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Linear codes over signed graphsApr 20 2019We give formulas, in terms of graph theoretical invariants, for the minimum distance and the generalized Hamming weights of the linear code generated by the rows of the incidence matrix of a signed graph over a finite field, and for those of its dual ... More
Two definable subcategories of maximal Cohen-Macaulay modulesApr 18 2019Over a Cohen-Macaulay ring we consider two extensions of the maximal Cohen-Macaulay modules from the viewpoint of definable subcategories, which are closed under direct limits, direct products and pure submodules. After describing these categories, we ... More
Generators of Koszul homology with coefficients in a $\underline{g}$-weak complete intersection moduleApr 17 2019We discuss a class of modules, which we call $\underline{g}$-weak complete intersection modules, inspired by the weak complete intersection ideals studied by Rahmati, Striuli, and Yang and we present explicit formulas for the generators of Koszul homology ... More
Higher dimensional connectivity and minimal degree of random graphs with an eye towards minimal free resolutionsApr 17 2019In this note we define and study graph invariants generalizing to higher dimension the maximum degree of a vertex and the vertex-connectivity (our $0$-dimensional cases). These are known to coincide almost surely in any regime for Erdoes-Renyi random ... More
The module of vector-valued modular forms is Cohen-MacaulayApr 17 2019Let $H$ denote a finite index subgroup of the modular group $\Gamma$ and let $\rho$ denote a finite-dimensional complex representation of $H.$ Let $M(\rho)$ denote the collection of holomorphic vector-valued modular forms for $\rho$ and let $M(H)$ denote ... More
Graded Quivers, Generalized Dimer Models and Toric GeometryApr 16 2019The open string sector of the topological B-model model on CY $(m+2)$-folds is described by $m$-graded quivers with superpotentials. This correspondence extends to general $m$ the well known connection between CY $(m+2)$-folds and gauge theories on the ... More
Computing the Lie algebra of the differential Galois group: the reducible caseApr 16 2019In this paper, we explain how to compute the Lie algebra of the differential Galois group of a reducible linear differential system. We achieve this by showing how to transform a block-triangular linear differential system into a Kolchin-Kovacic reduced ... More
Toric degenerations of flag varieties from matching field tableauxApr 16 2019We present families of tableaux which interpolate between the classical semi-standard Young tableaux and matching field tableaux. Algebraically, this corresponds to SAGBI bases of Pl\"ucker algebras. We show that each such family of tableaux leads to ... More
The 0-th Fitting ideal of the Jacobian module of a plane curveApr 16 2019We describe the 0-th Fitting ideal of the Jacobian module of a plane curve in terms of determinants involving the Jacobian syzygies of this curve. This leads to new characterizations of maximal Tjurina curves, that is of non free plane curves, whose global ... More
Free differential Galois groupsApr 16 2019We study the structure of the absolute differential Galois group of a rational function field over an algebraically closed field of characteristic zero. In particular, we relate the behavior of differential embedding problems to the condition that the ... More
A transformation rule for natural multiplicitiesApr 16 2019Apr 17 2019For multiplicities arising from a family of ideals we provide a general approach to transformation rules for a ring extension \'etale in codimension one. Our result can be applied to bound the size of the local \'etale fundamental group of a singularity ... More
A transformation rule for natural multiplicitiesApr 16 2019For multiplicities arising from a family of ideals we provide a general approach to transformation rules for a ring extension \'etale in codimension one. Our result can be applied to bound the size of the local \'etale fundamental group of a singularity ... More
