Mohammad T. Dibaei, Siamak Yassemi
Let $R$ be a commutative Noetherian ring, $\fa$ an ideal of $R$ and $M$ a finitely generated $R$--module. Let $t$ be a non-negative integer such that $\H^i_\fa(M)$ is $\fa$--cofinite for all $i<t$. It is well--known that $\Hom_R(R/\fa,\H^t_\fa(M))$ is finitely generated $R$--module. In this paper we study the finiteness of $\Ext^1_R(R/\fa,\H^t_\fa(M))$ and $\Ext^2_R(R/\fa,\H^t_\fa(M))$.
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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 Principle for minimax local cohomology modules is shown. If $\LC^{i}_{\fa}(M)$ is coatomic for $i\leq r$ ($M$ finite) then $\LC^{i}_{\fa}(M)$ is finite for $i\leq r$. We give conditions for a module, which is locally minimax to be a minimax module. A non-vanishing theorem and some vanishing theorems are proved for local cohomology modules.
In this paper we present a condition on a local Cohen-Macaulay F-injective ring of positive characteristic $p > 2$ which implies that its top local cohomology module with support in the maximal ideal has finitely many Frobenius compatible submodules.
We investigate Matlis duals of local cohomology modules and prove that, in general, their zeroth Bass number with respect to the zero ideal is not finite. We also prove that, somewhat surprisingly, if we apply local cohomology again (i. e. to the Matlis dual of the local cohomology module), we get (under certain hypotheses) either zero or $E$, an $R$-injective hull of the residue field of the local ring $R$.
Let $R$ be a commutative noetherian ring, $\fa$ an ideal of $R$ and $M,N$ finite $R$--modules. We prove that the following statements are equivalent. \begin{enumerate} \item[(i)] $\lc^{i}_{\fa}(M,N)$ is finite for all $i< n$. \item[(ii)] $\Coass_R(\lc^{i}_{\fa}(M,N)) \subset \V{(\fa)}$ for all $i< n$. \item[(iii)] $\lc^{i}_{\fa}(M,N)$ is coatomic for all $i< n$. \end{enumerate} If $\pd M$ is finite and $r$ be a non-negative integer such that $r>\pd M$ and $\lc^{i}_{\fa}(M,N)$ is finite (resp. minimax) for all $i\geq r$, then $\lc^{i}_{\fa}(M,N)$ is zero (resp. artinian) for all $i\geq r$.
Local cohomology modules, even over a Noetherian ring $R$, are typically unwieldly. As such, it is of interest whether or not they have finitely many associated primes. We prove the affirmative in the case where $R$ is a Stanley-Reisner ring over a field, whose associated simplicial complex is a $T$-space.
Let (R,m) be a complete local ring, a an ideal of R and M a finitely generated R-module. The aim of this paper is to show that for any non-negative integer n, the least integer i such that the i-th local cohomology with respect to a is not in dimension <n, is equal to the n-th finiteness dimension of M relative to a. This generalizes the main result of Quy and Brodmann-Lashgari.
The purpose of this paper is to give some equivalent conditions to the socle and Bass numbers' conjectures which raised by C. Huneke in (Problems on local cohomology, Free resolutions in commutative algebra and algebraic geometry, Res. Notes Math. 1992, pp. 93-108). In addition, some results about certain Gorenstein ideals are included.
Let \fa be an ideal of a commutative Noetherian ring R and M a finitely generated R-module. We explore the behavior of the two notions f_{\fa}(M), the finiteness dimension of M with respect to \fa, and, its dual notion q_{\fa}(M), the Artinianess dimension of M with respect to \fa. When (R,\fm) is local and r:=f_{\fa}(M) is less than f_{\fa}^{\fm}(M), the \fm-finiteness dimension of M relative to \fa, we prove that H^r_{\fa}(M) is not Artinian, and so the filter depth of \fa on M doesn't exceeds f_{\fa}(M). Also, we show that if M has finite dimension and H^i_{\fa}(M) is Artinian for all i>t, …
Cofiniteness of the generalized local cohomology modules Hai(M, N) of two R-modules M and N with respect to an ideal a is studied for some i,s with a specified property. Furthermore, Artinianness of Hbj 0(Hai(M, N)) is investigated by using the above result, in certain graded situations, where b 0 is an ideal of R 0 such that b 0 + a 0 is an m 0-primary ideal.
