Ryo Takahashi, Yuji Yoshino, Takeshi Yoshizawa
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.
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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
We characterize the class of ideals of a polynomial ring such that the hilbert series of their graded local cohomology modules is maximal.
We first provide here a very short proof of a refinement of a theorem of Kodiyalam and Cutkosky, Herzog and Trung on the regularity of powers of ideals. This result implies a conjecture of Hà and generalizes a result of Eisenbud and Harris concerning the case of ideals primary for the graded maximal ideal in a standard graded algebra over a field. It also implies a new result on the regularities of powers of ideal sheaves. We then compare the cohomology of the stalks and the cohomology of the fibers of a projective morphism to the effect of comparing the maximum over fibers and over stalks of …
The study of Frobenius actions on local cohomology modules over a local ring of prime characteristic has interesting connections with the theory of tight closure. This paper establishes new connections by developing the notion of relative Frobenius actions on local cohomology. As an application, we show that a ring has $F$-nilpotent singularities if and only if the tight closure of every parameter ideal is equal to its Frobenius closure.
Let $k$ be a field of characteristic zero and I an ideal defining an arrangement of linear subspaces in the affine space $A^n_k$. We compute the D-module theoretic characteristic cycle of the local cohomology modules $H^r_I(k[x_1,...,x_n])$ in terms of the poset defined by the arrangement. In case I is a monomial ideal, we relate the multiplicities of the characteristic cycle with the Betti numbers of the Alexander dual ideal of I, we also study some extension problems attached to the modules $H^r_I(k[x_1,...,x_n])$. If $k$ is the field of complex numbers we study the ubiquity of these local c…
For positive integers m >= n >= p, we compute the GL_m x GL_n-equivariant description of the local cohomology modules of the polynomial ring S of functions on the space of m x n matrices, with support in the ideal of p x p minors. Our techniques allow us to explicitly compute all the modules Ext_S(S/I_x,S), for x a partition and I_x the ideal generated by the irreducible sub-representation of S indexed by x. In particular we determine the regularity of the ideals I_x, and we deduce that the only ones admitting a linear free resolution are the powers of the ideal of maximal minors of the generi…
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…
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.
We prove an inequality between Hilbert functions of local cohomology modules supported in the homogeneous maximal ideal of standard graded algebras over a field, within the framework of embeddings of posets of Hilbert functions. As a main application, we prove an analogue for local cohomology of Evans' Lex-Plus-Power Conjecture for Betti numbers. This results implies some cases of the classical Lex-Plus-Power Conjecture, namely an inequality between extremal Betti numbers. In particular, for the classes of ideals for which the Eisenbud-Green-Harris Conjecture is currently known, the projective…
We compute the local cohomology modules H_Y^(X,O_X) in the case when X is the complex vector space of n x n symmetric, respectively skew-symmetric matrices, and Y is the closure of the GL-orbit consisting of matrices of any fixed rank, for the natural action of the general linear group GL on X. We describe the D-module composition factors of the local cohomology modules, and compute their multiplicities explicitly in terms of generalized binomial coefficients. One consequence of our work is a formula for the cohomological dimension of ideals of even minors of a generic symmetric matrix: in the…
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 Hilb^p be the Hilbert scheme parametrizing the closed subschemes of P^n with Hilbert polynomial p \in Q[t] over a field K of characteristic zero. By bounding below the cohomological Hilbert functions of the points of Hilb^p we define locally closed subspaces of the Hilbert scheme. The aim of this thesis is to show that some of these subspaces are connected. For this we exploit the ideals constructed by D. Mall. It turns out that these ideals are sequentially Cohen-Macaulay and that their initial ideals and their generic initial ideals coincide for any admissible term order.
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 Hilb^p be the Hilbert scheme parametrizing the closed subschemes of P^n with Hilbert polynomial p\in Q[t] over a field K of characteristic zero. By bounding below the cohomological Hilbert functions of the points of Hilb^p we define locally closed subspaces of the Hilbert scheme. The aim of this paper is to show that some of these subspaces are connected. For this we exploit the edge ideals constructed by D. Mall. It turns out that these ideals are sequentially Cohen-Macaulay and that their initial ideals with respect to the reverse lexicographic term order are generic initial ideals.
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.
