Tameness of Local cohomology of monomial ideals with respect to monomial prime ideals

Graded components of local cohomology modules supported on $\mathfrak{C}$-monomial ideals

Let $A$ be a Dedekind domain of characteristic zero such that its localization at every maximal ideal has mixed characteristic with finite residue field. Let $R=A[X_1,\ldots, X_n]$ be a polynomial ring and $I=(a_1U_1, \ldots, a_c U_c)\subseteq R$ an ideal, where $a_j \in A$ (not necessarily units) and $U_j$'s are monomials in $X_1, \ldots, X_n$. We call such an ideal $I$ as a $\mathfrak{C}$-monomial ideal. Consider the standard multigrading on $R$. We produce a structure theorem for the multigraded components of the local cohomology modules $H^i_I(R)$ for $i \geq 0$. We further analyze the tor…

On graded local cohomology modules defined by a pair of ideals

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 $H^{i}_{R_{+},J}(M)$.

Cohomology on Toric Varieties and Local Cohomology with Monomial Supports

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…

特征零下$\mathfrak{C}$-单项式理想的局部上同调模的分次分量

Graded components of local cohomology modules of $\mathfrak{C}$-monomial ideals in characteristic zero

本文研究特征零情形下$\mathfrak{C}$-单项式理想的局部上同调模的分次分量，证明其性质仅依赖于指标的负坐标位置，并刻画了Bass数、内射维数、相伴素理想等不变量，在正则基环和一元形式幂级数环情形下给出有限性、维数上界与结构定理。

Local Cohomology, Arrangement of Subspaces and Monomial Ideals

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…

Lyubeznik numbers of monomial ideals

We study Bass numbers of local cohomology modules supported on squarefree monomial ideals paying special attention to Lyubeznik numbers. We build a dictionary between local cohomology modules and minimal free resolutions that allow us to interpret Lyubeznik numbers as the obstruction to the acyclicity of the linear strands of the Alexander dual ideals. The methods we develop also help us to give a bound for the injective dimension of the local cohomology modules in terms of the dimension of the small support.

Lengths of Local Cohomology of Thickenings

Let $R$ be a standard graded polynomial ring that is finitely generated over a field of characteristic $0$, let $\mathfrak{m}$ be the homogeneous maximal ideal of $R$, and let $I$ be a homogeneous prime ideal of $R$. Dao and Montaño defined an invariant that, in the case that $\operatorname{Proj}(R/I)$ is lci and for cohomological index less than $\dim(R/I)$, measures the asymptotic growth of lengths of local cohomology modules of thickenings. They showed its existence and rationality for certain classes of monomial ideals $I$. The following affirms that the invariant exists and is rational fo…

Graded components of Local cohomology modules II

Let $A$ be a commutative Noetherian ring containing a field of characteristic zero. Let $R= A[X_1, \ldots, X_m]$ be a polynomial ring and $A_m(A) = A \langle X_1, \ldots, X_m, \partial_1, \ldots, \partial_m \rangle$ be the $m^{th}$ Weyl algebra over $A$, where $\partial_i=\partial/\partial X_i$. Consider both $R$ and $A_m(A)$ as standard graded with $\mathrm{deg}~ z=0$ for all $z \in A$, $\mathrm{deg}~ X_i=1$, and $\mathrm{deg}~ \partial_i =-1$ for $i=1, \ldots, m$. We present a few results about the behavior of the graded components of local cohomology modules $H_I^i(R)$, where $I$ is an arbi…

Local cohomology and pure morphisms

We study a question raised by Eisenbud, Mustata, and Stillman regarding the injectivity of natural maps from Ext modules to local cohomology modules. We obtain some positive answers to this question which extend earlier results of Lyubeznik. In the process, we also prove a vanishing theorem for local cohomology modules which connects theorems previously known in the case of positive characteristic and in the case of monomial ideals.

