Let denote an ideal in a Noetherian ring R, and M a finitely generated R-module. We introduce the concept of the cohomological dimension filtration $={M_i}_{i=0}^c$, where c = cd(,M) and $M_i$ denotes the largest submodule of M such that $cd(,M_i)\le i$. Some properties of this filtration are investigated. In particular, if (R,) is local and c = dim M, we are able to determine the annihilator of the top local cohomology module $H_{}^c\left(M\right)$, namely $An{n}_R\left(H_{}^c\left(M\right)\right)=An{n}_R(M/M_{c-1})$. As a consequence, there exists an ideal of R such that $An{n}_R\left(H_{}^c\left(M\right)\right)=An{n}_R(M/H{⁰}_{}\left(M\right))$. This generalizes the...

Let $𝔞$ be an ideal of Noetherian local ring $(R,𝔪)$ and $M$ a finitely generated $R$-module of dimension $d$. In this paper we investigate the Artinianness of formal local cohomology modules under certain conditions on the local cohomology modules with respect to $𝔪$. Also we prove that for an arbitrary local ring $(R,𝔪)$ (not necessarily complete), we have ${\mathrm{Att}}_R\left({𝔉}_{𝔞}^d\left(M\right)\right)=\mathrm{Min}\mathrm{V}\left({\mathrm{Ann}}_R{𝔉}_{𝔞}^d\left(M\right)\right).$

Let $I$ and $J$ be ideals of a Noetherian local ring $(R,𝔪)$ and let $M$ be a nonzero finitely generated $R$-module. We study the relation between the vanishing of $H_{I,J}^{dimM}\left(M\right)$ and the comparison of certain ideal topologies. Also, we characterize when the integral closure of an ideal relative to the Noetherian $R$-module $M/JM$ is equal to its integral closure relative to the Artinian $R$-module $H_{I,J}^{dimM}\left(M\right)$.

Let $(R,𝔪)$ be a commutative Noetherian regular local ring of dimension $d$ and $I$ be a proper ideal of $R$ such that ${\mathrm{mAss}}_R(R/I)={\mathrm{Assh}}_R\left(I\right)$. It is shown that the $R$-module $H_I^{\mathrm{ht}\left(I\right)}\left(R\right)$ is $I$-cofinite if and only if $\mathrm{cd}(I,R)=\mathrm{ht}\left(I\right)$. Also we present a sufficient condition under which this condition the $R$-module $H_I^i\left(R\right)$ is finitely generated if and only if it vanishes.

Let $𝔞$ be an ideal of a commutative Noetherian ring $R$ and $t$ be a nonnegative integer. Let $M$ and $N$ be two finitely generated $R$-modules. In certain cases, we give some bounds under inclusion for the annihilators of ${\mathrm{Ext}}_R^t(M,N)$ and ${\mathrm{H}}_{𝔞}^t\left(M\right)$ in terms of minimal primary decomposition of the zero submodule of $M$, which are independent of the choice of minimal primary decomposition. Then, by using those bounds, we compute the annihilators of local cohomology and Ext modules in certain cases.

Let $𝔞$, $I$, $J$ be ideals of a Noetherian local ring $(R,𝔪,k)$. Let $M$ and $N$ be finitely generated $R$-modules. We give a generalized version of the Duality Theorem for Cohen-Macaulay rings using local cohomology defined by a pair of ideals. We study the behavior of the endomorphism rings of $H_{I,J}^t\left(M\right)$ and $D\left(H_{I,J}^t\left(M\right)\right)$, where $t$ is the smallest integer such that the local cohomology with respect to a pair of ideals is nonzero and $D(-):={\mathrm{Hom}}_R(-,E_R\left(k\right))$ is the Matlis dual functor. We show that if $R$ is a $d$-dimensional complete Cohen-Macaulay...

Let $R$ be a commutative Noetherian ring, $I$ an ideal of $R$ and $M$ an $R$-module. We wish to investigate the relation between vanishing, finiteness, Artinianness, minimaxness and $𝒞$-minimaxness of local cohomology modules. We show that if $M$ is a minimax $R$-module, then the local-global principle is valid for minimaxness of local cohomology modules. This implies that if $n$ is a nonnegative integer such that ${\left(H_I^i\left(M\right)\right)}_{𝔪}$ is a minimax $R_{𝔪}$-module for all $𝔪\in \mathrm{Max}\left(R\right)$ and for all $i<n$, then the set ${\mathrm{Ass}}_R\left(H_I^n\left(M\right)\right)$ is finite. Also, if $H_I^i\left(M\right)$ is...