Invariants of the symbolic powers of edge idealsApr 16 2019Let $G$ be a graph and $I=I(G)$ be its edge ideal. When $G$ is the clique sum of two different length odd cycles joined at single vertex then we give an explicit description of the symbolic powers of $I$ and compute the Waldschmidt constant. When $G$ ... More
The stable category of Gorenstein flat sheaves on a noetherian schemeApr 16 2019For a semi-separated noetherian scheme, we show that the category of cotorsion Gorenstein flat quasi-coherent sheaves is Frobenius and a natural non-affine analogue of the category of Gorenstein projective modules over a noetherian ring. We show that ... More
Depth functions of powers of homogeneous idealsApr 16 2019We settle a conjecture of Herzog and Hibi, which states that the function depth $S/Q^n$, $n \ge 1$, where $Q$ is a homogeneous ideal in a polynomial ring $S$, can be any convergent numerical function. We also give a positive answer to a long-standing ... More
On the computation of Castelnuovo-Mumford regularity of the Rees algebra and of the fiber ringApr 16 2019We present algorithms for the computation of the Castelnuovo-Mumford regularity of the Rees algebra and of the fiber ring of equigenerated $\mm$-primary ideals in two variables. Applying these algorithms, we find a counter-example to a conjecture of Eisenbud ... More
Almost Gorenstein rings arising from fiber productsApr 15 2019The purpose of this paper is, as part of the stratification of Cohen-Macaulay rings, to investigate the question of when the fiber products are almost Gorenstein rings. We show that the fiber product $R \times_T S$ of Cohen-Macaulay local rings $R$, $S$ ... More
Minimal System of Generators and Syzygies of Centro-Affine Invariants and Covariants for Homogeneous Planar Cubic Differential Systems with Free Terms and Linear PartApr 14 2019A minimal system of generators of the algebra of the centro-affine covariants for homogeneous planar cubic differential systems with linear part is known. With the help of the Gurevich theorem avoiding the Aronhold's identities based on the calculation ... More
Connected Sums of Graded Artinian Gorenstein Algebras and Lefschetz PropertiesApr 12 2019A connected sum construction for local rings was introduced in a paper by H. Ananthnarayan, L. Avramov, and W.F. Moore. In the graded Artinian Gorenstein case, this can be viewed as an algebraic analogue of the topological construction of the same name. ... More
A topology on the set of isomorphism classes of maximal Cohen--Macaulay modulesApr 12 2019In this paper, we introduce a topology on the set of isomorphism classes of finitely generated modules over an associative algebra. Then we focus on the relative topology on the set of isomorphism classes of maximal Cohen--Macaulay modules over a Cohen--Macaulay ... More
Implications of positive formulas in modules (RIMS)Apr 12 2019In this survey the role of implications of positive formulas -- finitary and infinitary -- is dicussed, in general and in module categories, where they seem of particular importance. A list of algebraic examples is given, some old, some rather new, and ... More
What Makes a Complex VirtualApr 12 2019Virtual resolutions are homological representations of finitely generated $\text{Pic}(X)$-graded modules over the Cox ring of a smooth projective toric variety. In this paper, we identify two algebraic conditions that characterize when a chain complex ... More
Resurgence and Castelnuovo-Mumford regularity of certain monomial curves in ${\mathbb A}^3$Apr 11 2019Let ${\mathfrak p}$ be the defining ideal of the monomial curve ${\mathcal C}(2q+1, 2q+1+m, 2q+1+2m)$ in the affine space ${\mathbb A}_k^3$ parameterized by $(x^{2q +1}, x^{2q +1 + m}, x^{2q +1 +2 m})$ where $gcd( 2q+1,m)=1$. In this paper we compute ... More
The Hochschild cohomology ring of the numerical semigroup algebras of embedding dimension twoApr 11 2019Let $a$ and $b$ be two coprime positive integers and $k$ an arbitrary field. We determine the ring structure of the Hochschild cohomology of the numerical semigroup algebras $k[s^{a},s^{b}]$ of embedding dimension two (thus also complete intersections) ... More