The Auslander-Reiten translate on monomial quotient rings

For a multidegree t in N^n, E.Miller has defined a category of positively t-determined modules over the polynomial ring S in n variables. We consider the Auslander-Reiten translate, Na_t, on the (derived) category of such modules. A monomial ideal I is positively t-determined if every generator x^a has a \leq t. We compute the multigraded cohomology- and betti spaces of Na_t^k(S/I) for every iterate k, and also the S-module structure of these cohomology modules. This comprehensively generalizes results of Hochster and Gräbe on local cohomology of Stanley-Reisner rings.

相对Hochster—Takayama公式与Cohen—Macaulay单项理想商理想

Relative Hochster--Takayama formula and Cohen--Macaulay monomial ideal quotients

本文提出相对Hochster—Takayama公式，用于研究单项理想商理想，给出Cohen-Macaulay性判别准则及符号幂滤相关模的性质。

Limit Behavior of the Rational Powers of Monomial Ideals

We investigate the rational powers of ideals. We find that in the case of monomial ideals, the canonical indexing leads to a characterization of the rational powers yielding that symbolic powers of squarefree monomial ideals are indeed rational powers themselves. Using the connection with symbolic powers techniques, we use splittings to show the convergence of depths and normalized Castelnuovo-Mumford regularities. We show the convergence of Stanley depths for rational powers, and as a consequence of this we show the before-now unknown convergence of Stanley depths of integral closure powers. …

Derived Category of Squarefree Modules and Local Cohomology with Monomial Ideal Support

A "squarefree module" over a polynomial ring $S = k[x_1, .., x_n]$ is a generalization of a Stanley-Reisner ring, and allows us to apply homological methods to the study of monomial ideals systematically. Let $Sq$ be the category of squarefree modules. Then the derived category $D^b(Sq)$ of $Sq$ has three duality functors which act on $D^b(Sq)$ just like three transpositions of the symmetric group $S_3$ (up to translation). This phenomenon is closely related to the Koszul dulaity (in particular, the Bernstein-Gel'fand-Gel'fand correspondence). We also study the local cohomology module $H_{I_Δ}…

Torsion functors with monomial support

The dependence of torsion functors on their supporting ideals is investigated, especially in the case of monomial ideals of certain subrings of polynomial algebras over not necessarily Noetherian rings. As an application it is shown how flatness of quasicoherent sheaves on toric schemes is related to graded local cohomology.

Local Cohomology at Monomial Ideals

For a reduced monomial ideal B in R=k[X_1,...,X_n], we write H^i_B(R) as the union of {Ext^i(R/B^[d],R)}_d, where {B^[d]}_d are the "Frobenius powers of B". We describe H^i_B(R)_p, for every p in Z^n, in the spirit of the Stanley-Reisner theory. As a first application we give an isomorphism Tor_i(B', k)_p\iso Ext^{|p|-i}(R/B,R)_{-p} for all p in {0,1}^n, where B' is the Alexander dual ideal of B. We deduce a canonical filtration of Ext^i(R/B,R) with succesive quotients of the form R/(X_{j_1},...,X_{j_i}) suitably shifted, the multiplicities being computed from the Betti numbers of B'. As a fin…

Algorithms for graded injective resolutions and local cohomology over semigroup rings

Let Q be an affine semigroup generating Z^d, and fix a finitely generated Z^d-graded module M over the semigroup algebra k[Q] for a field k. We provide an algorithm to compute a minimal Z^d-graded injective resolution of M up to any desired cohomological degree. As an application, we derive an algorithm computing the local cohomology modules H^i_I(M) supported on any monomial (that is, Z^d-graded) ideal I. Since these local cohomology modules are neither finitely generated nor finitely cogenerated, part of this task is defining a finite data structure to encode them.

