Stefania Gabelli, Moshe Roitman
We give conditions for a maximal divisorial ideal to be t-maximal and show with examples that, even in a completely integrally closed domain, maximal divisorial ideals need not be t-maximal.
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It is well known that a domain without proper strongly divisorial ideals is completely integrally closed. In this paper we show that a domain without {\em prime} strongly divisorial ideals is not necessarily completely integrally closed, although this property holds under some additional assumptions.
We study the class of domains in which each w-ideal is divisorial, extending several properties of divisorial and totally divisorial domains to a much wider class of domains. In particular we consider PvMDs and Mori domains.
Residual functions and divisorial ideals
该研究提出拓扑空间上剩余函数的定义，并将其用于研究Prüfer整环上除子理想群的自由性，为相关代数结构研究提供新工具。
The symbolic and divisorial analytic spreads are finite
该研究证明域上本质有限型环的理想符号幂最小生成元个数为多项式有界，整环的ℝ-除子滤过理想的最小生成元个数也呈多项式有界，明确符号与除子解析扩张有限性。
Toric rings associated with root systems and conic divisorial ideals via matroid theory
从拟阵理论视角研究圆锥除子理想，并将其应用于根系环面环，通过拟阵描述除子类对应的多面体，计算符号偏序集P对应的环面环R_P的除子类群并刻画其Q-Gorenstein性质。
Factoring ideals in integral domains is a central topic in multiplicative ideal theory. In the present paper we study monoids of ideals and consider factorizations of ideals into multiplicatively irreducible ideals. The focus is on the monoid of nonzero divisorial ideals and on the monoid of $v$-invertible divisorial ideals in weakly Krull Mori domains. Under suitable algebraic finiteness conditions we establish arithmetical finiteness results, in particular for the monotone catenary degree and for the structure of sets of lengths and of their unions.
Conic divisorial ideals of toric rings and applications to Hibi rings and stable set rings
本文研究环面环的锥除子理想，提出确定方法并描述了完美图的Hibi环与稳定集环的锥除子理想，刻画了其对称性，并构造了特殊稳定集环族的NCCR。
In this paper, we study divisorial ideals of a Hibi ring which is a toric ring arising from a partially ordered set. We especially characterize the special class of divisorial ideals called conic using the associated partially ordered set. Using our description of conic divisorial ideals, we also construct a module giving a non-commutative crepant resolution (= NCCR) of the Segre product of polynomial rings. Furthermore, applying the operation called mutation, we give other modules giving NCCRs of it.
We investigate the minimal number of generators $μ$ and the depth of divisorial ideals over normal semigroup rings. Such ideals are defined by the inhomogeneous systems of linear inequalities associated with the support hyperplanes of the semigroup. The main result is that for every bound $C$ there exist, up to isomorphism, only finitely divisorial ideals $I$ such that $μ(I)\le C$. It follows that there exist only finitely many Cohen--Macaulay divisor classes. Moreover we determine the minimal depth of all divisorial ideals and the behaviour of $μ$ and depth in ``arithmetic progressions'' in t…
Suppose that $R$ is an excellent local domain with maximal ideal $m_R$. The theory of multiplicities and mixed multiplicities of $m_R$-primary ideals extends to (possibly non Noetherian) filtrations of $R$ by $m_R$-primary ideals, and many of the classical theorems for $m_R$-primary ideals continue to hold for filtrations. The celebrated theorems involving inequalities continue to hold for filtrations, but the good conclusions that hold in the case of equality for $m_R$-primary ideals do not hold for filtrations. In this article, we consider multiplicities and mixed multiplicities of $R$ by $m…
Degree functions of graded families of ideals
本文研究优正规局部环中分次理想族的重数与度数函数，将其表示为交积极限，并在二维情形下对除子滤过给出更精细结果及关于 Rees 赋值集合有限性与无限性的例子。