In this paper, we show that for an $F$-pure local ring $(R,\m)$, all local cohomology modules $H_{\m}^i(R)$ have finitely many Frobenius compatible submodules. This answers positively an open question raised by F.Enescu and M.Hochster. We also prove that if $(R,\m)$ is excellent and is $F$-pure on the punctured spectrum, then all local cohomology modules have finite length in the category of $R$-modules with Frobenius action. Finally, we show that the property that all $H_{\m}^(R)$ have finitely many Frobenius compatible submodules passes to localizations.
We introduce a formalism to produce several families of spectral sequences involving the derived functors of the limit and colimit functors over a finite partially ordered set. The first type of spectral sequences involves the left derived functors of the colimit of the direct system that we obtain applying a family of functors to a single module. For the second type we follow a completely different strategy as we start with the inverse system that we obtain by applying a covariant functor to an inverse system. The spectral sequences involve the right derived functors of the corresponding limi…
Through a study of torsion functors of local cohomology modules we improve some non-finiteness results on the top non-zero local cohomology modules with respect to an ideal.
Finiteness of associated primes for local cohomology modules of excellent locally unramified regular rings of finite Krull dimension
本文证明了Huneke和Lyubeznik关于局部上同调模相伴素理想有限性的猜想，在有限Krull维数的优秀局部无分歧正则环等条件下取得了实质性新进展。
Let $R$ be a commutative Noetherian ring with non-zero identity, $\fa$ an ideal of $R$, and $X$ an $R$--module. In this paper, for fixed integers $s, t$ and a finite $\fa$--torsion $R$--module $N$, we first study the membership of $\Ext^{s+t}_{R}(N, X)$ and $\Ext^{s}_{R}(N, H^{t}_{\fa}(X))$ in Serre subcategories of the category of $R$--modules. Then we present some conditions which ensure the existence of an isomorphism between them. Finally, we introduce the concept of Serre cofiniteness as a generalization of cofiniteness and study this property for certain local cohomology modules.
Let $(R,\mathfrak{m})$ be a commutative Noetherian local ring, $M$ be a finitely generated $R$-module and $\mathfrak{a}$, $I$ and $J$ be ideals of $R$. We investigate the structure of formal local cohomology modules of $\mathfrak{F}^i_{\mathfrak{a},I,J}(M)$ and $\check{\mathfrak{F}}^i_{\mathfrak{a},I,J}(M)$ with respect to a pair of ideals, for all $i\geq 0$. The main subject of the paper is to study the finiteness properties and Artinianness of $\mathfrak{F}^i_{\mathfrak{a},I,J}(M)$ and $\check{\mathfrak{F}}^i_{\mathfrak{a},\mathfrak{m},J}(M)$. We study the maximum and minimum integer $i\in \…
We propose to define the notion of abstract local cohomology functors. The derived functors of the ordinary local cohomology functor with support in the closed subset defined by an ideal and the generalized local cohomology functor associated with a given pair of ideals are characterized as elements of the set of all the abstract local cohomology functors.
