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Binomial edge ideals of cographsJun 13 2019We determine the Castelnuovo--Mumford regularity of binomial edge ideals of complement reducible graphs (cographs). On $n$ vertices the maximum regularity is essentially $2n/3$. Independently of the number of vertices, we also bound the regularity by ... More
Analysis of linear systems over idempotent semifieldsJun 11 2019In this paper, we present and analyze methods for solving a system of linear equations over idempotent semifields. The first method is based on the pseudo-inverse of the system matrix. We then present a specific version of Cramer's rule which is also ... More
Linear resolutions over Koszul complexes and Koszul homology algebrasJun 10 2019Let $R$ be a standard graded commutative algebra over a field $k$, let $K$ be its Koszul complex viewed as a differential graded $k$-algebra, and let $H$ be the homology algebra of $K$. This paper studies the interplay between homological properties of ... More
$\mathbf{A}_{\text{inf}}$ is infinite dimensionalJun 09 2019Given a perfect valuation ring $R$ of characteristic $p$ that is complete with respect to a rank-$1$ nondiscrete valuation, we show that the ring $\mathbf{A}_{\text{inf}}$ of Witt vectors of $R$ has infinite Krull dimension.
Singularities and radical initial idealsJun 07 2019What kind of reduced monomial schemes can be obtained as a Gr\"obner degeneration of a smooth projective variety? Our conjectured answer is: only Stanley-Reisner schemes associated to acyclic Cohen-Macaulay simplicial complexes. This would imply, in particular, ... More
Auslander-Reiten conjecture for non-Gorenstein Cohen-Macaulay ringsJun 06 2019Let $R$ be a Cohen-Macaulay local ring and $Q$ be a parameter ideal of $R$. Due to M. Auslander, S. Ding, and \O. Solberg, the Auslander-Reiten conjecture holds for $R$ if and only if it holds for the residue ring $R/Q$. In the former part of this paper, ... More
On some ideals with linear free resolutionsJun 06 2019Given $\Sigma\subset\mathbb K[x_1,\ldots,x_k]$, any finite collection of linear forms, some possibly proportional, and any $1\leq a\leq |\Sigma|$, it has been conjectured that $I_a(\Sigma)$, the ideal generated by all $a$-fold products of $\Sigma$, has ... More
Locality and causality in perturbative AQFTJun 05 2019In this paper we introduce a notion of \textit{a group with causality}, which is a natural generalization of \textit{a locality group}, introduced by P.~Clavier, L.~Guo, S.~Paycha, and B.~Zhang. We also propose a generalization of the \textit{Hammerstein ... More
Locally Heavy Hyperplanes in MultiarrangementsJun 05 2019Hyperplane Arrangements of rank $3$ admitting an unbalanced Ziegler restriction are known to fulfil Terao's conjecture. This long-standing conjecture asks whether the freeness of an arrangement is determined by its combinatorics. In this note we prove ... More
A geometrical characterization of proportionally modular affine semigroupsJun 04 2019A proportionally modular affine semigroup is the set of nonnegative integer solutions of a modular Diophantine inequality $f_1x_1+\cdots +f_nx_n \mod b \le g_1x_1+\cdots +g_nx_n$ where $g_1,\dots,g_n,$ $f_1,\ldots ,f_n\in \mathbb{Z}$ and $b\in\mathbb{N}$. ... More
Union of sets of lengths of numerical semigroupsJun 04 2019Let $S=\langle a_1,\ldots,a_p\rangle$ be a numerical semigroup, $s\in S$ and ${\sf z}(s)$ its set of factorizations. The set of length is denoted by ${\mathcal L}(s)=\{{\tt L}(x_1,\dots,x_p)\mid (x_1,\dots,x_p)\in{\sf Z}(s)\}$ where ${\tt L}(x_1,\dots,x_p)=x_1+\ldots+x_p$. ... More
Mixed Multiplicities of Maximal Degrees (J. Korean Math. Soc. 55 (2018), No. 3, 605-622)Jun 04 2019The original mixed multiplicity theory considered the class of mixed multiplicities concerning the terms of highest total degree in the Hilbert polynomial. This paper defines a broader class of mixed multiplicities that concern the maximal terms in this ... More
Equimultiplicity Theory of Strongly $F$-regular ringsJun 04 2019We explore the equimultiplicity theory of the $F$-invariants Hilbert--Kunz multiplicity, $F$-signature, Frobenius Betti numbers, and Frobenius Euler characteristic over strongly $F$-regular rings. Techniques introduced in this article provide a unified ... More
Regularity and projective dimension of powers of edge ideal of the disjoint union of some weighted oriented gap-free bipartite graphsJun 04 2019In this paper we provide some precise formulas for regularity of powers of edge ideal of the disjoint union of some weighted oriented gap-free bipartite graphs. For the projective dimension of such an edge ideal, we give its exact formula. Meanwhile, ... More
The Hard Lefschetz Theorem for PL spheresJun 03 2019Jun 10 2019We provide a simpler proof of the hard Lefschetz Theorem for face rings of PL spheres: While the algebraic theory remains the same, we replace the geometric constructions by Pachner's Theorem. This simplifies the reasoning for an important special case ... More
The Hard Lefschetz Theorem for PL spheresJun 03 2019Jun 04 2019We provide a simpler proof of the hard Lefschetz Theorem for face rings of PL spheres: While the algebraic theory remains the same, we replace the geometric constructions by Pachner's Theorem. This simplifies the reasoning for an important special case ... More
Tilting Modules Over Gorensetein $T_n^d$-Injective Gorensetein $T_n^d$-flat ModulesJun 03 2019Let T be a tilting module.In this paper, some relative Gorenstein projective and Gortenstein injective modules are studied.