A Generalization of the Ishida Complex with applications

We construct a generalize Ishida complex to compute the local cohomology with monomial support of modules over quotients of polynomial rings by cellular binomial ideals. As a consequence, we obtain a combinatorial criterion to determine when such a quotient is Cohen--Macaulay. In particular, this gives a Cohen--Macaulayness criterion for lattice ideals. We also prove a result relating the local cohomology with radical monomial ideal support of an affine semigroup ring to the local cohomology with maximal ideal support of the quotient of the affine semigroup ring by the radical monomial ideal. …

Behavior of local cohomology modules under polarization

Let $S=k[x_1,..., x_n]$ be a polynomial ring over a field $k$ with $n$ variables $x_1$, ..., $x_n$, $\mmmm$ the irrelevant maximal ideal of $S$, $I$ a monomial ideal in $S$ and $I'$ the polarization of $I$ in the polynomial ring $S'$ with $ρ$ variables. We show that each graded piece $H_\mmmm^i(S/I)_\aaa$, $\aaa\in\ZZZ^n$, of the local cohomology module $H_\mmmm^i(S/I)$ is isomorphic to a specific graded piece $H_{\mmmm'}^{i+ρ-n}(S'/I')_\alphaaa$, $\alphaaa\in\ZZZ^ρ$, of the local cohomology module $H_{\mmmm'}^{i+ρ-n}(S'/I')$, where $\mmmm'$ is the irrelevant maximal ideal of $S'$.

The Golod property for products and high symbolic powers of monomial ideals

We show that for any two proper monomial ideals I and J in the polynomial ring S = k[x_1, ..., x_n] the ring S/IJ is Golod. We also show that if I is squarefree then for large enough k the quotient S/I^{(k)} of S by the kth symbolic power of I is Golod. As an application we prove that the multiplication on the cohomology algebra of some classes of moment-angle complexes is trivial.

On monomial ideal rings and a theorem of Trevisan

A direct proof is presented of a form of Alvise Trevisan's result, that every monomial ideal ring is represented by the cohomology of topological space. Certain of these rings are shown to be realized by polyhedral products indexed by simplicial complexes.

A-graded methods for monomial ideals

We use \ZZ^d-gradings to study d-dimensional monomial ideals. The Koszul functor is employed to interpret the quasidegrees of local cohomology in terms of the geometry of distractions and to explicitly compute the multiplicities of exponents. These multigraded techniques originate from the study of hypergeometric systems of differential equations.

D-module structure of local cohomology modules of toric algebras

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.

On the Local Cohomology of Reflexive Modules of Rank One over Normal Semigroup Rings

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.

Klyachko diagrams of monomial ideals

In this paper, we introduce the notion of a Klyachko diagram for a monomial ideal $I$ in a certain multi-graded polynomial ring, namely the Cox ring $R$ of a smooth complete toric variety, with irrelevant maximal ideal $B$. We present procedures to compute the Klyachko diagram of $I$ from its monomial generators, and to retrieve the $B-$saturation $I^{\mathrm{sat}}$ of $I$ from its Klyachko diagram. We use this description to compute the first local cohomology module $H^{1}_{B}(I)$. As an application, we find a formula for the Hilbert function of $I^{\mathrm{sat}}$, and a characterization of m…

Zero-generic initial ideals

Given a homogeneous ideal I of a polynomial ring A=K[X_1,...,X_n] and a monomial order, we construct a new monomial ideal of A associated with I. We call it the zero-generic initial ideal of I with respect to the order and denote it with gin_0(I), or with Gin_0(I) whenever the order is the reverse-lexicographic order. When the characteristic of K is zero, a zero-generic initial ideal is the usual generic initial ideal. We show that Gin_0(I) is endowed with many interesting properties, some of which are easily seen, e.g., it is a strongly stable monomial ideal in any characteristic, and has the…

关于 $\mathfrak{C}$-单项理想在混合特征上局部余同调模的格雷德成分的巴斯数

On Bass numbers of graded components of local cohomology modules supported on $\mathfrak{C}$-monomial ideals in mixed characteristic

本文研究了混合特征下局部余同调模的格雷德成分巴斯数的恒定性，通过分析格雷德结构定理证明了在特定子集下巴斯数的恒定性。

Lifting Monomial Ideals

We show how to lift any monomial ideal J in n variables to a saturated ideal I of the same codimension in n+t variables. We show that I has the same graded Betti numbers as J and we show how to obtain the matrices for the resolution of I. The cohomology of I is described. Making general choices for our lifting, we show that I is the ideal of a reduced union of linear varieties with singularities that are `as small as possible' given the cohomological constraints. The case where J is Artinian is the nicest. In the case of curves we obtain stick figures for I, and in the case of points we obtain…