In this article we extend a previous definition of Castelnuovo-Mumford regularity for modules over an algebra graded by a finitely generated abelian group. Our notion of regularity is based on Maclagan and Smith's definition, and is extended first by working over any commutative base ring, and second by considering local cohomology with support in an arbitrary finitely generated graded ideal $B$, obtaining, for each $B$, a $B$-regularity region. The first extension provides a natural approach for working with families of sheaves or of graded modules, while the second opens new applications. We…
Let $R$ be a polynomial ring over a field $k$ with irrelevant ideal $\frak m$ and dimension $d$. Let $I$ be a homogeneous ideal in $R$. We study the asymptotic behavior of the length of the modules $H^{i}_{\frak m}(R/I^n)$ for $n\gg 0$. We show that for a fixed number $α\in \mathbb Z$, $\limsup_{n\rightarrow \infty}\frac{λ(H^{i}_{\frak m}(R/I^n)_{\geq -αn})}{n^d}<\infty.$ Combining this with recent strong vanishing results gives that $\limsup_{n\rightarrow \infty}\frac{λ(H^{i}_{\frak m}{R/I^n})}{n^d}<\infty$ in many situations. We also establish that the actual limit exists and is rational for…
We provide a Hochster type formula for the local cohomology modules of binomial edge ideals. As a consequence we obtain a simple criterion for the Cohen-Macaulayness of these ideals and we describe their Castelnuovo-Mumford regularity and their Hilbert series. We also prove a conjecture of Conca, De Negri and Gorla relating the graded components of the local cohomology modules of binomial edge ideals and their generic initial ideals.
We study the deformation theory of quotients of polynomial rings by quadratic monomial ideals. More precisely we compute the first cotangent cohomology module of such rings. We also give a criterion for vanishing of second cotangent cohomology module.
Some affirmative answers are given to Huneke's problems. The calculation of local cohomology modules with respect to an arbitrary pair of ideals $I,J$ can be reduced to calculation of local cohomology modules with respect to a pair of ideala whose first ideal is generated by any $k$-regular sequence in $I$.
Epimorphisms of local cohomology modules, a general Peskine-Szpiro theorem, and an application to sheaf cohomology vanishing for thickenings
本文研究局部上同调模的满射性质，推广Peskine-Szpiro定理，并证明了增厚子方案上同调的Kodaira型消失定理。
Let $R_0$ be any domain, let $R=R_0[U_1, ..., U_s]/I$, where $U_1, ..., U_s$ are indeterminates of some positive degrees, and $I\subset R_0[U_1, ..., U_s]$ is a homogeneous ideal. The main theorem in this paper is states that all the associated primes of $H:=H^s_{R_+}(R)$ contain a certain non-zero ideal $c(I)$ of $R_0$ called the ``content'' of $I$. It follows that the support of $H$ is simply $V(\content(I)R + R_+)$ (Corollary 1.8) and, in particular, $H$ vanishes if and only if $c(I)$ is the unit ideal. These results raise the question of whether local cohomology modules have finitely many …
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.
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 number of equations needed to cut out a variety given by an ideal is called the arithmetic rank (of the ideal). It was shown in [8] that the notion of arithmetic rank is strongly related to the concept of regular sequences on the Matlis duals of certain local cohomology modules and thus is related to the set of associated primes of such Matlis duals. While in [8] the general situation with no restrictions on arithmetic rank and cohomological dimension was investigated, this work concentrates on the case ``cohomological dimension of the given ideal is (at most) one'' and also on the interpl…
We introduce a notion of generalized local cohomology modules with respect to a pair of ideals $(I,J)$ which is a generalization of the concept of local cohomology modules with respect to $(I,J).$ We show that generalized local cohomology modules $H^i_{I,J}(M,N)$ can be computed by the Čech cohomology modules. We also study the artinianness of generalized local cohomology modules $H^i_{I,J}(M,N).$
Let $A$ be a noetherian ring, $\fa$ an ideal of $A$, and $M$ an $A$--module. Some uniform theorems on the artinianness of certain local cohomology modules are proven in a general situation. They generalize and imply previous results about the artinianness of some special local cohomology modules in the graded case.
Let $S$ be the coordinate ring of the space of $n\times n$ complex skew-symmetric matrices. This ring has an action of the group $\textrm{GL}_n(\mathbb{C})$ induced by the action on the space of matrices. For every invariant ideal $I\subseteq S$, we provide an explicit description of the modules $\textrm{Ext}^{\bullet}_S(S/I,S)$ in terms of irreducible representations. This allows us to give formulas for the regularity of basic invariant ideals and (symbolic) powers of ideals of Pfaffians, as well as to characterize when these ideals have a linear free resolution. In addition, given an inclusi…
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.