Rees Algebra of a Squarefree Monomial IdealJun 03 2019Let $S={\sf k}[X_1,\dots, X_n]$ be a polynomial ring, where ${\sf k}$ is a field. This article deals with the defining ideal of the Rees algebra of squarefree monomial ideal generated in degree $n-2$. As a consequence, we prove that Betti numbers of powers ... More
The module of Valabrega-Valla of the Jacobian ideal of points in projective planeJun 03 2019The module of Valabrega-Valla of the Jacobian ideal of a reduced projective variety $V$ is the torsion of the Aluffi algebra. One considers the problem of its vanishing in the case of where $V$ is a reduced set of points in the projective plane. It is ... More
Unique Factorization in Polynomial Rings with Zero DivisorsJun 03 2019Given a certain factorization property of a ring $R$, we can ask if this property extends to the polynomial ring over $R$ or vice versa. For example, it is well known that $R$ is a unique factorization domain if and only if $R[X]$ is a unique factorization ... More
Virtual Resolutions of Monomial Ideals on Toric VarietiesJun 03 2019We use cellular resolutions of monomial ideals to prove an analog of Hilbert's syzygy theorem for virtual resolutions of monomial ideals on smooth toric varieties.
Mixed multiplicities and the multiplicity of rees modules of reductions (published in j. Algebra appl.)Jun 02 2019This paper shows that mixed multiplicities and the multiplicity of Rees modules of good filtrations and that of their reductions are the same. As an application of this result, we obtain interesting results on mixed multiplicities and the multiplicity ... More
On derived functors of graded local cohomology modules-IIJun 02 2019Let $R=K[X_1,\ldots, X_n]$ where $K$ is a field of characteristic zero, and let $A_n(K)$ be the $n^{th}$ Weyl algebra over $K$. We give standard grading on $R$ and $A_n(K)$. Let $I$, $J$ be homogeneous ideals of $R$. Let $M = H^i_I(R)$ and $N = H^j_J(R)$ ... More
On the Stanley depth of powers of monomial idealsJun 01 2019Let $\mathbb{K}$ be a field and $S=\mathbb{K}[x_1,\dots,x_n]$ be the polynomial ring in $n$ variables over $\mathbb{K}$. In 1982, R. Stanley associated a combinatorial invariant to any finitely generated $\mathbb{Z}^n$-graded $S$-module which is now called ... More
Elimination ideals and Bezout relationsJun 01 2019Let $k$ be an infinite field and $I\subset k [x_1, \ldots ,x_n]$ be an ideal such that dim $V(I)=q$. Denote by $(f_1, \ldots, f_s)$ a set of generators of $I$. One can see that in the set $I\cap k [x_{1},...,x_{q+1}]$ there exist non-zero polynomials, ... More
$I$-Cohen Macaulay modulesJun 01 2019A finitely generated module $M$ over a commutative Noetherian ring $R$ is called an $I$-Cohen Macaulay module, if \[ \grade(I,M) + \dim(M/IM)= \dim(M), \] where $I$ is a proper ideal of $R$. The aim of this paper is to study the structure of this class ... More
Separating Invariants for Two Copies of the Natural $S_n$-ActionMay 31 2019This note provides a set of separating invariants for the ring of vector invariants $K[V^2]^{S_n}$ of two copies of the natural $S_n$-representation $V = K^n$ over a field of characteristic 0. This set is much smaller than generating sets of $K[V^2]^{S_n}$. ... More
Matrix factorizations for self-orthogonal categories of modulesMay 31 2019For a commutative ring $S$ and self-orthogonal subcategory $\mathsf{C}$ of $\mathsf{Mod}(S)$, we consider matrix factorizations whose modules belong to $\mathsf{C}$. Let $f\in S$ be a regular element. If $f$ is $M$-regular for every $M\in \mathsf{C}$, ... More
Affine equivalences, isometries and symmetries of ruled rational surfacesMay 30 2019A method is presented for computing all the affine equivalences between two rational ruled surfaces defined by rational parametrizations that works directly in parametric rational form, i.e. without computing or making use of the implicit equation of ... More
The SchurVeronese package in Macaulay2May 29 2019This note introduces the Macaulay2 package SchurVeronese, which gathers together data about Veronese syzygies and makes it readily accessible in Macaulay2. In addition to standard Betti tables, the package includes information about the Schur decompositions ... More