Let $(R, \frak m)$ denote a local Cohen-Macaulay ring and $I$ a non-nilpotent ideal of $R$. The purpose of this article is to investigate Faltings' finiteness dimension $f_I(R)$ and equidimensionalness of certain homomorphic image of $R$. As a consequence we deduce that $f_I(R)={\rm max}\{1, {\rm ht}\ I\}$ and if ${\frak m}\mathrm{Ass}_R(R/I)$ is cotained in Ass$_R(R)$, then the ring $R/ I+\cup_{n\geq 1}(0:_RI^n)$ is equidimensional of dimension $\dim R-1$. Moreover, we will obtain a lower bound for injective dimension of the local cohomology module $H^{{\rm ht}\ I}_I(R)$, in the case $(R, \fr…
We study the behaviour of the finiteness of the set of associated primes of local cohomology modules, more generally of Lyubeznik functors, under various ring extensions. At first, we review the results for flat and faithfully flat extensions and we present new applications of them. Then, we focus how the finiteness property of the set of associated primes of local cohomology modules and Lyubeznik functors is transferred from extended ring to the base ring of pure and cyclically pure ring extensions. We show that finiteness property can be transferred from a ring to its pure local subring and …
Let $R$ denote a commutative Noetherian ring. Brodmann et al. defined and studied the concept of the local-global principle for annihilation of local cohomology modules at level $r\in\mathbb{N}$ for the ideals $\frak a$ and $\frak b$ of $R$. It was shown that this principle holds at levels 1,2, over $R$ and at all levels whenever $\dim R\leq 4$. The goal of this paper is to show that, if the set $\Ass_R(H_{\fa}^{f_{\fa}^{\fb}(M)}(M))$ is finite or $f_{\fa}(M)\neq c_{\fa}^{\fb}(M)$, then the local-global principle holds at all levels $r\in\mathbb{N}_0$, for all ideals $\fa, \fb$ of $R$ and each…
Finiteness of the set of associated primes for local cohomology modules of ideals via properties of almost factorial rings
研究通过比较局部上同调模H_I^{d+1}(J)与H_I^d(R/J)的性质，探讨理想J由R序列生成时关联素理想集的有限性，并在适当条件下证明其等价性及对所有i的有限性条件。
Generalized Hartshorne's problem on finiteness properties of local cohomology modules
本文利用谱序列及上同调维数、Gorenstein 投射维数等不变量，证明局部上同调模余有限性广义 Hartshorne 问题的新正面情形，并推出 Huneke 相伴素理想有限性猜想的相关结论。
Let $(R,\fr m)$ be a Noetherian local ring, $I$ an ideal of $R$ and $M, N$ two finitely generated $R$-modules. The first result of this paper is to prove a vanishing theorem for generalized local cohomology modules which says that $H^j_I(M,N)=0$ for all $j>\dim(R)$, provided $M$ is of finite projective dimension. Next, we study and give characterizations for the least and the last integer $r$ such that $\Supp(H^r_I(M,N))$ is infinite.
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 finiteness properties of formal local cohomology modules which indicate that these modules behave very similar to local cohomology modules. Among other things, we prove that if $\dim R\leq 2$ or either $\fa$ is principal or $\dim R/\fa\leq 1$, then $\Tor_j^R(R/\fa,{\vpl}_nH^i_{\fm}(M/\fa^n M))$ is Artinian for all $i$ and $j$. Also, we examine the notion $\fgrad…
Let $\fa$ be an ideal of a Noetherian local ring $(R,\fm)$ and $M$ a finitely generated $R$-module. In this paper we introduce some criterions on Artinianness of formal local cohomology, in particular vanishing and finiteness of local cohomology modules. We find out the lower and upper bound for Artinianness of formal local cohomology
Let $R$ be a commutative Noetherian ring with non-zero identity and $\fa$ an ideal of $R$. Let $M$ be a finite $R$--module of of finite projective dimension and $N$ an arbitrary finite $R$--module. We characterize the membership of the generalized local cohomology modules $\lc^{i}_{\fa}(M,N)$ in certain Serre subcategories of the category of modules from upper bounds. We define and study the properties of a generalization of cohomological dimension of generalized local cohomology modules. Let $\mathcal S$ be a Serre subcategory of the category of $R$--modules and $n \geqslant \pd M$ be an inte…
Let $(R,\m,k)$ be a commutative noetherian local ring of Krull dimension $d$. We prove that the cohomology annihilator $\ca(R)$ of $R$ is $\m$-primary if and only if for some $n\ge0$ the $n$-th syzygies in $\mod R$ are constructed from syzygies of $k$ by taking direct sums/summands and a fixed number of extensions. These conditions yield that $R$ is an isolated singularity such that the bounded derived category $\db(R)$ and the singularity category $\ds(R)$ have finite dimension, and the converse holds when $R$ is Gorenstein. We also show that the modules locally free on the punctured spectrum…
Let $R = \bigoplus_{n \in \mathbb{N}_0} R_n$ be a Noetherian homogeneous ring with local base ring $(R_0, \mathfrak{m}_0)$ and let $M$ and $N$ be finitely generated graded $R$-modules. Let $i,j\in\mathbb{N}_0$. In this paper we will study Artinianess of $Γ_{\mathfrak m_0R}(H_{R_+}^i(M,N)), H_{\mathfrak m_0R}^1(H_{R_+}^i(M,N)), H_{R_+}^i(M,N)/{\mathfrak m_0}H_{R_+}^i(M,N), H_{R_+}^j(M,H_{\mathfrak m_0R}^i(N)), H_{\mathfrak m_0R}^j(M,H_{R_+}^i(N))$, where $R_+$ denotes the irrelevant ideal of $R$.
