Mathematics - Commutative Algebra Publications (50)

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Mathematics - Commutative Algebra Publications

A semigroup prime of a commutative ring $R$ is a prime ideal of the semigroup $(R,\cdot)$. One of the purposes of this paper is to study, from a topological point of view, the space $\scal(R)$ of prime semigroups of $R$. We show that, under a natural topology introduced by B. Read More


This is a survey article on Newton-Okounkov bodies in projective geometry focusing on the relationship between positivity of divisors and Newton-Okounkov bodies. Read More


Let $(R, \mathfrak m)$ be a Noetherian local ring and $I$ a $\mathfrak m$-primary ideal. In this paper, we study an inequality involving the number of generators, the Loewy length and the multiplicity of $I$. There is strong evidence that the inequality holds for all integrally closed ideals of finite colength if and only if $R$ has sufficiently nice singularities. Read More


For the polynomial ring over an arbitrary field with twelve variables, there exists a prime ideal whose symbolic Rees algebra is not finitely generated. Read More


In 2002, Terao showed that every reflection multi-arrangement of a real reflection group with constant multiplicity is free by providing a basis of the module of derivations. We first generalize Terao's result to multi-arrangements stemming from well-generated unitary reflection groups, where the multiplicity of a hyperplane depends on the order of its stabilizer. Here the exponents depend on the exponents of the dual reflection representation. Read More


In the first section of the present work, we introduce the concept of pseudocomplementation for semirings and show semiring version of some known results in lattice theory. We also introduce semirings with pc-functions and prove some interesting results for minimal prime ideals of such semirings. In the second section, some classical results for minimal prime ideals in ring theory are generalized in the context of semiring theory. Read More


Recent important and powerful frameworks for the study of differential forms by Huber-Joerder and Huber-Kebekus-Kelly based on Voevodsky's h-topology have greatly simplified and unified many approaches. This article builds towards the goal of putting Illusie's de Rham-Witt complex in the same framework by exploring the h-sheafification of the rational de Rham-Witt differentials. Assuming resolution of singularities in positive characteristic one recovers a complete cohomological h-descent for all terms of the complex. Read More


We give a negative answer to a question proposed in [3], regarding the $h$-vector of ($S_r$) simplicial complexes. Read More


In this article, we prove commutativity principal for linear, symplectic and transvection groups. This principle is a consequence of Quillen-Suslin local global principle and using a non-symmetric application of it as done by A. Bak. Read More


We prove a version of weakly functorial big Cohen-Macaulay algebras that suffices to establish Hochster-Huneke's vanishing conjecture for maps of Tor in mixed characteristic. As a corollary, we prove an analog of Boutot's theorem that direct summands of regular rings are pseudo-rational in mixed characteristic. Our proof uses perfectoid spaces and is inspired by the recent breakthroughs on the direct summand conjecture by Andr\'{e} and Bhatt. Read More


Let $G = (V,E)$ be a simple graph. We give a necessary condition for the toric ring $k[G]$ associated to $G$ to be Cohen-Macaulay. Particularly, we investigate a "forbidden" structure in $G$ that prevents $k[G]$ from being Cohen-Macaulay. Read More


These lectures discuss recent advances on syzygies on algebraic curves, especially concerning the Green, the Prym-Green and the Green-Lazarsfeld Secant Conjectures. The methods used are largely geometric and variational, with a special emphasis on examples and explicit calculations. The notes are based on series of lectures given in Daejeon (March 2013), Rome (November-December 2015) and Guanajuato (February 2016). Read More


In this paper we classify the monomial complete intersection algebras, in two variables, and of positive characteristic, which has the strong Lef- schetz property. Together with known results, this gives a complete classi- fication of the monomial complete intersections with the strong Lefschetz property. Read More


Let $(R,\mathfrak{m})$ be a $d$-dimensional Cohen-Macaulay local ring, $I$ an $\mathfrak{m}$-primary ideal of $R$ and $J=(x_1,... Read More


Syzygies capture intricate geometric properties of a subvariety in projective space. However, when the ambient space is a product of projective spaces or a more general smooth projective toric variety, minimal free resolutions over the Cox ring are too long and contain many geometrically superfluous summands. In this paper, we construct some much shorter free complexes that better encode the geometry. Read More


We give a detailed proof for Gordan-Noether's results in "Ueber die algebraischen Formen, deren Hesse'sche Determinante identisch verschwindet." C. Lossen has written a paper in a similar direction as the present paper. Read More