The Jacobian algebra $\mathsf{A}$ arising from a consistent dimer model is derived equivalent to crepant resolutions of a $3$-dimensional Gorenstein toric singularity $R$, and it is also called a non-commutative crepant resolution of $R$. This algebra $\mathsf{A}$ is a maximal Cohen-Macaulay (= MCM) module over $R$, and it is a finite direct sum of rank one MCM $R$-modules. In this paper, we observe a relationship between properties of a dimer model and those of MCM modules appearing in the decomposition of $\mathsf{A}$ as an $R$-module. More precisely, we take notice of isoradial dimer models…
The first goal of the present paper is to study the class groups of the edge rings of complete multipartite graphs, denoted by $\Bbbk[K_{r_1,\ldots,r_n}]$, where $1 \leq r_1 \leq \cdots \leq r_n$. More concretely, we prove that the class group of $\Bbbk[K_{r_1,\ldots,r_n}]$ is isomorphic to $\mathbb{Z}^n$ if $n =3$ with $r_1 \geq 2$ or $n \geq 4$, while it turns out that the excluded cases can be deduced into Hibi rings. The second goal is to investigate the special class of divisorial ideals of $\Bbbk[K_{r_1,\ldots,r_n}]$, called conic divisorial ideals. We describe conic divisorial ideals fo…
In this paper we prove that a classical theorem by McAdam about the analytic spread of an ideal in a Noetherian local ring continues to be true for divisorial filtrations on a two dimensional normal excellent local ring $R$, and that the Hilbert polynomial of the fiber cone of a divisorial filtration on $R$ has a Hilbert function which is the sum of a linear polynomial and a bounded function. We prove these theorems by first studying asymptotic properties of divisors on a resolution of singularities of the spectrum of $R$. The filtration of the symbolic powers of an ideal is an example of a di…
We introduce a new variant of tight closure and give an interpretation of adjoint ideals via this tight closure. As a corollary, we prove that a log pair $(X,Δ)$ is plt if and only if the modulo $p$ reduction of $(X,Δ)$ is divisorially F-regular for all large $p \gg 0$. Here, divisorially F-regular pairs are a class of singularities in positive characteristic introduced by Hara and Watanabe in terms of Frobenius splitting.
The Rees algebra and analytic spread of a divisorial filtration
本文研究除子滤链的Rees代数与解析展度，将McAdam定理推广到$\mathbb Q$-除子滤链，证明其一阶差分函数的上极限有限但极限可能不存在，并构造了Rees赋值集合无限的符号滤链例子。
Let R=K[M] be a normal affine monoid algbera over a field K.Up to isomorphism the conic ideals are exactly the direct summands ofthe extension R^{1/n} of R. We show that the classes of the conic divisorial ideals can be identified with the full-dimensional open cells in a decomposition of a torus naturally associated with M. Furthermore, they can be characterized by the relative compactness of a certain group associated with them. Baetica has given examples of Cohen-Macaulay divisorial ideals that are not conic. Wereview his construction and streamline the arguments somewhat. In the last part …
Quasi-maximal ideals and ring extensions
本文研究了准极大理想与环扩张的关系，引入次极大理想并探讨其与2-吸收理想之间的联系。
In commutative ring theory, there is a theorem of Cohen which states that if in a commutative ring all prime ideals are finitely generated then every ideal is finitely generated. However, it is known that having only maximal ideals finitely generated doesn't imply all ideals be finitely generated. In his article we ask the question that what happens if we assume all non-maximal ideals are finitely generated, and we answer the question by showing that indeed then all maximal ideals are also finitely generated i.e. the ring becomes Noetherian.
We characterize componentwise linear monomial ideals with minimal Taylor resolution and consider the lower bound for the Betti numbers of componentwise linear ideals.
A commutative noetherian local ring $(R,\mathfrak{m})$ is Gorenstein if and only if every parameter ideal of $R$ is irreducible. Although irreducible parameter ideals may exist in non-Gorenstein rings, Marley, Rogers, and Sakurai show there exists an integer $\ell$ (depending on $R$) such that $R$ is Gorenstein if and only if there exists an irreducible parameter ideal contained in $\mathfrak{m}^\ell$. We give upper bounds for $\ell$ that depend primarily on the existence of certain systems of parameters in low powers of the maximal ideal.
