Behruz Sadeqi
Let $\mathfrak{a}$ be an ideal of a commutative Noetherian ring $R$ and $M$ a finitely generated $R$-module. In this paper we proved that if $\operatorname{Supp}\mathfrak{F}_\mathfrak{a}^i(M)$ is finite for all $i<t$, then so is $\operatorname{Ass}(\mathfrak{F}_{\mathfrak{a}}^t(M))$.
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In this paper we consider the local cohomology of monomial ideals with respect to monomial prime ideals and show that all these local cohomology modules are tame.
Infinitely many associated primes of local cohomology modules of ramified regular local rings
研究有限正则局部环的局部余同调模的关联素数和Bass数的无限性问题
Let $J$ be an ideal of a noetherian local ring $R$. We show new results on the set of attached primes $\Att_R(\LCMo ^l_J(R))$ of a local cohomology module $\LCMo ^l_J(R)$. To prove our results we establish and use new relations between the set $\Att_R(\LCMo ^l_J(R))$ of attached primes of a local cohomology module and the set $\Ass_R(D(\LCMo ^l_J(R)))$ of associated primes of the Matlis dual of the same local cohomology module.
This paper at first concerns some criteria on Artinianness and vanishing of formal local cohomology modules. Then we consider the cosupport and the set of coassociated primes of these modules more precisely.
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.
Let $R$ be a commutative Noetherian ring and $M$ be a finitely generated $R$-module. Considering the new concept of linkage of ideals over a module, we study associated prime ideals, cofiniteness and Artinianness of local cohomology modules of $M$ with respect to some linked ideals over it.
After motivating the question we prove various results about the set of associated primes of Matlis duals of top local cohomology modules. In some cases we can calculate this set, for the general situation we present a conjecture. An easy application of this theory is the well-known fact that Krull dimension can be expressed by the vanishing of local cohomology modules.
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.
We construct normal hypersurfaces whose local cohomology modules have infinitely many associated primes. These include unique factorization domains of characteristic zero with rational singularities, as well as F-regular unique factorization domains of positive characteristic. As a consequence, we answer a question on the associated primes of Frobenius powers of ideals, which arose from the localization problem in tight closure theory.
Lyubeznik's conjecture, (\cite{Ly1}, Remark 3.7) asserts the finiteness of the set ssociated primes of local cohomology modules for regular rings. But, in the case of ramified regular local ring, it is open. Recently, in Theorem 1.2 of \cite{Nu1}, it is proved that in any Noetherian regular local ring $S$ and for a fixed ideal $J\subset S$, associated primes of local cohomology $H^i_J(S)$ for $i\geq 0$ is finite, if it does not contain $p$. In this paper, we use this result to construct examples of local cohomology modules for ramified regular local ring so that they have finitely many associa…
Depth and Associated Primes in Group Cohomology
本文证明了Connor猜想等价于存在初等阿贝尔子群E使得群上同调深度等于中心化子上同调深度。
On local cohomology modules over ramified regular local rings
本文将分歧正则局部环视为非分歧正则局部环的艾森斯坦扩张，利用迈耶-维托里斯谱序列构造出具有有限相伴素理想集合的非扩张理想局部上同调模例子，其中部分例子的相伴素理想还包含素数p。
Let (R,m) be a complete Noetherian local ring and let M be a finite R--module of positive Krull dimension n. It is shown that any subset T of Assh_R(M) can be expressed as the set of attached primes of the top local cohomology module H^n_a(M) for some ideal a of R. Moreover if a is an ideal of R such that the set of attached primes of H^n_a(M) is a non--empty proper subset of Assh_R(M), then H^n_a(M)=H^n_b(M) for some ideal b of R with dim_R (R/b)=1.
Let $I$ and $J$ be two ideals of a commutative Noetherian ring $R$ and $M$ be an $R$-module of dimension $d$. If $R$ is a complete local ring and $M$ is finite, then attached prime ideals of $H^{d-1}_{I,J}(M)$ are computed by means of the concept of co-localization. Also, we illustrate the attached prime ideals of $H^{t}_{I,J}(M)$ on a non-local ring $R$, for $t= \dim M$ and $t= cd(I,J,M)$.
Let $R$ be an excellent regular ring of dimension $d$ containing a field $K$ of characteristic zero. Let $I$ be an ideal in $R$. We show that $Ass \ H^{d-1}_I(R)$ is a finite set. As an application we show that if $I$ is an ideal of height $g$ with $height \ Q = g$ for all minimal primes of $I$ then for all but finitely many primes $P \supseteq I$ with $height \ P \geq g +2$, the topological space $Spec^\circ(R_P/IR_P)$ is connected.