In this article, we study the behaviour of smooth algebra $R$ over local Noetherian local ring $A$. At first, we observe that for every $f\in R$, $R_f$ has finite length in the category of $D(R,A)$-module if dimension of $A$ is zero. This extends the result of Theorem 2 of \cite{Ly3}. We use this fact to generalize the result of Theorem 4.1 of \cite{BBLSZ}, from the finiteness of the set of associated primes of local cohomology module to that of Lyubeznik functor. Finally, we introduce the definition of $Σ$-finite $D$-modulue for smooth algebra and we extend the result of Theorem 1.3 of \cite{…
Gorenstein homological dimensions are refinements of the classical homological dimensions, and finiteness singles out modules with amenable properties reflecting those of modules over Gorenstein rings. As opposed to their classical counterparts, these dimensions do not immediately come with practical and robust criteria for finiteness, not even over commutative noetherian local rings. In this paper we enlarge the class of rings known to admit good criteria for finiteness of Gorenstein dimensions: It now includes, for instance, the rings encountered in commutative algebraic geometry and, in the…
Let $\frak a$ denote an ideal in a regular local (Noetherian) ring $R$ and let $N$ be a finitely generated $R$-module with support in $V(\frak a)$. The purpose of this paper is to show that all homomorphic images of the $R$-modules $\Ext^j_R(N, H^i_{\frak a}(R))$ have only finitely many associated primes, for all $i, j\geq 0$, whenever $\dim R \leq4$ or $\dim R/ \frak a \leq 3$ and $R$ contains a field. In addition, we show that if $\dim R=5$ and $R$ contains a field, then the $R$-modules $\Ext^j_R(N, H^i_{\frak a}(R))$ have only finitely many associated primes, for all $i, j\geq 0$.