In [9], Migliore, Mir\'o-Roig and Nagel, proved that if $R = \mathbb{K}[x,y,z]$, where $\mathbb{K}$ is a field of characteristic zero, and $I=(L_1^{a_1},\dots,L_r^{a_4})$ is an ideal generated by powers of 4 general linear forms, then the multiplication by the square $L^2$ of a general linear form $L$ induces an homomorphism of maximal rank in any graded component of $R/I$. More recently, Migliore and Mir\'o-Roig proved in [8] that the same is true for any number of general linear forms, as long the powers are uniform. In addition, they conjecture that the same holds for arbitrary powers. Read More


If $I$ is an ideal generated by any powers of any set of general linear forms in the ring $R = K[x,y,z]$ over a field of characteristic zero, and $L$ is a general linear form, we show that multiplication by $L^2$ in any degree always has maximal rank. We also give a complete description of when multiplication by $L^3$ has maximal rank (and its failure when it does not). As a consequence we show that if $I$ contains the square of a linear form then multiplication by any power of $L$ has maximal rank in any degree, that is, $I$ has the Strong Lefschetz Property. Read More


We show that the associated graded of the test module filtration $\tau(M, f^\lambda)_{\lambda \geq 0}$ admits natural Cartier structures. If $\lambda$ is an $F$-jumping number, then these Cartier structures are nilpotent on $\tau(M, f^{\lambda -\varepsilon})/\tau(M, f^\lambda)$ if and only if the denominator of $\lambda$ is divisible by $p$. We also show that these Cartier structures coincide with certain Cartier structures that are obtained by considering certain $\mathcal{D}$-modules associated to $M$ that were used to construct Bernstein-Sato polynomials. Read More


Let $F$ be a homogeneous polynomial in $n$ variables of degree $d$ over a field $K$. Let $A(F)$ be the associated Artinian graded $K$-algebra. If $B \subset A(F)$ is a subalgebra of $A(F)$ which is Gorenstein with the same socle degree as $A(F)$, we describe the Macaulay dual generator for $B$ in terms of $F$. Read More


We relate the Weyr structure of a square matrix $B$ to that of the $t \times t$ block upper triangular matrix $C$ that has $B$ down the main diagonal and first superdiagonal, and zeros elsewhere. Of special interest is the case $t = 2$ and where $C$ is the $n$th Sierpinski matrix $B_n$, which is defined inductively by $B_0 = 1$ and $B_n = \left[\begin{array}{cc} B_{n-1} & B_{n-1} 0 & B_{n-1} \end{array} \right]$. This yields an easy derivation of the Weyr structure of $B_n$ as the binomial coefficients arranged in decreasing order. Read More


We resolve a conjecture of Li and Ramos that relates the regularity of an FI-module to its local cohomology groups. This is an analogue of the familiar relationship between regularity and local cohomology in commutative algebra. Read More


We analyze the space of differentiable functions on a quad-mesh $\cM$, which are composed of 4-split spline macro-patch elements on each quadrangular face. We describe explicit transition maps across shared edges, that satisfy conditions which ensure that the space of differentiable functions is ample on a quad-mesh of arbitrary topology. These transition maps define a finite dimensional vector space of $G^{1}$ spline functions of bi-degree $\le (k,k)$ on each quadrangular face of $\cM$. Read More


We give a category theoretic approach to several known equivalences from tilting theory and commutative algebra. Furthermore we apply our main results to study the category of relative Cohen-Macaulay modules. Read More


We study properties of the Stanley-Reisner rings of simplicial complexes with isolated singularities modulo two generic linear forms. Miller, Novik, and Swartz proved that if a complex has homologically isolated singularities, then its Stanley-Reisner ring modulo one generic linear form is Buchsbaum. Here we examine the case of non-homologically isolated singularities, providing many examples in which the Stanley-Reisner ring modulo two generic linear forms is a quasi-Buchsbaum but not Buchsbaum ring. Read More


The celebrated Auslander-Reiten Conjecture, on the vanishing of self extensions of a module, is one of the long-standing conjectures in ring theory. Although it is still open, there are several results in the literature that establish the conjecture over Gorenstein rings under certain conditions. The purpose of this article is to obtain extensions of such results over Cohen-Macaulay local rings that admit canonical modules. Read More