Let $R=k[x_1,\dots,x_n]$ be a ring of polynomials over a field $k$ of characteristic $p>0$. There is an algorithm due to Lyubeznik for deciding the vanishing of local cohomology modules $H^i_I(R)$ where $I\subset R$ is an ideal. This algorithm has not been implemented because its complexity grows very rapidly with the growth of $p$ which makes it impractical. In this paper we produce a modification of this algorithm that consumes a modest amount of memory.
We find sufficient conditions which imply equality of the finitistic test ideal and test ideal in rings of prime characteristic. Utilizing recent progress from the prime characteristic minimal model program we equate the notions of $F$-regular and strongly $F$-regular for 4-dimensional rings essentially of finite type over a field of prime characteristic $p>5$.
Let $\fa$ be an ideal of a commutative Noetherian ring $R$ and $M$ a finitely generated $R$-module. Let $t$ be a natural integer. It is shown that there is a finite subset $X$ of $\Spec R$, such that $\Ass_R(H_{\fa}^t(M))$ is contained in $X$ union with the union of the sets $\Ass_R(\Ext_R^j(R/\fa,H_{\fa}^i(M)))$, where $0\leq i<t$ and $0\leq j\leq t^2+1$. As an immediate consequence, we deduce that the first non $\fa$-cofinite local cohomology module of $M$ with respect to $\fa$ has only finitely many associated prime ideals.
For a finitely generated module $M$, over a commutative Noetherian local ring $(R, \mathfrak{m})$, it is shown that there exist only a finite number of non--isomorphic top local cohomology modules $\mathrm{H}_{\mathfrak{a}}^{\mathrm{dim} (M)}(M)$, for all ideals $\mathfrak{a}$ of $R$. We present a reduced secondary representation for the top local cohomology modules with respect to an ideal. It is also shown that for a given integer $r\geq 0$, if $\mathrm{H}_{\mathfrak{a}}^{r}(R/\mathfrak {p})$ is zero for all $\mathfrak{p}$ in $\mathrm{Supp}(M)$, then $\mathrm{H}_{\mathfrak{a}}^{i}(M)= 0$ for…
We construct an example where a local cohomology module of a hypersurface has p-torsion elements for every prime integer p, and consequently has infinitely many associated prime ideals. We also answer a related question of Hochster.
Let $\mathfrak{a}$ be an ideal of Noetherian ring $R$ and let $M$ be an $R$-module such that $\mathrm{Ext}^i_R(R/\mathfrak{a},M)$ is finite $R$-module for every $i$. If $s$ is the first integer such that the local cohomology module $\mathrm{H}^s_\mathfrak{a}(M)$ is non $\mathfrak{a}$-cofinite, then we show that $\mathrm{Hom}_{R}(R/\mathfrak{a}, \mathrm{H}^s_\mathfrak{a}(M))$ is finite. Specially, the set of associated primes of $\mathrm{H}^s_\mathfrak{a}(M)$ is finite. Next assume $(R,\mathfrak{m})$ is a local Noetherian ring and $M$ is a finitely generated module. We study the last integer $n…
For a Noetherian regular ring $S$ and for a fixed ideal $J\subset S$, assume that the associated primes of local cohomology module $H^i_J(S)$ does not contain $p$ for some $i\geq 0$, and we call this as a property $\textit{\textbf{P}}^{i,p}_{J,S}$ or $\textit{\textbf{P}}$ for brevity. Recently, in Theorem 1.2 of \cite{Nu1}, it is proved that in a Noetherian regular local ring $S$ and for a fixed ideal $J\subset S$, associated primes of local cohomology module $H^i_J(S)$ for $i\geq 0$ is finite, if it does not contain $p$. In this paper, we study how the property $\textit{\textbf{P}}$ (as menti…
The notion of weakly Laskerian modules was introduced recently by the authors. Let $R$ be a commutative Noetherian ring with identity, $\fa$ an ideal of $R$, and $M$ a weakly Laskerian module. It is shown that if $\fa$ is principal, then the set of associated primes of the local cohomology module $H_{\fa}^i(M)$ is finite for all $i\geq 0$. We also prove that when $R$ is local, then $\Ass_R(H_{\fa}^i(M))$ is finite for all $i\geq 0$ in the following cases: (1) $\dim R\leq 3$, (2) $\dim R/\fa\leq 1$, (3) $M$ is Cohen-Macaulay and for any ideal $\fb$, with $l=\grade(\fb,M)$, $\Hom_R(R/\fb,H_{\fb}…