Let $R$ be a commutative Noetherian ring with non-zero identity, $\mathfrak{a}$ and ideal of $R$, $M$ a finite $R$--module, and $n$ a non-negative integer. In this paper, for an arbitrary $R$--module $X$ which is not necessarily finite, we study the finiteness dimension $f_\mathfrak{a}(M,X)$ and the $n$-th finiteness dimension $f^n_\mathfrak{a}(M,X)$ of $M$ and $X$ with respect to $\mathfrak{a}$. Assume that $\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},X)$ is finite for all $i\leq f^2_\mathfrak{a}(M,X)$ (resp. $i< f^1_\mathfrak{a}(M,X)$). We show that $\operatorname{H}^{i}_{\mathfrak{a}}(M,X)$ i…
Let $R$ be a commutative $F$-algebra, where $F$ is a field of characteristic 0, satisfying the following conditions: $R$ is equidimensional of dimension $n$, every residual field with respect to a maximal ideal is an algebraic extension of $F,$ and $\Der_F (R)$ is a finitely generated projective $R$-module of rank $n$ such that $R_m\otimes_R \Der_F (R)=\Der_F(R_m)$. We show that the associated graded ring of the ring of differentiable operators, $D(R,F)$, is a commutative Noetherian regular with unity and pure graded dimension equal to $2\dim(R)$. Moreover, we prove that $D(R,F)$ has weak glob…
Let $R$ denote a commutative Noetherian (not necessarily local) ring, $\frak a$ an ideal of $R$ and $M$ a finitely generated $R$-module. The purpose of this paper is to show that $f^n_{\frak a}(M)=\inf \{0\leq i\in\mathbb{Z}|\, \dim H^{i}_{\frak a}(M)/N \geq n\, \, \text{for any finitely generated submodule}\,\, N \subseteq H^{i}_{\frak a}(M)\}$, where $n$ is a non-negative integer and the invariant $f^n_{\frak a}(M):=\inf\{f_{\frak a R_{\frak p}}(M_{\frak p})\,\,|\,\,{\frak p}\in \Supp M/\frak a M\,\,{\rm and}\,\,\dim R/{\frak p}\geq n\}$ is the $n$-th finiteness dimension of $M$ relative to …
We compare and contrast various relative cohomology theories that arise from resolutions involving semidualizing modules. We prove a general balance result for relative cohomology over a Cohen-Macaulay ring with a dualizing module, and we demonstrate the failure of the naive version of balance one might expect for these functors. We prove that the natural comparison morphisms between relative cohomology modules are isomorphisms in several cases, and we provide a Yoneda-type description of the first relative Ext functor. Finally, we show by example that each distinct relative cohomology constru…
Let $R$ be a commutative Noetherian ring and $\fa$ an ideal of $R$. We intend to establish the dual of two Faltings' Theorems for local homology modules of an Artinian module. As a consequence of this, we show that, if $A$ is an Artinian module over semi-local complete ring $R$ and $j$ is an integer such that $H_i^{\fa}(A)$ is Artinian for all $i<j$, then the set $\Coass_R(H_j^{\fa}(A))$ is finite. We also introduce the notion of the $n$th Artinianness dimension $g_n^\fa(A)=\inf\{g^{\fa R_{\fp}}(^\fp A): \fp\in\Spec(R) \ \ \text{and} \ \ \dim R/\fp\geq n\}$, for all $n\in\mathbb{N}_{0}$ and pr…
Let $\k$ be a commutative ring, and let $(A,\mfrak{a})$ be an adic ring which is a $\k$-algebra. We study complete and torsion versions of the derived Hochschild homology and cohomology functors of $A$ over $\k$. To do this, we first establish weak proregularity of certain ideals in flat base changes of noetherian rings. Next, we develop a theory of DG-affine formal schemes, extending the Greenlees-May duality and the MGM equivalence to this setting. Finally, we define complete and torsion derived Hochschild homology and cohomology functors in this setting, and show that if $\k$ is noetherian …
Let $R$ be a Gorenstein local ring, $\frak{a}$ an ideal in $R$, and $M$ an $R$-module. The local cohomology of $M$ supported at $\frak{a}$ can be computed by applying the $\frak{a}$-torsion functor to an injective resolution of $M$. Since $R$ is Gorenstein, $M$ has a complete injective resolution, so it is natural to ask what one gets by applying the $\frak{a}$-torsion functor to it. Following this lead, we define stable local cohomology for modules with complete injective resolutions. This gives a functor to the stable category of Gorenstein injective modules. We show that in many ways this b…
Let $\mathcal{A}$ be a locally noetherian Grothendieck category. In this paper we define and study the section functor on $\mathcal{A}$ with respect to an open subset of ASpec$\mathcal{A}$. Next we define and study local cohomology theory in $\mathcal{A}$ in terms of the section functors. Finally we study abstract local cohomology functor on the derived category $\mathcal{D}^+(\mathcal{A})$.