The sequence of mixed Łojasiewicz exponents associated to pairs of idealsApr 10 2019We analyze the sequence $\mathcal L^*_J(I)$ of mixed \L ojasiewicz exponents attached to any pair $I,J$ of monomial ideals of finite colength of the ring of analytic function germs $(\mathbb C^n,0)\to \mathbb C$. In particular, we obtain a combinatorial ... More
Tensor Representation of Rank-Metric CodesApr 10 2019We present the theory of rank-metric codes with respect to the 3-tensors that generate them. We define the generator tensor and the parity check tensor of a matrix code, and describe the properties of a code through these objects. We define the tensor ... More
Characterizations of derivationsApr 10 2019The main purpose of this work is to characterize derivations through functional equations. This work consists of five chapters. In the first one, we summarize the most important notions and results from the theory of functional equations. In Chapter 2 ... More
On the fiber cone of monomial idealsApr 10 2019We consider the fiber cone of monomial ideals. It is shown that for monomial ideals $I\subset K[x,y]$ of height $2$, generated by $3$ elements, the fiber cone $F(I)$ of $I$ is a hypersurface ring, and that $F(I)$ has positive depth for interesting classes ... More
Asymptotic Syzygies in the Setting of Semi-Ample GrowthApr 09 2019We study the asymptotic non-vanishing of syzygies for products of projective spaces. Generalizing the monomial methods of Ein, Erman, and Lazarsfeld \cite{einErmanLazarsfeld16} we give an explicit range in which the graded Betti numbers of $\mathbb{P}^{n_1}\times ... More
Almost complete intersection binomial edge ideals and their Rees algebrasApr 09 2019Let $G$ be a simple graph on $n$ vertices and $J_G$ denote the corresponding binomial edge ideal in the polynomial ring $S = K[x_1, \ldots, x_n, y_1, \ldots, y_n].$ In this article, we compute the second Betti number and obtain a minimal presentation ... More
Some remarks about trunks and morphisms of neural codesApr 09 2019We give intrinsic characterizations of neural rings and homomorphisms between them. We also characterize monomial code maps as compositions of basic monomial code maps. Our work is based on two theorems by Curto and Youngs from 2015 and the notions of ... More
Loose edges and factorization theoremsApr 08 2019Let $ R $ be a regular local ring with maximal ideal $ \mathfrak{m} $. We consider elements $ f \in R $ such that their Newton polyhedron has a loose edge. We show that if the symbolic restriction of $f$ to such an edge is a product of two coprime polynomials, ... More
Projective dimension and regularity of edge ideals of some vertex-weighted oriented $m$-partite graphsApr 08 2019In this paper we provide some exact formulas for the projective dimension and the regularity of edge ideals associated to three special types of vertex-weighted oriented $m$-partite graphs. These formulas are functions of the weight and number of vertices. ... More
Cohen Macaulay Hybrid GraphsApr 08 2019We introduce a new family of graphs, namely, hybrid graphs. There are infinitely many hybrid graphs associated to a single graph. We show that every hybrid graph associated to a given graph is Cohen Macaulay. Furthermore, we show that every CohenMacaulay ... More
On the Saito's basis and the Tjurina Number for Plane BranchesApr 07 2019Apr 09 2019We introduce the concept of good Saito's basis for a plane curve $S$ and we explore it to obtain a formula for the minimal Tjurina number in a topological class. In particular, we present a positive answer for a question of Dimca and Greuel relating the ... More
On the Saito's basis and the Tjurina Number for Plane BranchesApr 07 2019We introduce the concept of good Saito's basis for a plane curve $S$ and we explore it to obtain a formula for the minimal Tjurina number in a topological class. In particular, we present a positive answer for a question of Dimca and Gruel relating the ... More
Cohen-Macaulay homological dimensionsApr 07 2019We introduce new homological dimensions, namely the Cohen-Macaulay projective, injective and flat dimensions for homologically bounded complexes. Among other things we show that (a) these invariants characterize the Cohen-Macaulay property for local rings, ... More