Let Φ be a system of ideals on a commutative Noetherian ring R, and let S be a multiplicatively closed subset of R. The first result shows that the topologies defined by ${I_a}_{I\in \Phi }$ and ${S\left(I_a\right)}_{I\in \Phi }$ are equivalent if and only if S is disjoint from the quintasymptotic primes of Φ. Also, by using the generalized Lichtenbaum-Hartshorne vanishing theorem we show that, if (R,) is a d-dimensional local quasi-unmixed ring, then $H_{\Phi }^d\left(R\right)$, the dth local cohomology module of R with respect to Φ, vanishes if and only if there...

Let be a proper ideal of a commutative Noetherian ring R of prime characteristic p and let Q() be the smallest positive integer m such that ${{(}^F)}^{\left[p^m\right]}{=}^{\left[p^m\right]}$, where ${}^F$ is the Frobenius closure of . This paper is concerned with the question whether the set $Q{(}^{\left[p^m\right]}):m\in \mathbb{N}₀$ is bounded. We give an affirmative answer in the case that the ideal is generated by an u.s.d-sequence c₁,..., cₙ for R such that (i) the map $R/{\sum }_{j=1}^{n}R{c}_j\to R/{\sum }_{j=1}^{n}Rc{²}_j$ induced by multiplication by c₁...cₙ is an R-monomorphism; (ii) for all $\in ass(c{₁}^j,...,c{ₙ}^j)$, c₁/1,..., cₙ/1 is a $R_{}$-filter...

Let $(R,𝔪)$ be a commutative Noetherian local ring, $𝔞$ be an ideal of $R$ and $M$ a finitely generated $R$-module such that $𝔞M\ne M$ and $\mathrm{cd}(𝔞,M)-\mathrm{grade}(𝔞,M)\le 1$, where $\mathrm{cd}(𝔞,M)$ is the cohomological dimension of $M$ with respect to $𝔞$ and $\mathrm{grade}(𝔞,M)$ is the $M$-grade of $𝔞$. Let $D(-):={\mathrm{Hom}}_R(-,E)$ be the Matlis dual functor, where $E:=E(R/𝔪)$ is the injective hull of the residue field $R/𝔪$. We show that there exists the following long exact sequence $$\begin{array}{cc}\hfill 0⟶& H_{𝔞}^{n-2}\left(D\left(H_{𝔞}^{n-1}\left(M\right)\right)\right)⟶H_{𝔞}^n\left(D\left(H_{𝔞}^n\left(M\right)\right)\right)⟶D\left(M\right)\\ \hfill ⟶& H_{𝔞}^{n-1}\left(D\left(H_{𝔞}^{n-1}\left(M\right)\right)\right)⟶H_{𝔞}^{n+1}\left(D\left(H_{𝔞}^n\left(M\right)\right)\right)\\ \hfill ⟶& H_{𝔞}^n\left(D\left(H_{(x_1,...,x_{n-1})}^{n-1}\left(M\right)\right)\right)⟶H_{𝔞}^n\left(D\left(H_{(}^{n-1}M\right)\right))⟶...,\end{array}$$ where $n:=\mathrm{cd}(𝔞,M)$ is a non-negative integer, $x_1,...,x_{n-1}$ is a regular sequence in $𝔞$ on $M$ and, for an $R$-module $L$, $H_{𝔞}^i\left(L\right)$ is the $i$th local cohomology module...

Let $I$ be an ideal of a commutative Noetherian ring $R$. It is shown that the $R$-modules $H_I^j\left(M\right)$ are $I$-cofinite for all finitely generated $R$-modules $M$ and all $j\in {\mathbb{N}}_0$ if and only if the $R$-modules ${\mathrm{Ext}}_R^i(N,H_I^j\left(M\right))$ and ${\mathrm{Tor}}_i^R(N,H_I^j\left(M\right))$ are $I$-cofinite for all finitely generated $R$-modules $M$, $N$ and all integers $i,j\in {\mathbb{N}}_0$.

