated $R$-module. We investigate the structure of the formal local cohomology modules ${\vpl}_nH^i_{\fm}(M/\fa^n M)$, $i\geq 0$. We prove several results concerning…

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…