Topics on Smooth Commutative AlgebraApr 04 2019We present, in the same vein as in [20] and [21], some results of the so-called "Smooth (or $\mathcal{C}^\infty$) Commutative Algebra", a version of Commutative Algebra of $\mathcal{C}^{\infty}-$rings instead of ordinary commutative unital rings, looking ... More
The minimal Tjurina number of irreducible germs of plane curve singularitiesApr 04 2019In this paper we provide a closed formula for the minimal Tjurina number of an equisingularity class in terms of the sequence of multiplicities of the strict transform along a resolution. As a consequence, we give a positive answer to a question of Dimca ... More
The minimal Tjurina number of irreducible germs of plane curve singularitiesApr 04 2019Apr 10 2019In this paper we provide a closed formula for the minimal Tjurina number of an equisingularity class in terms of the sequence of multiplicities of the strict transform along a resolution. As a consequence, we give a positive answer to a question of Dimca ... More
Projective dimension and regularity of powers of edge ideals of vertex-weighted rooted forestsApr 04 2019In this paper we provide some exact formulas for projective dimension and the regularity of powers of edge ideals of vertex-weighted rooted forests. These formulas are functions of the weight of the vertices and the number of edges. We also give some ... More
Regularity of powers of edge ideals of vertex-weighted oriented unicyclic graphsApr 04 2019In this paper we provide some exact formulas for the regularity of powers of edge ideals of vertex-weighted oriented cycles and vertex-weighted unicyclic graphs. These formulas are functions of the weight of vertices and the number of edges. We also give ... More
On the Betti numbers and Rees algebras of ideals with linear powersApr 03 2019An ideal $I \subset \mathbb{k}[x_1, \ldots, x_n]$ is said to have linear powers if $I^k$ has a linear minimal free resolution, for all $k$. In this paper we study the Betti numbers of $I^k$, for ideals $I$ with linear powers. The Betti numbers are computed ... More
Strict log-concavity of the Kirchhoff polynomial and its applications to the strong Lefschetz propertyApr 03 2019Anari, Gharan, and Vinzant proved (completely) log-concavity of the basis generating functions for all matroids. From the viewpoint of combinatorial Hodge theory, it is natural to ask whether these functions are "strict" log-concave for simple matroids. ... More
Strict log-concavity of the Kirchhoff polynomial and its applications to the strong Lefschetz propertyApr 03 2019Apr 04 2019Anari, Gharan, and Vinzant proved (completely) log-concavity of the basis generating functions for all matroids. From the viewpoint of combinatorial Hodge theory, it is natural to ask whether these functions are "strict" log-concave for simple matroids. ... More
Frobenius and Homological Dimensions of ComplexesApr 01 2019It is proved that a module $M$ over a Noetherian local ring $R$ of prime characteristic and positive dimension has finite flat dimension if Tor$_i^R({}^e R, M)=0$ for dim $R$ consecutive positive values of $i$ and infinitely many $e$. Here ${}^e R$ denotes ... More
Depth and Extremal Betti Number of Binomial Edge IdealsApr 01 2019Let $G$ be a simple graph on the vertex set $[n]$ and $J_G$ be the corresponding binomial edge ideal. Let $G=v*H$ be the cone of $v$ on $H$. In this article, we compute all the Betti numbers of $J_G$ in terms of Betti number of $J_H$ and as a consequence, ... More
An application of Liaison theory to zero-dimensional schemesApr 01 2019Given a 0-dimensional scheme X in a n-dimensional projective space P^n_K over an arbitrary field K, we use Liaison theory to characterize the Cayley-Bacharach property of X. Our result extends the result for sets of K-rational points given in [7]. In ... More
Symbolic blowup algebras and invariants associated to certain monomial curves in ${\mathbb P}^3$Apr 01 2019In this paper we explicitly describe the symbolic powers of curves ${\mathcal C}(q,m)$ in ${\mathbb P}^3$ parametrized by $( x^{d+2m}, x^{d+m} y^m, x^{d} y^{2m}, y^{d+2m})$, where $q,m$ are positive integers, $d=2q+1$ and $\gcd(d,m)=1$. The defining ideal ... More
