Karen E. Smith, Irena Swanson
We find explicit bounds on the primary components and on the Castelnuovo-Mumford regularity of powers of monomial ideals. We also analyze the primary decompositions of Katzman's example.
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We construct families of prime ideals in polynomial rings for which the number of associated primes of the second power (or higher powers) is exponential in the number of variables in the ring. We give a lower bound on the Ananyan-Hochster constant for the number of associated primes.
We specify a class of graphs, $H_t$, and characterize the irreducible decomposition of all powers of the cover ideals. This gives insight into the structure and stabilization of the corresponding associated primes; specifically, providing an answer to the question "For each integer $t\geq 0$, does there exist a (hyper) graph $H_t$ such that stabilization of associated primes occurs at $s\geq (χ(H_t)-1)+t$?" asked by Francisco, Hà, and Van Tuyl. For each $t$, $H_t$ has chromatic number 3 and associated primes that stabilize at $s=2+t$.
It is shown that any set of nonzero monomial prime ideals can be realized as the stable set of associated prime ideals of a monomial ideal. Moreover, an algorithm is given to compute the stable set of associated prime ideals of a monomial ideal.
It is shown that a squarefree principal Borel ideal satisfies the persistence property for the associated prime ideals. For the graded maximal ideal we compute the index of stability with respect to squarefree principal Borel ideals and determine their stable set of associated prime ideals.
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
Associated primes of powers of edge ideals of edge-weighted trees
该研究针对递增加权树的边理想，完整刻画了其各次幂的伴随素数，丰富了交换代数中边理想幂的相关理论。
There are several theorems describing the intricate relationship between flatness and associated primes over commutative Noetherian rings. However, associated primes are known to act badly over non-Noetherian rings, so one needs a suitable replacement. In this paper, we show that the behavior of strong Krull primes most closely resembles that of associated primes over a Noetherian ring. We prove an analogue of a theorem of Epstein and Yao characterizing flat modules in terms of associated primes by replacing them with strong Krull primes. Also, we partly generalize a classical equational theor…
We prove a doubly exponential bound for the Castelnuovo-Mumford regularity of prime ideals defining varieties with polynomial parametrisation.
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.
We present combinatorial characterizations for the associated primes of the second power of squarefree monomial ideals and criteria for this power to have positive depth or depth greater than one.
Let $R$ be a commutative ring with identity. For a finitely generated $R$-module $M$, the notion of associated prime submodules of $M$ is defined. It is shown that this notion inherits most of essential properties of the usual notion of associated prime ideals. In particular, it is proved that for a Noetherian multiplication module $M$, the set of associated prime submodules of $M$ coincides with the set of $M$-radicals of primary submodules of $M$ which appear in a minimal primary decomposition of the zero submodule of $M$. Also, Anderson's theorem [{\bf 2}] is extended to minimal prime submo…
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.
For the edge ideal I of an arbitrary simple graph G we describe the monomials of the saturation of a power of I in terms of (vertex) weighted graphs associated with the monomials. This description allows us to characterize the embedded associated primes of a power of I as covers of G which contain certain types of subgraphs of G. As an application, we completely classify the associated primes of the second and the third power of I in terms of G.
It was previously known, by work of Smith-Swanson and of Sharp-Nossem, that the linear growth property of primary decompositions of Frobenius powers of ideals in rings of prime characteristic has strong connections to the localization problem in tight closure theory. The localization problem has recently been settled in the negative by Brenner and Monsky, but the linear growth question is still open. We study growth of primary decompositions of Frobenius powers of dimension one homogeneous ideals in graded rings over fields. If the ring is positively graded we prove that the linear growth prop…
Let $A, B$ be polynomial rings over a field $k$, and $I\subseteq A, J\subseteq B$ proper homogeneous ideals. We analyze the associated primes of powers of $I+J\subseteq A\otimes_k B$ given the data on the summands. The associated primes of large enough powers of $I+J$ are determined. We then answer positively a question about the persistence property of $I+J$ in many new cases.