Cohomology of flag supervarieties and resolutions of determinantal ideals. II
本文计算了复periplectic Grassmannian结构层的凝聚上同调，证明其可分解为辛或正交Grassmannian上同调环与periplectic Lie超群半单表示的张量积，并建立了与行列式理想syzygy的联系。
In this paper, we prove some well-known results on local cohomology with respect to a pair of ideals in graded version, such as, Independence Theorem, Lichtenbaum-Harshorne Vanishing Theorem, Basic Finiteness and Vanishing Theorem, among others. Besides, we present a generalized version of Melkersson Theorem about Artinianess of modules and a result concerning Artinianess of local cohomology modules.
Let $R$ be a commutative Noetherian ring, $Φ$ a system of ideals of $R$, $\fa \in Φ$, $M$ an arbitrary $R$-module and $t$ a non-negative integer. Let $\mathcal{S}$ be a Melkersson subcategory of $R$-modules. Among other things, we prove that if $\lc^{i}_Φ(M)$ is in $\mathcal{S}$ for all $i < t$ then $\lc^{i}_\fa(M)$ is in $\mathcal{S}$ for all $i < t$ and for all $\fa \in Φ$. If $\mathcal{S}$ is the class of $R$-modules $N$ with $\dim N \leq k$ where $k \geq -1$, is an integer, then $\lc^{i}_Φ(M)$ is in $\mathcal{S}$ for all $i < t$ (if and only if) $\lc^{i}_\fa(M)$ is in $\mathcal{S}$ for all…
We generalize some known results on the relation between the cohomological and projective dimension. Then we examine the set-theoretically Cohen-Macaulay ideals to find some cohomological characterization of these kind of ideals.
Let $R$ be a noetherian local ring. We consider the following quastion: Does there exist an integer $n$ such that all idelas generated by a system of parameters contained in the $n$-th power of the maximal ideal have the same Betti numbers? We obtain a positive answer for some rings with finite local cohomology.
Let k be a field of characteristic 0, R = k[x_1, ..., x_d] be a polynomial ring, and m its maximal homogeneous ideal. Let I be a homogeneous ideal in R. In this paper we investigate asymptotic behaviour of the quotient between the length of local cohomology group H^0_m(R/I^n) and n^d. We show that this quantity always has a limit as n goes to infinity. We also give an example for which the limit is irrational; in particular, this proves that the length of H^0_m(R/I^n) is not asymptotically a polynomial in n.
Cohomology of flag supervarieties and resolutions of determinantal ideals
研究广义旗超簇的凝聚层上同调，揭示其与行列式理想自由消解的关联；在超Grassmann流形情形完全计算结构层上同调，并借助Grothendieck–Springer消解类似物的几何给出Pragacz–Weyman结果的概念性解释。
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 $p>5$.
We find conditions on the local cohomology modules of multi-Rees algebras of admissible filtrations which enable us to predict joint reduction numbers. As a consequence we are able to prove a generalisation of a result of Reid-Roberts-Vitulli in the setting of analytically unramified local rings for completeness of power products of complete ideals.
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 S be a toric algebra over a field K of characteristic 0 and let I be a monomial ideal of S. We show that the local cohomology modules H^i_I(S) are of finite length over the ring of differential operators D(S;K), generalizing the classical case of a polynomial algebra S. As an application, we compute the characteristic cycles of some local cohomology modules.
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 $t= cd(I,J,M)$.
Annihilation of cohomology over one dimensional almost Gorenstein rings
本文研究一维几乎Gorenstein环的上同调湮灭理想，给出其与稳定范畴湮灭理想相等的充分条件，并证明其与导子理想一致。
Vanishing of local cohomology in unramified mixed characteristic
本文在非分歧混合特征环境下扩展了 Faltings 关于理想上同调维数的上界结果，并证明该上界是紧致的。
In this work we describe the local cohomology of reflexive modules of rank one over normal semigroup rings with respect to monomial ideals. Using our description we show that the problem of classifying maximal Cohen-Macaulay modules of rank one can be rephrased in terms of finding integral solutions to certain sets of linear inequalities.