Lyubeznik numbers of irreducible projective varieties depend on the embeddingMar 31 2019We construct irreducible complex projective varieties such that the Lyubeznik numbers of their affine cones depend on the choices of projective embeddings. The main ingredient is the recent work of Reichelt-Saito-Walther, where the Lyubeznik numbers are ... More
Factorization invariants of Puiseux monoids generated by geometric sequencesMar 30 2019We study here some of the factorization invariants of the class of Puiseux monoids generated by geometric sequences, and we compare and contrast them with the known results for numerical monoids generated by arithmetic sequences. The class we study here ... More
Large lower bounds for the betti numbers of graded modules with low regularityMar 29 2019Suppose that $M$ is a finitely-generated graded module of codimension $c\geq 3$ over a polynomial ring and that the regularity of $M$ is at most $2a-2$ where $a\geq 2$ is the minimal degree of a first syzygy of $M$. Then we show that the sum of the betti ... More
Extremal growth of Betti numbers and rigidity of (co)homologyMar 29 2019A Cohen-Macaulay local ring $R$ satisfies trivial vanishing if $\operatorname{Tor}_i^R(M,N)=0$ for all large $i$ implies $M$ or $N$ has finite projective dimension. If $R$ satisfies trivial vanishing then we also have that $\operatorname{Ext}^i_R(M,N)=0$ ... More
Extremal growth of Betti numbers and rigidity of (co)homologyMar 29 2019Apr 03 2019A Cohen-Macaulay local ring $R$ satisfies trivial vanishing if $\operatorname{Tor}_i^R(M,N)=0$ for all large $i$ implies $M$ or $N$ has finite projective dimension. If $R$ satisfies trivial vanishing then we also have that $\operatorname{Ext}^i_R(M,N)=0$ ... More
Expected resurgences and symbolic powers of idealsMar 28 2019We give explicit criteria that imply the resurgence of a self-radical ideal in a regular ring is strictly smaller than its codimension, which in turn implies that the stable version of Harbourne's conjecture holds for such ideals. This criterion is used ... More
Notes on polynomials $(1+X)^n + (-1)^n(X^n+1)$ concerning the regularity problem for symmetric power sums in 3 variablesMar 27 2019Let $K$ be a field and $f _{n}(X) = (X + 1) ^{n} + (-1) ^{n}(X ^{n} + 1) \in K[X]$, for each $n \in \mathbb N$. This note shows that the polynomials $f _{m}(X)$ and $f _{m'}(X)$ are relatively prime, for some distinct indices $m$ and $m ^{\prime}$ at ... More
The level of pairs of polynomialsMar 27 2019Given a polynomial $f$ with coefficients in a field of prime characteristic $p$, it is known that there exists a differential operator that raises $1/f$ to its $p$th power. We first discuss a relation between the `level' of this differential operator ... More
Full rank valuations and toric initial idealsMar 26 2019Let $V(I)$ be a polarized projective variety or a subvariety of a product of projective spaces and let $A$ be its (multi-)homogeneous coordinate ring. Given a full-rank valuation $\mathfrak v$ on $A$ we associate weights to the coordinates of the projective ... More
Ideal containment vs. powersMar 26 2019Let $R$ be a commutative ring with identity. In this note, we study the property: If $ I \subsetneqq J$ are ideals in $R$, then $ I^n \subsetneqq J^n$ for all $ n\geq 1$. We define the notion of a big ideal (Definition 1.2). It is noted that the property ... More
On pseudo-Frobenius elements of submonoids of $\mathbb{N}^d$Mar 26 2019In this paper we study those submonoids of $\mathbb{N}^d$ which a non-trivial pseudo-Frobenius set. In the affine case, we prove that they are the affine semigroups whose associated algebra over a field has maximal projective dimension possible. We prove ... More
Powers Vs. PowersMar 26 2019Let $ A \subset B$ be rings. An ideal $ J \subset B$ is called power stable in $A$ if $ J^n \cap A = (J\cap A)^n$ for all $ n\geq 1$. Further, $J$ is called ultimately power stable in $A$ if $ J^n \cap A = (J\cap A)^n$ for all $n$ large i.e., $ n \gg ... More
Regularity of symbolic powers of edge ideals of unicyclic graphsMar 26 2019Let $G$ be a unicyclic graph with edge ideal $I(G)$. For any integer $s\geq 1$, we denote the $s$-th symbolic power of $I(G)$ by $I(G)^{(s)}$. It is shown that ${\rm reg}(I(G)^{(s)})={\rm reg}(I(G)^s)$, for every $s\geq 1$.