We prove that each positive power of the maximal ideal of a commutative Noetherian local ring is Tor-rigid, and strongly-rigid. This gives new characterizations of regularity and, in particular, shows that such ideals satisfy the torsion condition of a long-standing conjecture of Huneke and Wiegand.
We generalize a result of Eisenbud-Huneke-Ulrich on the maximal graded shifts of a module with prescribed annihilator and prove a linear regularity bound for ideals in a polynomial ring depending only on the first $p - c$ steps in the resolution, where $p = \mathrm{pd}(S/I)$ and $c = \mathrm{codim}(I)$.
Let R be a Cohen-Macaulay local ring possessing a canonical module. In this paper we consider when the maximal ideal of R is self-dual, i.e. it is isomorphic to its canonical dual as an R-module. local rings satisfying this condition are called Teter rings, and studied by Teter, Huneke-Vraciu, Ananthnarayan-Avramov-Moore, and so on. On the positive dimensional case, we show such rings are exactly the endomorphism rings of the maximal ideals of some Gorenstein local rings of dimension one. We also provide some connection between the self-duality of the maximal ideal and near Gorensteinness.
We study a notion called $n$-standardness (defined by M. E. Rossi and extended in this paper) of ideals primary to the maximal ideal in a Cohen-Macaulay local ring and some of its consequences. We further study conditions under which the maximal ideal is three-standard, first proving results when the residue field has prime characteristic and then using the method of reduction to prime characteristic to extend the results to the equicharacteristic zero case. As an application, we extend a result due to T. Puthenpurakal and show that a certain length associated to a minimal reduction of the max…
We show that, given integers $n_1,n_2, \ldots ,n_k$ with $2 < n_1 < n_2 < \cdots < n_k$, there exists a local (Noetherian) unique factorization domain that has maximal chains of prime ideals of lengths $n_1, n_2, \ldots ,n_k$ which are disjoint except at their minimal and maximal elements. In addition, we demonstrate that unique factorization domains can have other unusual prime ideal structures.
We study a large family of products of Borel fixed ideals of maximal minors. We compute their initial ideals and primary decompositions, and show that they have linear free resolutions. The main tools are an extension of straightening law and a very surprising primary decomposition formula. We study also the homological properties of associated multi-Rees algebra which are shown to be Cohen-Macaulay, Koszul and defined by a Gröbner basis of quadrics.
In this paper, we study the two natural polarizations, namely the standard polarization and the box polarization, of the d-th power of the maximal ideal in a polynomial ring. We show that these polarizations correspond to smooth points in the Hilbert scheme, and we calculate the dimension of their component which shows that they lie on different components. When d=2, we show that all maximal polarizations are smooth points, and we give a simple method for calculating the dimension of their component.
In this paper, we introduce the notion of the strong Rees property (SRP) for $\mathfrak{m}$-primary ideals of a Noetherian local ring and prove that any power of the maximal ideal $\mathfrak{m}$ has its property if the associated graded ring $G$ of $\mathfrak{m}$ satisfies $\text{depth} \ G \ge 2$. As its application, we characterize two-dimensional excellent normal local domains so that $\mathfrak{m}$ is a $p_g$-ideal. Finally we ask what $\mathfrak{m}$-primary ideals have SRP and state a conjecture which characterizes the case when $\mathfrak{m}^n$ are the only ideals which have SRP.
A graded ideal $I$ in $\mathbb{K}[x_1,\ldots,x_n]$, where $\mathbb{K}$ is a field, is said to have almost maximal finite index if its minimal free resolution is linear up to the homological degree $\mathrm{pd}(I)-2$, while it is not linear at the homological degree $\mathrm{pd}(I)-1$, where $\mathrm{pd}(I)$ denotes the projective dimension of $I$. In this paper we classify the graphs whose edge ideals have this property. This in particular shows that for edge ideals the property of having almost maximal finite index does not depend on the characteristic of $\mathbb{K}$. We also compute the non…
We consider ideals $I$ in a Stanley-Reisner ring $k[Δ]$ over the simplical complex $Δ$, such that the tight closure of $I$, $I^*$, is equal to $\mathfrak{m}$, the standard graded maximal ideal of $k[Δ]$. We determine the minimal number of generators of $I$ to be the $\dim Δ+1$ and note the important role this value plays in bounding the intersection of all such ideals $I$. We make mention of this intersection in special cases of Stanley-Reisner rings. We conclude with a description of how this work relates to integral closure.