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…
Let R be a commutative Noetherian d-dimensional complete equicharacterisitc regular local ring and let I be an ideal of R such that every minimal prime over I has height at most c. Let v=d - [(d-2)/c]-1 and v'=d - [(d-1)/c]. It has been known that the i-th local cohomology module of any R-module M with support in I vanishes for i>v', and if I is prime, for i>v; both results are sharp. The purpose of this paper is to prove a necessary and sufficient condition, for a not necessarily prime I, for the vanishing in the range i>v, i.e. in the same range as for a prime ideal. The condition is in term…
Bass numbers of local cohomology modules at the first and last non-vanishing levels
本文研究局部余同调模在特定条件下的Bass数，探讨了在不同层次下的计算方法和贡献。
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.
Cofiniteness of generalized loca cohomology modules with respect to a system of ideals
本文研究了交换Noether环上相对于理想系统的广义局部上同调模$H^i_Φ(M,X)$的Artin性与余有限性，该模扩展了Herzog的广义局部上同调模。
Stability of annihilators of cohomology and closed subsets defined by Jacobian ideals
本文证明了交换 Noether 环中上同调零化子幂次对 Ext 模的零化性质，并利用上同调零化子与 Jacobson 理想定义的闭子集刻画了特定环的等维性。
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 $R = \bigoplus_{n \in \mathbb{N}_{0}} R_{n}$ be a standard graded ring, $M$ be a finite graded $R$-module and $J$ be a homogenous ideal of $R$. In this paper we study the graded structure of the $i$-th local cohomology module of $M$ defined by a pair of ideals $(R_{+},J)$, i.e. $H^{i}_{R_{+},J}(M)$. More precisely, we discuss finiteness property and vanishing of the graded components $H^{i}_{R_{+},J}(M)_{n}$. Also, we study the Artinian property and tameness of certain submodules and quotient modules of $H^{i}_{R_{+},J}(M)$.
We consider the ring S=C[x_ij] of polynomial functions on the vector space C^(m x n) of complex m x n matrices. We let GL= GL_m x GL_n and consider its action via row and column operations on C^(m x n) (and the induced action on S). For every GL-invariant ideal I in S and every j>=0, we describe the decomposition of the modules Ext^j_S(S/I,S) into irreducible GL-representations. For any inclusion I into J of GL-invariant ideals we determine the kernels and cokernels of the induced maps Ext^j_S(S/I,S) -> Ext^j_S(S/J,S). As a consequence of our work, we give a formula for the regularity of the p…
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 $i=1,2$.
This paper examines the dimension of the graded local cohomology $H_\mathfrak{m}^p(S/K^s)_γ$ and $H_\mathfrak{m}^p(S/K^{(s)})$ for a monomial ideal $K$. This information is encoded in the reduced homology of a simplicial complex called the degree complex. We explicitly compute the degree complexes of ordinary and symbolic powers of sums and fiber products of ideals, as well as the degree complex of the mixed product, in terms of the degree complexes of their components. We then use homological techniques to discuss the cohomology of their quotient rings. In particular, this technique allows fo…
We prove that if M is a finitely-generated module of dimension d with finite local cohomologies over a Noetherian local ring, and if the ith local cohomology module of M is zero unless i = d, i = 0, and i = r for some r strictly between 0 and d, then there is an integer l such that every parameter ideal for M contained in the lth power of the maximal ideal has the same index of reducibility on M. This theorem generalizes work of the second author (math.AC/0408143) and is closely related to work of S. Goto, H. Sakurai (math.AC/0306148), and N. Suzuki.
Homology of multiple complexes and Mayer-Vietoris spectral sequences
本文利用多个复形的上同调及其两种相关构造，统一推导两类多理想 Mayer–Vietoris 谱序列，并发现理想积上同调可由理想和 Čech 复形的子复形计算。
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.
Let $(R,\mathfrak{m},k)$ denote a local ring. For $I$ and $J$ ideals of $R$, for all integer $i$, let $H^i_{I,J}(-)$ denote the $i$-th local cohomology functor with respect to $(I,J)$. Here we give a generalized version of Local Duality Theorem for local cohomology defined by a pair of ideals. Also, for $M$ be a finitely generated $R$-module, we study the behavior of the endomorphism rings $H^t_{I,J}(M)$ and $D(H^t_{I,J}(M))$ where $t$ is the smallest integer such that the local cohomology with respect to a pair of ideals is non-zero and $D(-):= {\rm Hom}_R(-,E_R(k))$ is the Matlis dual functo…
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