Valuations and henselizationMar 26 2019We study the extension of valuations centered in a local domain to its henseliza-tion. We prove that a valuation $\nu$ centered in a local domain R uniquely determines a minimal prime H($\nu$) of the henselization R h of R and an extension of $\nu$ centered ... More
On the containment problem for fat points idealsMar 26 2019In this note we show that Harbourne's conjecture is true for symbolic powers of ideals of points, we check that the stable version of this conjecture is valid for ideals of very general points (resp. generic points) in $\mathbb P_{\mathbb K}^N$ (resp. ... More
On the central geometry of nonnoetherian dimer algebrasMar 25 2019Let $Z$ be the center of a nonnoetherian dimer algebra on a torus. Although $Z$ itself is also nonnoetherian, we show that it has Krull dimension $3$, and is locally noetherian on an open dense set of $\operatorname{Max}Z$. Furthermore, we show that the ... More
On the structure of the Sally module and the second normal Hilbert coefficientMar 25 2019The Hilbert coefficients of the normal filtration give important geometric information on the base ring like the pseudo-rationality. The Sally module was introduced by W.V. Vasconcelos and it is useful to connect the Hilbert coefficients to the homological ... More
Algorithms for Checking Zero-Dimensional Complete IntersectionsMar 22 2019Given a 0-dimensional affine K-algebra R=K[x_1,...,x_n]/I, where I is an ideal in a polynomial ring K[x_1,...,x_n] over a field K, or, equivalently, given a 0-dimensional affine scheme, we construct effective algorithms for checking whether R is a complete ... More
Constructing symplectomorphisms between symplectic torus quotientsMar 22 2019We identify a family of torus representations such that the corresponding singular symplectic quotients at the $0$-level of the moment map are graded regularly symplectomorphic to symplectic quotients associated to representations of the circle. For a ... More
Radical factorization in finitary ideal systemsMar 21 2019In this paper we investigate the concept of radical factorization with respect to finitary ideal systems of cancellative monoids. We present new characterizations for r-almost Dedekind r-SP-monoids and provide specific descriptions of t-almost Dedekind ... More
Regularity and Koszul property of symbolic powers of monomial idealsMar 21 2019Let $I$ be a homogeneous ideal in a polynomial ring over a field. Let $I^{(n)}$ be the $n$-th symbolic power of $I$. Motivated by results about ordinary powers of $I$, we study the asymptotic behavior of the regularity function $\text{reg}(I^{(n)})$ and ... More
Basis Criteria for Generalized Spline Modules via DeterminantMar 21 2019Given a graph whose edges are labeled by ideals of a commutative ring R with identity, a generalized spline is a vertex labeling by the elements of R such that the difference of the labels on adjacent vertices lies in the ideal associated to the edge. ... More
Syzygies in Hilbert schemes of complete intersectionsMar 20 2019Let $ d_1, \ldots, d_{c} $ be positive integers and let $ Y \subseteq \mathbb{P}^n$ be the monomial complete intersection defined by the vanishing of $x_1^{d_1}, \ldots, x_{c}^{d_{c}}$. For each Hilbert polynomial $p(\zeta)$ we construct a distinguished ... More
Quadratic Gorenstein rings and the Koszul property IIMar 19 2019In a previous paper, we use idealization to construct numerous examples of standard graded quadratic Gorenstein rings having regularity three which are not Koszul, negatively answering a question of Conca, Rossi, and Valla. This paper is the natural continuation ... More
Quadratic Gorenstein rings and the Koszul property IMar 19 2019Let $R$ be a standard graded Gorenstein algebra over a field presented by quadrics. Conca, Rossi, and Valla have shown that such a ring is Koszul if $\mathrm{reg}\, R \leq 2$ or if $\mathrm{reg}\, R = 3$ and $\mathrm{codim}\, R \leq 4$, and they ask whether ... More