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。
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。
The main goal of this paper is to prove, in positive characteristic $p$, stability behavior for the graded Betti numbers in the periodic tails of the minimal resolutions of Frobenius powers of the homogeneous maximal ideals for very general choices of hypersurface in three variables whose degree has the opposite parity to that of $p$. We also find some of the structure of the matrix factorization giving the resolution. We achieve this by developing a method for obtaining the degrees of the generators of the defining ideal of an $\mathfrak{c}$-compressed Gorenstein Artinian graded algebra from …
In this paper, we study different polarizations of powers of the maximal ideal m^d, and polarizations of its related square-free version I_d. For n = 3, we show that every minimal free cellular resolution of m^d comes from a certain polarization of the ideal m^d. When I is a square-free ideal, we show that the Alexander dual of any polarization of I is a polarization of the Alexander dual ideal of I. We apply this theorem and study different polarizations of I_d and its Alexander dual I_{n-d+1} simultaneously. We do this by giving a combinatorial description of the polarizations, which has a n…
We give a complete combinatorial characterization of all possible polarizations of powers of the graded maximal ideal $(x_1,x_2,\cdots,x_m)^n$ of a polynomial ring in $m$ variables. We also give a combinatorial description of the Alexander duals of such polarizations. In the three variable case $m=3$ and also in the power two case $n=2$ the descriptions are easily visualized and we show that every polarization defines a (shellable) simplicial ball. We give conjectures relating to topological properties and to algebraic geometry, in particular that any polarization of an Artinian monomial ideal…
The degree excess function $ε(I;n)$ is the difference between the maximal generating degree $d(I^n)$ of a homogeneous ideal $I$ of a polynomial ring and $p(I)n$, where $p(I)$ is the leading coefficient of the asymptotically linear function $d(I^n)$. It is shown that any non-increasing numerical function can be realized as a degree excess function, and there is a monomial ideal $I$ whose $ε(I;n)$ has exactly a given number of local maxima. In the case of monomial ideals, an upper bound on $ε(I;n)$ is provided. As an application it is shown that in the worst case, the so-called stability index o…
Let $B$ be a reduced local (Noetherian) ring with maximal ideal $M$. Suppose that $B$ contains the rationals, $B/M$ is uncountable and $|B| = |B/M|$. Let the minimal prime ideals of $B$ be partitioned into $m \geq 1$ subcollections $C_1, \ldots ,C_m$. We show that there is a reduced local ring $S \subseteq B$ with maximal ideal $S \cap M$ such that the completion of $S$ with respect to its maximal ideal is isomorphic to the completion of $B$ with respect to its maximal ideal and such that, if $P$ and $Q$ are prime ideals of $B$, then $P \cap S = Q \cap S$ if and only if $P$ and $Q$ are in $C_i…
Maximal Ideals in Commutative Rings and the Axiom of Choice
本文给出选择公理蕴含极大理想定理的逆向推导，旨在提供自洽且易于非专家理解的证明。
