Related papers: Characterizing local rings via homological dimensi…

Let (R,m) be an n-dimensional regular local ring, essentially of finite type over a field of characteristic zero. In this paper we study the relationship between the singularities of the scheme defined by an m-primary ideal I of R and the…

Let R be a local complete ring. For an R-module M the canonical ring map R\to End_R(M) is in general neither injective nor surjective; we show that it is bijective for every local cohomology module M := H^h_I(R) if H^l_I(R) = 0 for every…

Let $\mathfrak{q}$ denote an ideal of a local ring $(A,\mathfrak{m})$. For a system of elements $\underline{a} = a_1,\ldots,a_t$ such that $a_i \in \mathfrak{q}^{c_i}, i = 1, \ldots,t,$ and $n \in \mathbb{Z}$ we investigate a subcomplex…

We provide new results on the vanishing of local cohomology modules supported at ideals of minors of matrices over arbitrary commutative Noetherian rings. In the process, we compute the local cohomology of rings of polynomials with integer…

Let M be a filtered module. Some properties of elements of M are "generic" in the following sense: (being open/stable) if an element z of M has a property P then any approximation of z has P; (being dense) any element of M is approximated…

Let $\frak a$ denote an ideal in a commutative Noetherian ring $R$ and $M$ a finitely generated $R$-module. In this paper, we introduce the concept of the cohomological dimension filtration $\mathscr{M} =\{M_i\}_{i=0}^c$, where $ c={\rm cd}…

Let $(R,\frak{m})$ be a Noetherian local ring, $I$ an ideal of $R$ and $N$ a finitely generated $R$-module. Let $k{\ge}-1$ be an integer and $ r=\depth_k(I,N)$ the length of a maximal $N$-sequence in dimension $>k$ in $I$ defined by M.…

Let R be a commutative Noetherian ring, a a proper ideal of R and M a finite R-module. It is shown that, if (R;m) is a complete local ring, then under certain conditions a contains a regular element on DR(Hc a(M)), where c = cd(a;M). A…

Let $(R, \mathfrak m)$ denote an $n$-dimensional Gorenstein ring. For an ideal $I \subset R$ of height $c$ we are interested in the endomorphism ring $B = \Hom_R(H^c_I(R), H^c_I(R)).$ It turns out that $B$ is a commutative ring. In the case…

Let $(R,\mathfrak{m})$ be a Noetherian local ring and $\widehat{R}$ its $\mathfrak{m}$-adic completion. We study the problem of determining when a finitely generated $\widehat{R}$-module arises from an $R$-module, i.e., when it is…

Let A be the local ring at a point of a normal complex variety with completion R. Srinivas has asked about the possible images of the induced map from Cl A to Cl R over all geometric normal domains A with fixed completion R. We use…

Let $R$ be a commutative noetherian ring. The $n$-semidualizing modules of $R$ are generalizations of its semidualizing modules. We will prove some basic properties of $n$-semidualizing modules. Our main result and example shows that the…

Let R be a ring and X = SH(R)-{0} be the set of all non-zero strongly hollow ideals (briefly, sh-ideals) of R. We first study the concept SH-topology and investigate some of the basic properties of a topological space with this topology. It…

Let $\mathfrak a$ denote an ideal of a local ring $(R, \mathfrak m).$ Let $M$ be a finitely generated $R$-module. There is a systematic study of the formal cohomology modules $\varprojlim \HH^i(M/\mathfrak a^nM), i \in \mathbb Z.$ We…

Let $A$ be a commutative Noetherian ring containing a field of characteristic zero. Let $R= A[X_1, \ldots, X_m]$ be a polynomial ring and $A_m(A) = A \langle X_1, \ldots, X_m, \partial_1, \ldots, \partial_m \rangle$ be the $m^{th}$ Weyl…

Let $A$ be a commutative Noetherian ring of characteristic zero and $R=A[X_1, \ldots, X_d]$ be a polynomial ring over $A$ with the standard $\mathbb{N}^d$-grading. Let $I\subseteq R$ be an ideal which can be generated by elements of the…

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 work concerns the linearity defect of a module $M$ over a noetherian local ring $R$, introduced by Herzog and Iyengar in 2005, and denoted by $\text{ld}_R M$. Roughly speaking, $\text{ld}_R M$ is the homological degree beyond which the…

Let $T$ be a complete local (Noetherian) ring of characteristic zero. We find necessary and sufficient conditions for $T$ to be the completion of a quasi-excellent local domain. In the case that $T$ contains the rationals, we provide…

For a pair of finitely generated modules $M$ and $N$ over a codimension $c$ complete intersection ring $R$ with $\ell(M\otimes_RN)$ finite, we pay special attention to the inequality $\dim M+\dim N \leq \dim R +c$. In particular, we develop…