Let $R$ be a commutative Noetherian ring of prime characteristic $p$. In this paper we give a short proof using filter regular sequences that the set of associated prime ideals of $H^t_I(R)$ is finite for any ideal $I$ and for any $t \ge 0$ when $R$ has finite $F$-representation type or finite singular locus. This extends a previous result by Takagi-Takahashi and gives affirmative answers for a problem of Huneke in many new classes of rings in positive characteristic. We also give a criterion about the singularities of $R$ (in any characteristic) to guarantee that the set of associated primes …
Assume that $R = \bigoplus_{i\in \mathbb{N}_{0}} R_{i}$ is a homogeneous graded Noetherian ring, and that $M$ is a $ \mathbb{Z}$--graded $R$--module, where $ \mathbb{N}_{0}$ (resp. $ \mathbb{Z}$) denote the set all non--negative integers (resp. integers). The set of all homogeneous attached prime ideals of the top non--vanishing local cohomology module of a finitely generated module $M$, $\LH_{R_{+}}^{c}(M)$, with respect to the irrelevant ideal $R_{+}: =\bigoplus_{i\geq 1} R_{i}$ and the set of associated primes of $\LH _{R_{+}}^{i}(M)$ is studied. The asymptotic behavior of $\Hom_{R}(R/R_{+}…
In this article, we study the following question raised by Mel Hochster: let $(R,m,K)$ be a local ring and $S$ be a flat extension with regular closed fiber. Is $\cV(mS)\cap\Ass_S H^i_I(S)$ finite for every ideal $I\subset S$ and $i\in \NN?$ We prove that the answer is positive when $S$ is either a polynomial or a power series ring over $R$ and $\dim(R/I\cap R)\leq 1.$ In addition, we analyze when this question can be reduced to the case where $S$ is a power series ring over $R$. An important tool for our proof is the use of $Σ$-finite $D$-modules, which are not necessarily finitely generated …
In continuation of [1] we study associated primes of Matlis duals of local cohomology modules (MDLCM). We combine ideas from Helmut Zöschinger on coassociated primes of arbitrary modules with results from [1], [4], [5], [6] and obtain partial answers to questions which were left open in [1]. These partial answers give further support for conjecture $(*)$ from [1] on the set of associated primes of MDLCMs. In addition, and also inspired by ideas from Zöschinger, we prove some non-finiteness results of local cohomology.
Let R be a commutative Noetherian ring. We introduce a theory of formal local cohomology for complexes of R-modules. As an application, we establish some relations between formal local cohomology, local homology, local cohomology and local Tate cohomology through some natural isomorphisms. We investigate vanishing of formal local cohomology modules. Also, we give a characterization of Cohen-Macaulay complexes.
Let $R$ be a noetherian ring, $\fa$ an ideal of $R$ such that $\dim R/\fa=1$ and $M$ a finite $R$--module. We will study cofiniteness and some other properties of the local cohomology modules $\lc^{i}_{\fa}(M)$. For an arbitrary ideal $\fa$ and an $R$--module $M$ (not necessarily finite), we will characterize $\fa$--cofinite artinian local cohomology modules. Certain sets of coassociated primes of top local cohomology modules over local rings are characterized.
Let $R$ be a commutative Noetherian ring that is a smooth $\mathbb Z$-algebra. For each ideal $I$ of $R$ and integer $k$, we prove that the local cohomology module $H^k_I(R)$ has finitely many associated prime ideals. This settles a crucial outstanding case of a conjecture of Lyubeznik asserting this finiteness for local cohomology modules of all regular rings.
Let $I$ and $J$ be two ideals of a commutative Noetherian ring $R$ and $M$ be an $R$-module. For a non-negative integer $n$ it is shown that, if the sets $\Ass_R(\Ext^{n} _{R}(R/I,M))$ and $\Supp_R(\Ext^{i}_{R}(R/I,H^{j}_{I,J} (M)))$ are finite for all $i \leq n+1$ and all $j< n$, then so is \linebreak$\Ass_R(\Hom_{R}(R/I,H^{n}_{I,J}(M)))$. We also study the finiteness of $\Ass_R(\Ext^{i}_{R}(R/I,H^{n}_{I,J} (M)))$ for $i=1,2$.
Minimal Attached Primes of Local Cohomology Modules of Binomial Edge Ideals of Block Graphs
本文研究块图二项边理想的局部上同调模的极小附着素理想，给出非零模的组合刻画，并借助同期独立发表的相关结果给出该结论的简短替代证明。
Let $(R,m)$ be a local Noetherian ring, let $I\subset R$ be any ideal and let $M$ be a finitely generated $R$-module. In 1990 Craig Huneke conjectured that the local cohomology modules $H^i_I(M)$ have finitely many associated primes for all $i$. In this paper I settle this conjecture by constructing a local cohomology module of a local $k$-algebra with an infinite set of associated primes, and I do this for any field $k$.
