Mohsen Asgharzadeh
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.
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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.
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。
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.
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。
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].
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 $\operatorname{Ass}(\mathfrak{F}_{\mathfrak{a}}^t(M))$.
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 \…
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 $\mathfrak{a}$ be an ideal of local ring $(R,\mathfrak{m})$ and $M$ a finitely generated $R$-module and $n\in\Bbb N$. It is shown that some results concerning cominimaxness of formal local cohomology modules.
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 coefficients---supported at generic determinantal ideals---and also obtain results on F-modules and D-modules that are likely to be of independent interest.
This paper has been withdrawn by the author due to a crucial sign error in Theorem 3.4.
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 analyze their $R$-module structure, the upper and lower vanishing and non-vanishing in terms of intrinsic data of $M,$ and its functorial behavior. These cohomology modules occur in relation to the formal completion of the punctured spectrum $\Spec R \setminus V(\mathfrak m).$ As a new cohomological data there is a description on the formal grade $\fgrade(\mathfra…
The Frobenius depth denoted by F-depth defined by Hartshorne-Speiser in 1977 and later by Lyubeznik in 2006, in a different way, for rings of positive characteristic. The first aim of the present paper is to compare the F-depth with formal grade and reprove some results of Lyubeznik using formal local cohomology. Then the endomorphism rings of local cohomology modules will be considered. As an application, we reprove the results due to Huneke-Koh in positive characteristic and Lyubeznik in characteristic zero on the annihilators of local cohomology modules.
Let $(A,\mathfrak{m})$ be a complete intersection with $k = A/\mathfrak{m}$ algebraically closed. Let CMS(A) be the stable category of maximal CM $A$-modules. For a large class of thick subcategories $\mathcal{S}$ of CMS(A) we show that there is a theory of support varieties for the Verdier quotient $\mathcal{T} = $ CMS(A)$/\mathcal{S}$. As an application we show that the analogous version of Auslander-Reiten conjecture, Murthys result, Avramov-Buchweitz result on symmetry of vanishing of cohomology holds for $\mathcal{T}$.
Cohen-Macaulayness of formal fibers and dimension of local cohomology modules
本文研究了局部环中形式纤维的Cohen-Macaulay性质与局部上同调模维数的关系，证明了在特定条件下环的维数与模的支撑关系。
In this expository paper, we give a complete proof of van den Essen's theorem that the de Rham cohomology spaces of a holonomic D-module are finite-dimensional in the case of a formal power series ring over a field of characteristic zero. This proof requires results from at least five of van den Essen's papers as well as his unpublished thesis, and until now has not been available in a self-contained document.
We introduce an idea for generalization of a local cohomology module, which we call a local cohomology module with respect to a pair of ideals (I,J), and study their various properties. Some vanishing and nonvanishing theorems are given for this generalized version of local cohomology. We also discuss its connection with the ordinary local cohomology.
Local cohomology modules with nonclosed support
该研究构造了具有非闭支集的局部上同调模对应的诺特环，回答了Huneke-Lyubeznik提出的问题
In this paper we study cohomology and deformations of Jacobi-Jordan algebras. We develop their formal deformation theory. In particular, we introduce a method to construct a versal deformation for a given Jacobi-Jordan algebra, which can induce all deformations and is unique on the infinitesimal level. We construct formal 1-parameter deformations of Jacobi-Jordan algebras up to dimension 5 and versal deformations for 3-dimensional Jacobi-Jordan algebras.
These are the notes from an Oberwolfach Seminar which we ran from 23--29 May 2010.
We employ the inductive structure of determinantal varieties to calculate the mixed Hodge module structure of local cohomology modules with determinantal support. We show that the weight of a simple composition factor is uniquely determined by its support and cohomological degree. As a consequence, we obtain the equivariant structure of the Hodge filtration on each local cohomology module. Finally, as an application, we provide a formula for the generation level of the Hodge filtration on these modules.
We study the local cohomology modules H^i_B(R) for a reduced monomial ideal B in a polynomial ring R=k[X_1,...,X_n]. We consider a grading on R which is coarser than the Z^n-grading such that each component of H^i_B(R) is finite dimensional and we give an effective way to compute these components. Using Cox's description for sheaves on toric varieties, we apply these results to compute the cohomology of coherent sheaves on toric varieties. We give algorithms for this computation which have been implemented in the Macaulay 2 system. We obtain also a topological description for the cohomology of…
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.
For a polynomial ring $R=k[x_1,...,x_n]$, we present a method to compute the characteristic cycle of the localization $R_f$ for any nonzero polynomial $f\in R$ that avoids a direct computation of $R_f$ as a $D$-module. Based on this approach, we develop an algorithm for computing the characteristic cycle of the local cohomology modules $H^r_I(R)$ for any ideal $I\subseteq R$ using the Čech complex. The algorithm, in particular, is useful for answering questions regarding vanishing of local cohomology modules and computing Lyubeznik numbers. These applications are illustrated by examples of com…
We compute the GL-equivariant description of the local cohomology modules with support in the ideal of maximal minors of a generic matrix, as well as of those with support in the ideal of 2n x 2n Pfaffians of a (2n+1)x(2n+1) generic skew-symmetric matrix. As an application, we characterize the Cohen-Macaulay modules of covariants for the action of the special linear group SL(G) on G^m. The main tool we develop is a method for computing certain Ext modules based on the geometric technique for computing syzygies and on Matlis duality.
Suppose that k is a field of characteristic zero, X is an r by s matrix of indeterminates, where r \leq s, and R = k[X] is the polynomial ring over k in the entries of X. We study the local cohomology modules H^i_I(R), where I is the ideal of R generated by the maximal minors of X. We identify the indices i for which these modules vanish, compute H^i_I(R) at the highest nonvanishing index, i = r(s-r)+1, and characterize all nonzero ones as submodules of certain indecomposable injective modules. These results are consequences of more general theorems regarding linearly reductive groups acting o…
Noncommutative supports, local cohomology and spectral sequences
本文在 Artin-Zhang 非交换代数几何框架中，利用初等对象及其内射包建立支集与相伴素理想理论，并由此导出局部上同调相关导出函子的谱序列及广义 Nagata 理想变换。
In this article we study a theory of support varieties over a skew complete intersection $R$, i.e. a skew polynomial ring modulo an ideal generated by a sequence of regular normal elements. We compute the derived braided Hochschild cohomology of $R$ relative to the skew polynomial ring and show its action on $\mathrm{Ext}_R(M,N)$ is noetherian for finitely generated $R$-modules $M$ and $N$ respecting the braiding of $R$. When the parameters defining the skew polynomial ring are roots of unity we use this action to define a support theory. In this setting applications include a proof of the Gen…
We propose a new method for defining a notion of support for objects in any compactly generated triangulated category admitting small coproducts. This approach is based on a construction of local cohomology functors on triangulated categories, with respect to a central ring of operators. Suitably specialized one recovers, for example, the theory for commutative noetherian rings due to Foxby and Neeman, the theory of Avramov and Buchweitz for complete intersection local rings, and varieties for representations of finite groups according to Benson, Carlson, and Rickard. We give explicit examples…
Local cohomology with Schubert support on compact Hermitian symmetric spaces
本文为紧致埃尔米特对称空间上舒伯特簇支撑的局部上同调模提供了统一的根系构造，给出权滤过公式，并应用于对称行列式与Pfaffian簇，同时计算了霍奇同调与上同调亏格。
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