Related papers: A note on Deligne's formula

Let $R = k[x_1, \ldots, x_n]$ be a polynomial ring over a field $k$ of characteristic zero and $\cR$ be the formal power series ring $k[[x_1, \ldots, x_n]]$. If $M$ is a $\D$-module over $R$, then $\cR \otimes_R M$ is naturally a…

Let $R$ be an $F$-finite Noetherian regular ring containing an algebraically closed field $k$ of positive characteristic, and let $M$ be an $\F$-finite $\F$-module over $R$ in the sense of Lyubeznik (for example, any local cohomology module…

(1) Let $(A,\mathfrak{m})$ be complete Noetherian local ring of dimension $d$ and let $P$ be a prime ideal with $G_P(A) = \bigoplus_{n \geq 0}P^n/P^{n+1}$ a domain. Fix $r \geq 1$. If $J$ is a homogeneous ideal of $G_{P^r}(A)$ with…

Let $(R,\mathfrak{m})$ be a Noetherian local ring and $M$ a finitely generated $R$-module. We study the relations of the index of reducibility and the irreducible multiplicity of an $\mathfrak{m}$-primary ideal of $R$ and these of…

Let $R$ be a commutative ring and ${\Bbb{A}}(R)$ be the set of ideals with non-zero annihilators. The annihilating-ideal graph of $R$ is defined as the graph ${\Bbb{AG}}(R)$ with vertex set ${\Bbb{A}}(R)^*={\Bbb{A}}\setminus\{(0)\}$ such…

Let R be a ring of polynomials in a finite number of variables over a perfect field k of characteristic p>0 and let F:R\to R be the Frobenius map of R, i.e. F(r)=r^p. We explicitly describe an R-module isomorphism Hom_R(F_*(M),N)\cong…

Let $X \rightarrow S$ be a smooth projective surjective morphism, where $X$ and $S$ are integral schemes over complex numbers. Let L_0, L_1, .... L_{n-1}, L_{n} be line bundles over $X$. There is a natural isomorphism of the Deligne pairing…

A triangular matrix ring A is defined by a triplet (R,S,M) where R and S are rings and M is an S-R-bimodule. In the main theorem of this paper we show that if T is a tilting S-module, then under certain homological conditions on M as an…

We derive several tools for classifying tensor ideals in monoidal categories. We use these results to classify tensor ideals in Deligne's universal categories RepO, RepGL and RepP. These results are then used to obtain new insight into the…

Let $\hat{R}$ be the $I$-adic completion of a commutative ring $R$ with respect to a finitely generated ideal $I$. We give a necessary and sufficient criterion for the category of perfect complexes over $\hat{R}$ to be equivalent to the…

Let $A$ be a commutative noetherian ring, let $\mathfrak a$ be an ideal of $A$. In this paper, we extend Hartshorne's characterization of cofinite complexes to more general classes of rings. We also determine conditions under which…

Let $R$ be a commutative Noetherian ring, $\fa$ an ideal of $R$ and $\mathcal{D}(R)$ denote the derived category of $R$-modules. We investigate the theory of local homology in conjunction with Gorenstein flat modules. Let $X$ be a…

Let $R\to U$ be an associative ring epimorphism such that $U$ is a flat left $R$-module. Assume that the related Gabriel topology $\mathbb G$ of right ideals in $R$ has a countable base. Then we show that the left $R$-module $U$ has…

For two ideals $I$ and $J$ of a noetherian ring, we characterize, in terms of the vanishing of Tor modules, when the associated graded ring of the sum $I+J$ is isomorphic to the tensor product of the associated graded ring of $I$ and the…

Let (R,m) be a Noetherian local ring of depth d and C a semidualizing R-complex. Let M be a finite R-module and t an integer between 0 and d. If G_C-dimension of M/IM is finite for all ideals I generated by an R-regular sequence of length…

Let R be a commutative ring with identity and M be an R- module. The aim of this paper is to introduce and investigate the notions of nil-M-Noetherian and nil-M-Artinian modules as generalizations of Noetherian and Artinian modules. Also,…

This survey provides a comprehensive overview of the recent advancements in the theory of ``uniformly $S$''-algebraic structures in commutative ring theory. Originating from the classical concepts of Noetherian, coherent, von Neumann…

Let R be a commutative ring with identity and M be an R-module. A proper ideal I of R is said to be a $z^\circ$-ideal if for each $a \in I$ the intersection of all minimal prime ideals containing a is contained in I. The purpose of this…

To each finitely presented module $M$ over a commutative ring $R$ one can associate an $R$-ideal $\mathrm{Fitt}_{R}(M)$, which is called the (zeroth) Fitting ideal of $M$ over $R$. This is of interest because it is always contained in the…

Let M be a fixed left R-module. For a left R-module X, we introduce the notion of M-prime (resp. M-semiprime) submodule of X such that in the case M=R, which coincides with prime (resp. semiprime) submodule of X. Other concepts encountered…