In this work we compute the Chen–Ruan cohomology of the moduli spaces of smooth and stable $n$-pointed curves of genus $1$. In the first part of the paper we study and describe stack theoretically the twisted sectors of ${ℳ}_{1,n}$ and ${\overline{ℳ}}_{1,n}$. In the second part, we study the orbifold intersection theory of ${\overline{ℳ}}_{1,n}$. We suggest a definition for an orbifold tautological ring in genus $1$, which is a subring of both the Chen–Ruan cohomology and of the stringy Chow ring.

Let $R$ be a Noetherian ring, and $I$ and $J$ be two ideals of $R$. Let $S$ be a Serre subcategory of the category of $R$-modules satisfying the condition $C_I$ and $M$ be a $ZD$-module. As a generalization of the $S$-$\mathrm{depth}(I,M)$ and $\mathrm{depth}(I,J,M)$, the $S$-$\mathrm{depth}$ of $(I,J)$ on $M$ is defined as $S$-$\mathrm{depth}(I,J,M)=inf\{S$-$\mathrm{depth}(𝔞,M):𝔞\in \tilde{\mathrm{W}}(I,J)\}$, and some properties of this concept are investigated. The relations between $S$-$\mathrm{depth}(I,J,M)$ and $H_{I,J}^i\left(M\right)$ are studied, and it is proved that $S$-$\mathrm{depth}(I,J,M)=inf\{i:H_{I,J}^i\left(M\right)\notin S\}$, where $S$ is a Serre subcategory closed under taking injective hulls. Some conditions are provided that local cohomology...

We use known results on the characteristic rank of the canonical $4$–plane bundle over the oriented Grassmann manifold ${\tilde{G}}_{n,4}$ to compute the generators of the ${\mathbb{Z}}_2$–cohomology groups $H^j\left({\tilde{G}}_{n,4}\right)$ for $n=8,9,10,11$. Drawing from the similarities of these examples with the general description of the cohomology rings of ${\tilde{G}}_{n,3}$ we conjecture some predictions.

Let $(R,𝔪)$ be a local ring, $𝔞$ an ideal of $R$ and $M$ a nonzero Artinian $R$-module of Noetherian dimension $n$ with $\mathrm{hd}(𝔞,M)=n$. We determine the annihilator of the top local homology module ${\mathrm{H}}_n^{𝔞}\left(M\right)$. In fact, we prove that $${\mathrm{Ann}}_R\left({\mathrm{H}}_n^{𝔞}\left(M\right)\right)={\mathrm{Ann}}_R\left(N(𝔞,M)\right),$$ where $N(𝔞,M)$ denotes the smallest submodule of $M$ such that $\mathrm{hd}(𝔞,M/N(𝔞,M\left)\right)<n$. As a consequence, it follows that for a complete local ring $(R,𝔪)$ all associated primes of ${\mathrm{H}}_n^{𝔞}\left(M\right)$ are minimal.

The goal of this work is to construct, for a smooth variety $X$ over a perfect field k of finite characteristic $p\&gt;0$, an overconvergent de Rham-Witt complex $W^{†}{\Omega }_{X/k}$ as a suitable subcomplex of the de Rham-Witt complex of Deligne-Illusie. This complex, which is functorial in $X$, is a complex of étale sheaves and a differential graded algebra over the ring $W^{†}\left({𝒪}_X\right)$ of overconvergent Witt-vectors. If $X$ is affine one proves that there is an isomorphism between Monsky-Washnitzer cohomology and (rational) overconvergent...

Let $R$ be a local ring and $C$ a semidualizing module of $R$. We investigate the behavior of certain classes of generalized Cohen-Macaulay $R$-modules under the Foxby equivalence between the Auslander and Bass classes with respect to $C$. In particular, we show that generalized Cohen-Macaulay $R$-modules are invariant under this equivalence and if $M$ is a finitely generated $R$-module in the Auslander class with respect to $C$ such that $C{\otimes }_{R}M$ is surjective Buchsbaum, then $M$ is also surjective Buchsbaum. ...

We investigate some properties of $n$-submodules. More precisely, we find a necessary and sufficient condition for every proper submodule of a module to be an $n$-submodule. Also, we show that if $M$ is a finitely generated $R$-module and $\sqrt{{\mathrm{Ann}}_R\left(M\right)}$ is a prime ideal of $R$, then $M$ has $n$-submodule. Moreover, we define the notion of $G.n$-submodule, which is a generalization of the notion of $n$-submodule. We find some characterizations of $G.n$-submodules and we examine the way the aforementioned notions are related...