Assume $R$ is a local Cohen-Macaulay ring. It is shown that $\Ass_R (H^l_I(R))$ is finite for any ideal $I$ and any integer $l$ provided $\Ass_R (H^2_{(x,y)}(R))$ is finite for any $x,y\in R$ and $\Ass_R (H^3_{(x_1,x_2,y)}(R))$ is finite for any $y\in R$ and any regular sequence $x_1,x_2\in R$. Furthermore it is shown that $\Ass_R (H^l_I(R))$ is always finite if $\dim (R)\leq 3$. The same statement is even true for $\dim (R)\leq 4$ if $R$ is almost factorial.
Let $R$ be a polynomial ring in finitely many variables over the integers, and fix an ideal $I$ of $R$. We prove that for all but finitely prime integers $p$, the Bockstein homomorphisms on local cohomology, $H^k_I(R/pR)\to H^{k+1}_I(R/pR)$, are zero. This provides strong evidence for Lyubeznik's conjecture which states that the modules $H^k_I(R)$ have a finite number of associated prime ideals.
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。
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{…
We compute support of formal cohomology modules in a serial of non-trivial cases. Applications are given. For example, we compute injective dimension of certain local cohomology modules in terms of dimension of their's support.
Let R be a regular local ring of dimension d, I an ideal of R, and M a finitely generated R-module of dimension n. We prove that the set of associated primes of Ext^i_R(R/I,H^j_I(M)) is finite for all i and j in the following cases: (1) dim M\le 3; (2) dim R\le 4; (3) dim M/IM \le 2 and M satisfies Serre's condition S_{n-3}; (4) dim M/IM\le 3, R is unramified, and M is faithful and satisfies S_{n-3}. We also prove that if dim R/I\ge 2 and the punctured spectrum of R/I is disconnected then H^{d-1}_I(R) is not I-cofinite. This generalizes a result due to Huneke and Koh.
Let $(R,\m)$ be a Noetherian local ring and $M$ a finitely generated $R$-module. Following I. G. Macdonald \cite{Mac}, the set of all attached primes of the Artinian local cohomology module $H^i_{\m}(M)$ is denoted by $\Att_R(H^i_{\m}(M))$. In \cite[Theorem 3.7]{Sh}, R. Y. Sharp proved that if $R$ is a quotient of a Gorenstein local ring then the shifted localization principle always holds true, i.e. $$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \Att_{R_{\p}}\big(H^{i-\dim (R/\p)}_{\p R_{\p}}(M_{\p})\big)=\big\{\q R_{\p}\mid \q\in\Att_RH^i_{\m}(M), \q\subseteq \p\big\} \ \ \ \ \ \ \ \ \ \…
Let $R$ be a noetherian ring, $\fa$ an ideal of $R$, $M$ an $R$--module and $n$ a non-negative integer. In this paper we first will study the finiteness properties of the kernel and the cokernel of the natural map $f:\Ext^n_{R}(R/\fa,M)\lo \Hom_{R}(R/\fa,\lc^{n}_{\fa}(M))$. Then we will get some corollaries about the associated primes and artinianness of local cohomology modules. Finally we will study the asymptotic behaviour of the kernel and the cokernel of this natural map in the graded case.
Betti Numbers and Formal Local Cohomology Modules
本文研究诺特局部环上有限生成模的Betti数与形式局部上同调模的关系，建立了消失与深度的联系，推广经典结果，并应用于Cohen-Macaulay环。
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。
The Second Vanishing Theorem for Formal Local Cohomology Modules
本文在诺特局部环上引入形式维数，通过商环维数刻画高阶形式局部上同调消失，将经典定理推广至形式情形，并应用于导出范畴。
Let $(R,\m)$ be a Noetherian local ring $I, J$ two ideals of $R$ and $M$ a finitely generated $R-$module. It is first shown that for $k\geq -1$ the integer $r_k = \depth_k(I,J^nM/J^{n+1}M)$, it is the length of a maximal $(J^nM/J^{n+1}M)-$sequence in dimension $>k$ in $I$ defined by M. Brodmann and L. T. Nhan \cite{BN}, becomes for large $n$ independent of $n$. Then we prove in this paper that the sets $\bigcup_{j\le r_k}\Ass_R(H^j_I(J^nM/J^{n+1}M))$ with $k=-1$ or $k=0$, and $\bigcup_{j\le r_1}\Ass_R(H^j_I(J^nM/J^{n+1}M))\cup\{\m\}$ are stable for large $n$. We also obtain similar results for…
Let $\fa$ be an ideal of a commutative Noetherian ring $R$ with identity and let $M$ and $N$ be two finitely generated $R$-modules. Let $t$ be a positive integer. It is shown that $\Ass_R(H_{\fa}^t(M,N))$ is contained in the union of the sets $\Ass_R(\Ext_R^i(M,H_{\fa}^{t-i}(N)))$, where $0\leq i\leq t$. As an immediate consequence, it follows that if either $H_{\fa}^i(N)$ is finitely generated for all $i<t$ or $\Supp_R(H_{\fa}^i(N))$ is finite for all $i<t$, then $\Ass_R(H_{\fa}^t(M,N))$ is finite. Also, we prove that if $d=\pd(M)$ and $n=\dim(N)$ are finite, then $H_{\fa}^{d+n}(M,N)$ is Arti…
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].