Given the complement of a hyperplane arrangement, let $\Gamma$ be the closure of the graph of the map inverting each of its defining linear forms. The characteristic polynomial manifests itself in the Hilbert series of $\Gamma$ in two different-seeming ways, one due to Orlik and Terao and the other to Huh and Katz. We define an extension of the no broken circuit complex of a matroid and use it to give a direct Gr\"obner basis argument that the polynomials extracted from the Hilbert series in these two ways agree. Read More


The Cholesky decomposition plays an important role in finding the inverse of the correlation matrices. As it is a fast and numerically stable for linear system solving, inversion, and factorization compared to singular valued decomposition (SVD), QR factorization and LU decomposition. As different methods exist to find the Cholesky decomposition of a given matrix. Read More


We introduce pretty $k$-clean monomial ideals and $k$-decomposable multicomplexes, respectively, as the extensions of the notions of $k$-clean monomial ideals and $k$-decomposable simplicial complexes. We show that a multicomplex $\Gamma$ is $k$-decomposable if and only if its associated monomial ideal $I(\Gamma)$ is pretty $k$-clean. Also, we prove that an arbitrary monomial ideal $I$ is pretty $k$-clean if and only if its polarization $I^p$ is $k$-clean. Read More


In this article, we give a full description of the topology of the one dimensional affine analytic space $\mathbb{A}_R^1$ over a complete valuation ring $R$ (i.e. a valuation ring with "real valued valuation" which is complete under the induced metric), when its field of fractions $K$ is algebraically closed. Read More


Let K be a non-empty set of ideals of the commutative ring R, closed under taking smaller ideals. A subset X of the group ring R[Z^s] is called a K-set if the ideal generated by the coefficients of the elements of X is in K. For X not a K-set we investigate the set of those homomorphisms p from Z^s to Z^t such that p_*(X) is a K-set. Read More


Over an arbitrary field $\mathbb{F}$, Harbourne conjectured that the symbolic power $I^{(N (r-1)+1)} \subseteq I^r$ for all $r>0$ and all graded ideals $I$ in $S = \mathbb{F} [\mathbb{P}^N] = \mathbb{F}[x_0, \ldots, x_N]$ ($N \ge 2$). The conjecture has been disproven in both zero- and odd prime characteristic. However, the conjecture does hold over any field when, e. Read More


Twisted commutative algebras (tca's) have played an important role in the nascent field of representation stability. Let A_d be the complex tca freely generated by d indeterminates of degree 1. In a previous paper, we determined the structure of the category of A_1-modules (which is equivalent to the category of FI-modules). Read More


Reflexive polytopes form one of the distinguished classes of lattice polytopes. Especially reflexive polytopes which possess the integer decomposition property are of interest. In the present paper, by virtue of the algebraic technique on Gr\"onbner bases, a new class of reflexive polytopes arising from perfect graphs which possess the integer decomposition property will be presented. Read More


The definition of a pseudo-dualizing complex is obtained from that of a dualizing complex by dropping the injective dimension condition, while retaining the finite generatedness and homothety isomorphism conditions. In the specific setting of a pair of associative rings, we show that the datum of a pseudo-dualizing complex induces a triangulated equivalence between a pseudo-coderived category and a pseudo-contraderived category. The latter terms mean triangulated categories standing "in between" the conventional derived category and the coderived or the contraderived category. Read More


Let $M$ be an atomic monoid and let $x$ be a non-unit element of $M$. The elasticity of $x$, denoted by $\rho(x)$, is the ratio of its largest factorization length to its shortest factorization length, and it measures how far is $x$ from having a unique factorization. The elasticity $\rho(M)$ of $M$ is the supremum of the elasticities of all non-unit elements of $M$. Read More


We give a new proof of a polynomial identity involving the minors of a matrix, that originated in the study of integer torsion in a local cohomology module. Read More


We show that a monomial ideal $I$ has projective dimension $\leq$ 1 if and only if the minimal free resolution of $S/I$ is supported on a graph that is a tree. This is done by constructing specific graphs which support the resolution of the $S/I$. We also provide a new characterization of quasi-trees, which we use to give a new proof to a result by Herzog, Hibi, and Zheng which characterizes monomial ideals of projective dimension 1 in terms of quasi-trees. Read More