Given an integral domain $D$ with quotient field $K$, the ring of integer-valued polynomials on D is the subring $\{f (X) \in K[X]: f(D) \subset D\}$ of the polynomial ring $K[X]$. Using the related tools of $t$-closure and associated primes, we generalize some known results on integer-valued polynomial rings over Krull domains, PVMD's, and Mori domains.
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.
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.
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。
Gluing Associated Prime Ideals of Small Height
本文研究了局部环的相伴素理想粘合问题，提出了存在具有相同完备化且相伴素理想个数减少1的子环的充分条件，并给出了相关应用与推论。
In this paper we shall investigate the concepts of cofiniteness of local cohomology modules and Abelian categories of cofinite modules over arbitrary Noetherian rings. Then we shall improve some of the results given in the literature.
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…
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 $R$ 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 …
On the existence of the maximal ideal in the set of associated primes of monomial ideals
本文研究单项式理想幂的相伴素的性质，建立K[x,y,z]中极大理想是否属于其相伴素集的准则，证明n≥3时存在无穷多理想的相伴素集波动，并构造不满足持久性等性质的理想例子。
Let A be a Noetherian ring and B be a finitely generated A-algebra. Denote by A' the integral closure of A in B. We give necessary and sufficient conditions for prime ideals to be in Ass_{A}(B/A') and Ass_{A'}(B/A') generalizing and strengthening classical results for rings of special type.
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.
We prove that if the initial ideal of a prime ideal is Borel-fixed and the dimension of the quotient ring is less than or equal to two, then given any non-minimal associated prime ideal of the initial ideal it contains another associated prime ideal of dimension one larger.
We find an explicit expression of the associated primes of monomial ideals as a colon by an element $v$, using the unique irredundant irreducible decomposition whose irreducible components are monomial ideals (Theorem 3.1). An algorithm to compute $v$ is given using Macaulay2 (Section 7). For squarefree monomial ideals the problem is related to the combinatorics of the underlying clutter or graph (Proposition 4.3). For ideals of Borel type the monomial $f$ takes a simpler form (Proposition 5.2). The authors classify when $f$ is unique (Proposition 6.2).
Criteria for the presence of the maximal ideal in the set of associated primes
本文建立了多项式环中单项式理想幂的相伴素理想集中存在极大理想 (x1,...,xn) 的判别准则，并通过例子说明其适用性。
To compute the local cohomology of powers of edge ideals one needs to know their saturations. The saturation of the second and third powers has been described in terms of the graph in [13] and [10]. In this article, we give a combinatorial description of the generators of the saturation of the fourth power. As a consequence, we are able to give a complete classification of the associated primes of the fourth power of edge ideals in terms of the graph.
Fluctuations in depth and associated primes of powers of ideals
本文研究了单项式理想BHH(m,r,s)的幂的深度函数特性，揭示其周期性行为及变量优化优势。
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 …
Associated primes, witnesses, and omega invariants of monomial ideals
本文引入诺特环中理想的omega不变量（相伴素理想个数），通过单项式见证元与矩阵方法刻画单项式理想及其幂的相伴素，给出无需准素分解的公式与界，并推广至边理想。
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.
Let $R$ be a ring, let $\mathfrak{a}\subseteq R$ be an ideal, and let $M$ be an $R$-module. Let $Γ_{\mathfrak{a}}$ denote the $\mathfrak{a}$-torsion functor. Conditions are given for the (weakly) associated primes of $Γ_{\mathfrak{a}}(M)$ to be the (weakly) associated primes of $M$ containing $\mathfrak{a}$, and for the (weakly) associated primes of $M/Γ_{\mathfrak{a}}(M)$ to be the (weakly) associated primes of $M$ not containing $\mathfrak{a}$.
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…
In this paper, we study associated primes and depth of integral closures of powers of edge ideals. We provide sharp bounds on how big of powers for which the set of associated primes and the depth of integral closures of powers of edge ideals are stable.