Fibers of multi-graded rational maps and orthogonal projection onto rational surfacesMar 19 2019We contribute a new algebraic method for computing the orthogonal projections of a point onto a rational algebraic surface embedded in the three dimensional projective space. This problem is first turned into the computation of the finite fibers of a ... More
On commutative differential graded algebrasMar 18 2019In this paper we undertake a basic study on connective commutative differential graded algebras (CDGA), more precisely, piecewise Noetherian CDGA, which is a DG-counter part of commutative Noetherian algebra. We establish basic results for example, Auslaner-Buchsbaum ... More
Gorenstein $π[T]$-projectivity with respect to a tilting moduleMar 17 2019Let $T$ be a tilting module. In this paper, Gorenstein $\pi[T]$-projective modules are introduced and some of their basic properties are studied. Moreover, some characterizations of rings over which all modules are Gorenstein $\pi[T]$-projective are given. ... More
The Hilbert scheme of a pair of linear spacesMar 15 2019Let $H(c,d,n)$ be the component of the Hilbert scheme whose general point parameterizes a $c$-plane union a $d$-plane in $\mathbf{P}^n$. We show that $H(c,d,n)$ is non-singular and isomorphic to successive blow ups of $\mathbf{G}(c,n) \times \mathbf{G}(d,n)$ ... More
Measuring the non-Gorenstein locus of Hibi rings and normal affine semigroup ringsMar 14 2019The trace of the canonical module of a Cohen-Macaulay ring describes its non-Gorenstein locus. We study the trace of the canonical module of a Segre product of algebras, and we apply our results to compute the non-Gorenstein locus of toric rings. We provide ... More
On interplay between the Frobenius functor and its dualMar 13 2019For a commutative Noetherian ring $R$ of prime characteristic, denote by $^{f}R$ the ring $R$ with the left structure given by the Frobenius map. We develop Thomas Marley's work on the property of the Frobenius functor $\F(-) = - \otimes_R {^f}R$ and ... More
Regularity of Symbolic Powers of Edge IdealsMar 13 2019In this article, we prove that for several classes of graphs, the Castelnuovo-Mumford regularity of symbolic powers of their edge ideals coincide with that of their ordinary powers.
A new Homological Invariant for ModulesMar 13 2019Let $R$ be a commutative Noetherian local ring with residue field $k$. Using the structure of Vogel cohomology, for any finitely generated module $M$, we introduce a new dimension, called $\zeta$-dimension, denoted by $\zeta-dim_R M$. This dimension is ... More
So what is class number 2?Mar 11 2019Using factorization properties, we give several characterizations for an algebraic number ring to have class number 2.
Wilf's conjecture in fixed multiplicityMar 11 2019We give an algorithm to determine whether Wilf's conjecture holds for all numerical semigroups with a given multiplicity $m$, and use it to prove Wilf's conjecture holds whenever $m \le 17$. Our algorithm utilizes techniques from polyhedral geometry, ... More
Polyhedral products and features of their homotopy theoryMar 11 2019A polyhedral product is a natural subspace of a Cartesian product that is specified by a simplicial complex. The modern formalism arose as a generalization of the spaces known as moment-angle complexes which were developed within the nascent subject of ... More
A study of absolute, relative and Tate homology with respect to a semidualizing moduleMar 11 2019In this paper we are concerned with absolute, relative and Tate Tor modules with respect to a fixed semidualizing module over commutative Noetherian local rings. Motivated by a result of Avramov and Martsinkovsky, we obtain an exact sequence connecting ... More
On Large Homomorphisms of Local RingsMar 10 2019We study ideals in a local ring $R$ whose quotient rings induce large homomorphisms of local rings. We characterize such ideals over complete intersections, Koszul rings, and over some classes of Golod rings.