Let $(Q, q)$ be a quadratic space over a commutative ring $R$ in which $2$ is invertible, and consider the Dickson--Siegel--Eichler--Roy's subgroup $EO_{R}(Q, H(R)^{m})$ of the orthogonal group $O_R(Q \perp H(R)^m)$, with rank $Q= n \geq 1$ and $m\geq 2$. We show that $EO_{R}(Q, H(R)^{m})$ is a normal subgroup of $O_R(Q \perp H(R)^m)$, for all $m\geq 2$. We also prove that the DSER group $EO_{R}(Q, H(P))$ is a normal subgroup of $O_{R}(Q \perp H(P))$, where $Q$ and $H(P)$ are quadratic spaces over a commutative ring $R$, with rank $(Q) \ge 1$ and rank $(P) \ge 2$. Read More


In this paper we improve the recent results on the transfer of Pr\"ufer, Gaussian and arithmetical conditions on amalgamated constructions. As an application we provide an answer to a question posed by Chhiti, Jarrar, Kabbaj and Mahdou as well as we construct various examples. Read More


In this article, we give the explicit minimal free resolution of the associated graded ring of certain affine monomial curves in affine 4-space based on the standard basis theory. As a result, we give the minimal graded free resolution and compute the Hilbert function of the tangent cone of these families. Read More


We describe an algorithm to compute a presentation of the pushforward module $f_*{\mathcal O}_{\mathcal{X}}$ for a finite map germ $f\colon \mathcal{X}\to ({\mathbb{C}}^{n+1},0)$, where $\mathcal{X}$ is Cohen-Macaulay of dimension $n$. The algorithm is an improvement of a method by Mond and Pellikaan. We give applications to problems in singularity theory, computed by means of an implementation in the software Singular. Read More


We present an algorithm that, for a given vector $\mathbf{a}$ of $n$ relatively prime polynomials in one variable over an arbitrary field $\mathbb{K}$, outputs an $n\times n$ invertible matrix $P$ with polynomial entries such that it forms a $\mathit{\text{degree-optimal moving frame}}$ for the rational curve defined by $\mathbf{a}$. From an algebraic point of view, the first column of the matrix $P$ consists of a minimal-degree B\'ezout vector (a minimal-degree solution to the univariate effective Nullstellensatz problem) of $\mathbf{a}$, and the last $n-1$ columns comprise an optimal-degree basis, called a $\mu$-basis, of the syzygy module of $\mathbf{a}$. The main step of the algorithm is a partial row-echelon reduction of a $(2d+1)\times(nd+n+1)$ matrix over $\mathbb{K}$, where $d$ is the maximum degree of the input $\mathbf{a}$. Read More


Let K be a field and denote by K[t], the polynomial ring with coefficients in K. Set A = K[f1,. . Read More


We prove that for every indecomposable ordinal there exists a (transfinitely valued) Euclidean domain whose minimal Euclidean norm is of that order type. Conversely, any such norm must have indecomposable type, and so we completely characterize the norm complexity of Euclidean domains. Modifying this construction, we also find a finitely valued Euclidean domain with no multiplicative integer valued norm. Read More


Inspirited by the importance of the spectral theory of graphs, we introduce the spectral theory of valued cluster quiver of a cluster algebra. Our aim is to characterize a cluster algebra via its spectrum so as to use the spectral theory as a tool. First, we give the relations between exchange spectrum of a valued cluster quiver and adjacency spectrum of its underlying valued graph, and between exchange spectra of a valued cluster quiver and its full valued subquivers. Read More


The notion of an Ohm-Rush algebra, and its associated content map, has connections with prime characteristic algebra, polynomial extensions, and the Ananyan-Hochster proof of Stillman's conjecture. As further restrictions are placed (creating the increasingly more specialized notions of weak content, semicontent, content, and Gaussian algebras), the construction becomes more powerful. Here we settle the question in the affirmative over a Noetherian ring from our previous article of whether a faithfully flat weak content algebra is semicontent (and over an Artinian ring of whether such an algebra is content), though both questions remain open in general. Read More


In the interest of finding the minimum additive generating set for the set of $\boldsymbol{s}$-lecture hall partitions, we compute the Hilbert bases for the $\boldsymbol{s}$-lecture hall cones in certain cases. In particular, we compute the Hilbert bases for two well-studied families of sequences, namely the $1\mod k$ sequences and the $\ell$-sequences. Additionally, we provide a characterization of the Hilbert bases for $\boldsymbol{u}$-generated Gorenstein $\boldsymbol{s}$-lecture hall cones in low dimensions. Read More


In this paper we construct some regular sequences which arise naturally from determinantal conditions. Read More


We show that if $G$ is a gap-free and diamond-free graph, then $I(G)^s$ has a linear minimal free resolution for every $s\geq 2$. Read More