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}…
The associated prime ideals of powers of polymatroidal ideals are studied, including the stable set of associated prime ideals of this class of ideals. It is shown that polymatroidal ideals have the persistence property and for transversal polymatroids and polymatroidal ideals of Veronese type the index of stability and the stable set of associated ideals is determined explicitly.
This paper investigates the doubly exponential ideal membership property of the Mayr-Meyer ideals from the point of view of their associated primes. A doubly-exponential upper bound on the set of associated primes is proved. In the paper a new family of ideals emerges which also has the doubly exponential ideal membership property. More on the new family can be found in author's paper "A new family of ideals with the doubly exponential ideal membership property".
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_{+}…
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…
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 …
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.
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 $I$ 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 $i=1,2$.
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.
The aim of this paper is to study the associated primes of powers of squarefree monomial ideals. Hypergraphs and squarefree monomial ideals are strongly connected. The cover ideal $J(H)$ of a hypergraph $H$ is the intersection of the primes corresponding to the edges of $H$. We define the $shadow$ of $H$ as a certain set of smaller hypergraphs related to $H$. We then describe how the shadows of $H$ preserve information about the associated primes of the powers of $J(H)$. Some implications to the persistence property are studied.
Given a polynomial ring $C$ over a field and proper ideals $I$ and $J$ whose generating sets involve disjoint variables, we determine how to embed the associated primes of each power of $I+J$ into a collection of primes described in terms of the associated primes of select powers of $I$ and of $J$. We record two applications. First, in case the field is algebraically closed, we construct primary decompositions for powers of $I+J$ from primary decompositions for powers of $I$ and $J$. Separately, we attack the persistence problem for associated primes of powers of an ideal in case one of $I$ or…
The spline complex $\mathcal{R}/\mathcal{J}[Σ]$ whose top homology is the algebra $C^α(Σ)$ of mixed splines over the fan $Σ\subset\mathbb{R}^{n+1}$ was introduced by Schenck-Stillman in [Schenck-Stillman 97] as a variant of a complex $\mathcal{R}/\mathcal{I}[Σ]$ of Billera [Billera 88]. In this paper we analyze the associated primes of homology modules of this complex. In particular, we show that all such primes are linear. We give two applications to computations of dimensions. The first is a computation of the third coefficient of the Hilbert polynomial of $C^α(Σ)$, including cases where van…
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 $k$.
The purpose of this paper is to provide methods for determining the associated primes of the square of the Alexander dual of the edge ideal for an m-hypergraph H. We prove a general method for detecting associated primes of the square of the Alexander dual of the edge ideal based on combinatorial conditions on the m-hypergraph. Also, we demonstrate a more efficient combinatorial criterion for detecting the non-existence of non-minimal associated primes. In investigating 3-hypergraphs, we prove a surprising extension of the previously discovered results for 2-hypergraphs (simple graphs). For 2-…
For a reduced Noetherian ring $R$ of characteristic $p > 0$, in this paper we discuss an extension of $R$ called its perfect closure $R^\infty$. This extension contains all $p^e$-th roots of elements of $R$, and is usually non-Noetherian. We first define the generalized notions of associated primes of a module over a non-Noetherian ring. Then for any $R$-module $M$, we state a correspondence between certain generalized prime ideals of $(R^\infty \otimes_R M)/N$ over $R^\infty$, and the union of associated prime ideals of $F^e(M)/N_e$ as $e \in \mathbb{N}$ varies. Here $F$ refers to the Frobeni…
The stable set of associated primes of a complementary edge ideal
本文研究了互补边理想各次幂的关联素理想，证明其满足持久性性质并计算了v函数，同时探讨了高次生成的平方自由单项式理想同调性质。
Associated primes of the second power of closed neighborhood ideals of graphs
本文研究极大齐次理想作为闭邻域理想二次幂相伴素理想的简单图，证明此类图直径至多为6，且直径为2的图必为顶点直径-2-临界图。
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{…
Let $I$ be a monomial ideal of a polynomial ring $R$. In this paper we determine a number $B$ such that $\Ass (I^n/I^{n+1}) = \Ass (I^{B}/I^{B+1})$ for all $n\geq B$.
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.
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