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 $I$ an ideal of $R$. We first show that the Bass numbers $μ^0(\frak p, H^2_I(R))$ and $μ^1(\frak p, H^2_I(R))$ are finite for all $\frak p\in \Spec R$, whenever $R$ is regular. As a consequence, it follows that the Goldie dimension of $H^2_I(R)$ is finite. Also, for a finitely generated $R$-module $M$ of dimension $d$, it is shown that the Bass numbers of $H^…
Stable sheaf cohomology and Koszul--Ringel duality
本文建立稳定层上同调与Koszul--Ringel对偶的联系，证实周期猜想，发现钩子划分Schur函子$\Ext$群的不变性，并给出非零稳定上同调度数的锐界。
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. Brodmann and L. T. Nhan ({Comm. Algebra, 36 (2008), 1527-1536). For a subset $S\subseteq \Spec R$ we set $S_{{\ge}k}={\p\in S\mid\dim(R/\p){\ge}k}$. We first prove in this paper that $\Ass_R(H^j_I(N))_{\ge k}$ is a finite set for all $j{\le}r$}. Let $\fN=\oplus_{n\ge 0}N_n$ be a finitely generated graded $\fR$-module, where $\fR$ is a finitely generated standard…
This is an expanded version of a lecture on finiteness properties of local cohomology modules given at the 48th Algebra Symposium held at Nagoya University in August 2003.
In this paper we describe an inductive machinery to investigate asymptotic behaviors of homology groups and related invariants of representations of certain graded combinatorial categories over a commutative Noetherian ring $k$, via introducing inductive functors which generalize important properties of shift functors of $\mathrm{FI}$-modules. In particular, a sufficient criterion for finiteness of Castelnuovo-Mumford regularity of finitely generated representations of these categories is obtained. As applications, we show that a few important infinite combinatorial categories appearing in rep…
Let $R=K[X_1,\ldots, X_n]$ where $K$ is a field of characteristic zero, and let $A_n(K)$ be the $n^{th}$ Weyl algebra over $K$. We give standard grading on $R$ and $A_n(K)$. Let $I$, $J$ be homogeneous ideals of $R$. Let $M = H^i_I(R)$ and $N = H^j_J(R)$ for some $i, j$. We show that $\Ext_{A_n(K)}^ν(M,N)$ is concentrated in degree zero for all $ν\geq 0$, i.e., $\Ext_{A_n(K)}^ν(M,N)_l=0$ for $l \neq0$. This proves a conjecture stated in part I of this paper.
Cofiniteness and finiteness of associated prime ideals of generalized local cohomology modules
本文研究广义局部上同调模的余有限性及其相伴素理想的有限性，给出若干维数条件下的有限性与余有限性结论。
Let $R$ be a commutative Noetherian ring with non-zero identity, $\mathfrak{a}$ an ideal of $R$, $M$ a finitely generated $R$--module, and $X$ an arbitrary $R$--module. In this paper, for non-negative integers $s, t$ and a finitely generated $R$--module $N$, we study the membership of $\operatorname{Ext}_{R}^{s}(N, \operatorname{H}^{t}_{\mathfrak{a}}(M, X))$ in Serre subcategories of the category of $R$--modules and present some upper bounds for the injective dimension and the Bass numbers of $\operatorname{H}^{t}_{\mathfrak{a}}(M, X)$. We also give some results on cofiniteness and minimaxness…
We investigate relative cohomology functors on subcategories of abelian categories via Auslander-Buchweitz approximations and the resulting strict resolutions. We verify that certain comparison maps between these functors are isomorphisms and introduce a notion of perfection for this context. Our main theorem is a balance result for relative cohomology that simultaneously recovers theorems of Holm and the current authors as special cases.
We show under suitable finiteness conditions that a functor between abelian categories induces a (not necessarily additive) map between their Grothendieck groups. This is related to the derived functors of Dold and Puppe, and generalizes a theorem of Dold.
In this paper, we prove several results on the finiteness of local cohomology of polynomial and formal power series rings. In particular, we give a partial affirmative answer for a question of L. Núñez-Betancourt in [J. Algebra 399 (2014), 770--781].
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