关于Hellus--Lyubeznik--Yildirim局部上同调模猜想

On Hellus--Lyubeznik--Yildirim's conjecture of local cohomology modules

本文研究Hellus-Lyubeznik-Yildirim猜想，在深度为1、特定条件下深度为2、平方自由单项理想及混合特征离散赋值环形式幂级数环等情形下给出部分肯定答案。

Degree bounds for local cohomology

Let R be a non-negatively graded Cohen-Macaulay ring with R_0 a Cohen-Macaulay factor ring of a local Gorenstein ring. Let d be the dimension of R, m be the maximal homogeneous ideal of R, and M be a finitely generated graded R-module. It has long been known how to read information about the socle degrees of the local cohomology module H_m^0(M) from the twists in position d in a resolution of M by free R-modules. It has also long been known how to use local cohomology to read valuable information from complexes which approximate resolutions in the sense that they have positive homology of smal…

The support of top graded local cohomology modules

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 …

多分次局部上同调模的范畴：Serre 滤过与Nakayama对偶

Multigraded Local Cohomology Categories: Homological Structure and Canonical Sources

本文研究多分次局部上同调模的范畴结构，通过将其等同于表示范畴，建立Serre滤过、TTF三元组及Nakayama-Serre对偶，并构造秩层分辨率。

Nets in $\mathbb P^2$ and Alexander Duality

A net in $\mathbb{P}^2$ is a configuration of lines $\mathcal A$ and points $X$ satisfying certain incidence properties. Nets appear in a variety of settings, ranging from quasigroups to combinatorial design to classification of Kac-Moody algebras to cohomology jump loci of hyperplane arrangements. For a matroid $M$ and rank $r$, we associate a monomial ideal (a monomial variant of the Orlik-Solomon ideal) to the set of flats of $M$ of rank $\le r$. In the context of line arrangements in $\mathbb{P}^2$, applying Alexander duality to the resulting ideal yields insight into the combinatorial str…

Length of local cohomology of powers of ideals

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…

Local Picard Group of Binoids and Their Algebras

We begin by introducing schemes of binoids, invertible $\mathcal{O}_M$-sets and cohomology of sheaves of abelian groups defined on schemes of binoids. We define the so-called punctured combinatorial Čech-Picard complex, whose first cohomology computes $\mathrm{Pic}^{\mathrm{loc}}(M)$, the local Picard group of a binoid. We look then at simplicial binoids, whose spectrum presents very nice combinatorial properties. We prove that the cohomology of a constant sheaf on this punctutured spectra can be computed entirely in terms of simplicial cohomology. Through some other results, we prove that we …

塞雷深度与局部余弦学

Serre depth and local cohomology

本文提出塞雷深度作为同调不变量，研究其在不同环上的性质及应用，包括等维性、稳定性等贡献。

走向由simplicial poset给出的理想局部余维理论

Toward the theory on local cohomologies at the ideals given by simplicial posets

本文研究了由simplicial poset给出的理想局部余维理论，通过构造和分析渐进注射包络来建立局部余维的理论基础。

Generalized Lyubeznik numbers

Given a local ring containing a field, we define and investigate a family of invariants that includes the Lyubeznik numbers, but that captures finer information. These "generalized Lyubeznik numbers" are defined as lengths of certain iterated local cohomology modules in a category of D-modules, and in order to define them, we develop the theory of a functor Lyubeznik utilized in proving that his original invariants are well defined. In particular, this functor gives an equivalence of categories with a category of D-modules. These new invariants are indicators of F-regularity and F-rationality …

通过对导出范畴的分层来解析对称理想

Resolutions of symmetric ideals via stratifications of derived categories

提出通过对相关导出范畴分层的方法统一多项式环中对称理想的稳定性结果，证明了特定序列的猜想，并获得了有界导出范畴的半正交分解。