Let $(A,{\mathfrak m})$ be a Cohen-Macaulay local ring and let $I$ be an ideal of $A$. We prove that the Rees algebra ${\mathcal R}(I)$ is an almost Gorenstein ring in the following cases: (1) $(A,{\mathfrak m})$ is a two-dimensional excellent Gorenstein normal domain over an algebraically closed field $K \cong A/{\mathfrak m}$ and $I$ is a $p_g$-ideal; (2) $(A,{\mathfrak m})$ is a two-dimensional almost Gorenstein local ring having minimal multiplicity and $I={\mathfrak m}^{\ell}$ for all $\ell \ge 1$; (3) $(A,{\mathfrak m})$ is a regular local ring of dimension $d \ge 2$ and $I={\mathfrak m}…
$S$-Prime and $S$-maximal ideals in trivial ring extensions of commutative rings
本文研究了在交换环的平凡环扩张中$S$-素和$S$-极大理想的特点，揭示了其与同质性及$S_0$-可除模的关系，并探讨了相关环概念的扩展。
Ring homomorphisms and local rings with quasi-decomposable maximal ideal
该研究围绕具有拟可分解极大理想的局部环展开，进一步探索其同调性质、从组合视角引入新环族，并研究相关同调性质在局部环同态图表上的可推广程度。
Free divisors form a celebrated class of hypersurfaces which has been extensively studied in the past fifteen years. Our main goal is to introduce four new families of homogeneous free divisors and investigate central aspects of the blowup algebras of their Jacobian ideals. For instance, for all families the Rees algebra and its special fiber are shown to be Cohen-Macaulay -- a desirable feature in blowup algebra theory. Moreover, we raise the problem of when the analytic spread of the Jacobian ideal of a (not necessarily free) polynomial is maximal, and we characterize this property with tool…
On ideals of affine semigroups and affine semigroups with maximal embedding dimension
本文刻画了由半群理想构造的I(S)-半群及同时为C-半群的情形，给出计算满足性质的I(S)-半群的算法，并引入和刻画了具有最大嵌入维数的仿射半群。
We identify minimal cases in which a power $m^i\not=0$ of the maximal ideal of a local ring $R$ is not Golod, i.e.\ the quotient ring $R/m^i$ is not Golod. Complementary to a 2014 result by Rossi and Şega, we prove that for a generic artinian Gorenstein local ring with $m^4=0\not= m^3$, the quotient $R/m^3$ is not Golod. This is provided that $m$ is minimally generated by at least $3$ elements. Indeed, we show that if $m$ is $2$-generated, then every power $m^i\not= 0$ is Golod.
We relate the annihilators of graded components of the canonical module of a graded Cohen-Macaulay ring to colon ideals of powers of the homogeneous maximal ideal. In particular, we connect them to the core of the maximal ideal. An application of our results characterizes Cayley-Bacharach sets of points in terms of the structure of the core of the maximal ideal of their homogeneous coordinate ring. In particular, we show that a scheme is Cayley-Bacharach if and only if the core is a power of the maximal ideal.
Tor algebra of local rings with decomposable maximal ideal
本文研究具有可分解极大理想的交换诺特局部环，通过其商环的Tor代数刻画原环的Tor代数结构。
Let $(R, \m)$ be a commutative Noetherian local ring with $\m^3 =(0)$. We give a condition for $R$ to have a non-free module of G-dimension zero. We shall also construct a family of non-isomorphic indecomposable modules of G-dimension zero with parameters in an open subset of projective space. We shall finally show that the subcategory consisting of modules of G-dimension zero over $R$ is not necessarily a contravariantly finite subcategory in the category of finitely generated $R$-modules.