Bass Numbers of the First Nonzero Local Cohomology Module
本文在正则局部环中计算第一个非零局部上同调模的前两个Bass数，并通过高一阶局部上同调模的伴随素理想研究第三个Bass数。
Let $A$ be a ring and $R$ be a polynomial or a power series ring over $A$. When $A$ has dimension zero, we show that the Bass numbers and the associated primes of the local cohomology modules over $R$ are finite. Moreover, if $A$ has dimension one and $π$ is an nonzero divisor, then the same properties hold for prime ideals that contain $π.$ These results do not require that $A$ contains a field. As a consequence, we give a different proof for the finiteness properties of local cohomology over unramified regular local rings. In addition, we extend previous results on the injective dimension of…
In the first section of this paper we present generalizations of known results on the set of associated primes of Matlis duals of local cohomology modules; we prove these generalizations by using a new technique. In section 2 we compute the set of associated primes of the Matlis dual of $\LCMo ^{d-1}_J(R)$, where $R$ is a $d$-dimensional local ring and $J\subseteq R$ an ideal such that $\dim (R/J)=1$ and $\LCMo ^d_J(R)=0$.
The asymptotic stability of several homological invariants of the graded pieces of a graded module has attracted quite a lot of attention over the last decades. We provide in this text several stability results together with estimates of the degree from which it stabilizes. Before we establish these regularity results, we prove several facts about depth and cohomological dimension with respect to a finitely generated ideal and about Castelnuovo-Mumford regularity of a graded module.
Lyubeznik conjectured that local cohomology modules of regular rings have finitely many associated primes. We examine this conjecture for polynomial rings over the integers, and record some equational identities that arise from studying special cases.
The Second Vanishing Theorem for Local Cohomology Modules
本文证明了非分歧正则局部环局部上同调模的第二消失定理并给出素特征$p$下的新证明，将等特征局部环最高Lyubeznik数的拓扑刻画推广到混合特征，还得到了混合特征下局部上同调维数的上界。
Associate primes of local cohomology modules over certain quotients of regular rings
本文证明在正则环的商环 $R/(\mathbf{x})^m$ 上，对任意理想 $Q$，局部上同调模的相伴素理想集合在正特征或零特征（局部或光滑仿射）情形下是有限的。
Let \frak a be an ideal of a commutative Noetherian ring R and M a finitely generated R-module. It is shown that {\rm Ann}_R(H_{\frak a}^{{\dim M}({\frak a}, M)}(M))= {\rm Ann}_R(M/T_R({\frak a}, M)), where T_R({\frak a}, M) is the largest submodule of M such that {\rm cd}({\frak a}, T_R({\frak a}, M))< {\rm cd}({\frak a}, M). Several applications of this result are given. Among other things, it is shown that there exists an ideal \frak b of R such that {\rm Ann}_R(H_{\frak a}^{\dim M}(M))={\rm Ann}_R(M/H_{\frak b}^{0}(M)). Using this, we show that if H_{\frak a}^{\dim R}(R)=0, then {\rm Att}_…
In this paper we present a technical lemma about localization at countable infinitely many prime ideals. We apply this lemma to get many results about the finiteness of associated prime ideals of local cohomology modules.
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…
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 $(R,\mathfrak{m})$ 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 describe the support of $F$-finite $F$-modules over polynomial rings $R$ of prime characteristic. Our description yields an algorithm to compute the support of such modules; the complexity of our algorithm is also analyzed. To the best of our knowledge, this is the first algorithm to avoid extensive use of Gröbner bases and hence of substantial practical value. We also use the idea behind this algorithm to prove that the support of $H^j_I(S)$ is Zariski closed for each ideal $I$ of $S$ where $R$ is noetherian commutative ring of prime characteristic with finitely many isolated singular poin…
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