Irreducible divisor pair domainsMar 09 2019We introduce and study a new class of integral domains which we call irreducible divisor pair domains (IDPDs). In particular, we show how IDPDs fit in with other classes of integral domains defined in terms of factorization conditions. For instance, every ... More
Upper Characteristic Trees of a Lie AlgebraMar 09 2019In this paper, we introduce upper characteristic trees for finite dimensional Lie algebras. The ideals of a finite dimensional Lie algebra are distributed as nodes of some upper characteristic trees. A node is connected with an upper level node in an ... More
Toward Free Resolutions Over ScrollsMar 08 2019Let $R = k[x]/I$ where $I$ is the defining ideal of a rational normal $k$-scroll. We compute the Betti numbers of the ground field $\mathbb{k}$ as a module over $R$. For $k = 2$, we give the minimal free resolution of $\mathbb{k}$ over $R$.
Packing properties of cubic squarefree monomial idealsMar 08 2019Mar 12 2019Let $I$ be an ideal in a Noetherian ring $R$. The $n^{\text{th}}$ symbolic power of $I$ is defined as $$I^{(n)}=\displaystyle\bigcap_{p\in Ass(R)}(I^nR_p\cap R).$$ The symbolic powers, in general, are not equal to the ordinary powers. Therefore, one interesting ... More
Packing properties of cubic squarefree monomial idealsMar 08 2019In this paper, we study the equality between symbolic and ordinary powers for some classes of cubic squarefree monomial ideals.
Maximal Cohen-Macaulay modules that are not locally free on the punctured spectrumMar 08 2019We say that a Cohen-Macaulay local ring has finite $\operatorname{\mathsf{CM}}_+$-representation type if there exist only finitely many isomorphism classes of indecomposable maximal Cohen-Macaulay modules that are not locally free on the punctured spectrum. ... More
An algorithmic approach to the existence of ideal objects in commutative algebraMar 07 2019The existence of ideal objects, such as maximal ideals in nonzero rings, plays a crucial role in commutative algebra. These are typically justified using Zorn's lemma, and thus pose a challenge from a computational point of view. Giving a constructive ... More
A Kruskal-Katona type result and applicationsMar 07 2019Inspired by the Kruskal-Katona theorem a minimization problem is studied, where the role of the shadow is replaced by the image of the action of the monoid of increasing functions. One of our main results shows that compressed sets are a solution to this ... More
Linkage of Pfister forms over $\mathbb{C}(x_1,\ldots,x_n)$Mar 07 2019In this note, we prove the existence of a set of $n$-fold Pfister forms of cardinality $2^n$ over $\mathbb{C}(x_1,\dots,x_n)$ which do not share a common $(n-1)$-fold factor. This gives a negative answer to a question raised by Becher. The main tools ... More
Adelic models of tensor-triangulated categoriesMar 07 2019We show that a well behaved Noetherian, finite dimensional, stable, monoidal model category is equivalent to a model built from categories of modules over completed rings in an adelic fashion. For abelian groups this is based on the Hasse square, for ... More
Adelic cohomologyMar 07 2019The characteristic feature of the adeles is that they involve localizations of products (or equivalently restricted products of localizations). The point of this paper is to introduce an adelic style cohomological invariant of a partially ordered set ... More
Local cohomology in Grothendieck categoriesMar 06 2019Let $\mathcal{A}$ be a locally noetherian Grothendieck category. In this paper we define and study the section functor on $\mathcal{A}$ with respect to an open subset of ASpec$\mathcal{A}$. Next we define and study local cohomology theory in $\mathcal{A}$ ... More
Embedding dimension of a good semigroupMar 05 2019In this paper, we study good semigroups of \mathbb{N}^n, a class of semigroups that contains the value semigroups of algebroid curves with n branches. We give the definition of embedding dimension of a good semigroup showing that, in the case of good ... More