Prime ideals in infinite products of commutative rings
该研究刻画了交换环无限乘积中的素理想与极大理想，证明极大理想由特定布尔代数上的超滤子诱导，并针对两类环族完全刻画了对应乘积环的极大或全部素理想。
We introduce two new invariants of a Noetherian (standard graded) local ring $(R, \mathfrak m)$ that measure the number of generators of certain kinds of reductions of $\mathfrak m,$ and we study their properties. Explicitly, we consider the minimum among the number of generators of ideals $I$ such that either $I^2 = \mathfrak m^2$ or $I \supseteq \mathfrak m^2$ holds. We investigate subsequently the case that $R$ is the quotient of a polynomial ring $k[x_1, \dots, x_n]$ by an ideal $I$ generated by homogeneous quadratic forms, and we compute these invariants. We devote specific attention to t…
Let $\ast $ be a star operation of finite character. Call a $\ast $-ideal $I$ of finite type a $\ast $-homogeneous ideal if $I$ is contained in a unique maximal $\ast $-ideal $M=M(I).$ A maximal $\ast $-ideal that contains a $\ast $-homogeneous ideal is called $\ast $-potent and the same name bears a domain all of whose maximal $\ast $-ideals are $\ast $-potent. One among the various aims of this article is to indicate what makes a $\ast $-ideal of finite type a $\ast $-homogeneous ideal, where and how we can find one, what they can do and how this notion came to be. We also prove some results…
Let $m$, $n$, $a_1$, ..., $a_r$, $b_1$, ..., $b_r$ be integers with $1\leq a_1<...<a_r\leq m$ and $1\leq b_1<...<b_r\leq n$. And let $x$ be the universal $m\times n$ matrix with the property that $i$-minors of first $a_i-1$ rows and first $b_i-1$ columns are all zero, for $i=1$, ..., $r+1$ ($a_{r+1}=m+1$ and $b_{r+1}=n+1$). For an integer $u$ with $1\leq u\leq m$, we denote by $U$ the $u\times n$ matrix consisting of the first $u$ rows of $x$. In this paper, we consider the analytic spread and the reduction number of the ideal of maximal minors of $U$
We construct a local Cohen-Macaulay ring $R$ with a prime ideal $\mathfrak{p}\in\spec(R)$ such that $R$ satisfies the uniform Auslander condition (UAC), but the localization $R_{\mathfrak{p}}$ does not satisfy Auslander's condition (AC). Given any positive integer $n$, we also construct a local Cohen-Macaulay ring $R$ with a prime ideal $\mathfrak{p}\in\spec(R)$ such that $R$ has exactly two non-isomorphic semidualizing modules, but the localization $R_{\mathfrak{p}}$ has $2^n$ non-isomorphic semidualizing modules. Each of these examples is constructed as a fiber product of two local rings ove…
We study different form of boundness for ideals of almost Dedekind domains, generalizing the notions of critical ideals, radical factorization, and SP-domains. We show that every almost Dedekind domain has at least one noncritical maximal ideals and, indeed, the set of noncritical maximal ideals is dense in the maximal space, with respect to the constructible topology; as a consequence, we show that every almost Dedekind domain is SP-scattered, and in particular that the group $\mathrm{Inv}(D)$ of invertible ideals of an almost Dedekind domain $D$ is always free. If $D$ is an almost Dedekind d…
On the minimal generating sets of the Eulerian ideal
本文研究简单图欧拉理想的极小齐次生成集，证明其为格理想，给出生成二项式的刻画与显式生成集，并确定最大生成度的上界，特别对二部图与完全图给出精确结果。
We describe the universal Groebner basis of the ideal of maximal minors and the ideal of $2$-minors of a multigraded matrix of linear forms. Our results imply that the ideals are radical and provide bounds on the regularity. In particular, the ideals of maximal minors have linear resolutions. Our main theoretical contribution consists of introducing two new classes of ideals named after Cartwright and Sturmfels, and proving that they are closed under multigraded hyperplane sections. The gins of the ideals that we study enjoy special properties.
Prime and maximal ideals in Hurwitz polynomial rings
本文刻画了Hurwitz多项式环中R-不相交理想为素理想的条件，并研究了极大理想的存在性及其由最小次数多项式生成的判定方法。
We characterize the class of ideals of a polynomial ring such that the hilbert series of their graded local cohomology modules is maximal.
We study algebraic and homological properties of facet ideals of order complexes of posets which we call ideals of maximal flags of posets or simply flag ideals. We characterize the unmixed and Cohen-Macaulay flag ideals of graded posets. We also give structural results for the multigraded Betti numbers of flag ideals of such posets. The structural results on multigraded Betti numbers of flag ideals are used to characterize the class of flag ideals with linear resolutions.
Let $(R,\mm)$ be a Noetherian local ring and $M$ a finitely generated $R$-module. We say $M$ has maximal depth if there is an associated prime $\pp$ of $M$ such that $\depth M=\dim R/\pp$. In this paper we study squarefree monomial ideals which have maximal depth. Edge ideals of cycle graphs, transversal polymatroidal ideals and high powers of connected bipartite graph with this property are classified.
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