A prime-characteristic analogue of a theorem of Hartshorne-PoliniMay 29 2019Let $R$ be an $F$-finite Noetherian regular ring containing an algebraically closed field $k$ of positive characteristic, and let $M$ be an $\F$-finite $\F$-module over $R$ in the sense of Lyubeznik (for example, any local cohomology module of $R$). We ... More
Cohomological dimension and relative Cohen-MaculaynessMay 29 2019Let R be a commutative Noetherian (not necessarily local) ring with identity and a be a proper ideal of R. We introduce a notion of a-relative system of parameters and characterize them by using the notion of cohomological dimension. Also, we present ... More
On the generalized Hamming weights of certain Reed-Muller-type codesMay 28 2019There is a nice combinatorial formula of P. Beelen and M. Datta for the $r$-th generalized Hamming weight of an affine cartesian code. Using this combinatorial formula we give an easy to evaluate formula to compute the $r$-th generalized Hamming weight ... More
Interpolation over ZZ and torsion in class groupsMay 28 2019We prove an interpolation result over the integers, and use this to give a new proof that class groups of rings of integers are torsion.
Weak Lefschetz Theorem for Simplicial PL-spheresMay 28 2019We prove the Weak Lefschetz theorem for simplicial PL-spheres. This result is weaker than the Hard Lefschetz theorem for more general spheres proved by Adiprasito (arXiv:1812.10454), but the proof here involves simple algebra and avoids the more complicated ... More
Some combinatorial problems on finite abelian groups and the rational Dyck pathsMay 28 2019In this paper, we study some objects from combinatorial number theory and relate them to the study of the rational Dyck paths. Let $G$ and $H$ be finite abelian groups with $\gcd(|G|,|H|)=1$. For any positive integer $m$, let $\mathsf M(G,m)$ be the set ... More
On $\mathcal{H}_Y$-IdealsMay 28 2019In this article, we continue the studying of $\mathcal{H}_Y$-ideals. We introducing two notions fixed and free $\mathcal{H}_Y$-ideals as an extension of fixed and free z-ideals in C(X) and relative $\mathcal{H}_Y$-ideals as an extension of relative z-ideals. ... More
On $\mathcal{H}_Y$-IdealsMay 28 2019Jun 08 2019In this article, we continue the studying of $\mathcal{H}_Y$-ideals. We introducing two notions fixed and free $\mathcal{H}_Y$-ideals as an extension of fixed and free z-ideals in C(X) and relative $\mathcal{H}_Y$-ideals as an extension of relative z-ideals. ... More
Use DG-methods to build a matrix factorizationMay 27 2019Let P be a commutative Noetherian ring, K be an ideal of P which is generated by a regular sequence of length four, f be a regular element of P, and Pbar be the hypersurface ring P/(f). Assume that K:f is a grade four Gorenstein ideal of P. We give a ... More
Compatible algebras with straightening laws on distributive latticesMay 27 2019We characterize the finite distributive lattices on which there exists a unique compatible algebra with straightening laws.
Compatible algebras with straightening laws on distributive latticesMay 27 2019Jun 01 2019We characterize the finite distributive lattices on which there exists a unique compatible algebra with straightening laws.
On the structure of power set ringMay 25 2019Jun 02 2019In this paper, Stone's Representation Theorem is generalized from Boolean rings to arbitrary commutative rings, and the generalized form is proved by an easy and natural approach. We have also made new progresses in the understanding the structure of ... More
On the structure of power set ringMay 25 2019May 28 2019In this paper, Stone's Representation Theorem is generalized from Boolean rings to arbitrary commutative rings, and the generalized form is proved by an easy and natural approach. We have also made new progresses in the understanding the structure of ... More
Monoidal networksMay 24 2019In this paper we define and study the notion of a monoidal network, which consists of a commutative ring $R$ and a collection of groups $\Gamma_I$, indexed by the ideals of $R$, with $\Gamma_I$ acting on the quotient $R/I$ and satisfying a certain lifting ... More
Unmixedness and arithmetic properties of matroidal idealsMay 24 2019Let $R=k[x_1,...,x_n]$ be the polynomial ring in $n$ variables over a field $k$ and $I$ be a matroidal ideal of degree $d$. In this paper, we study the unmixedness properties and the arithmetical rank of $I$. Moreover, we show that $ara(I)=n-d+1$. This ... More
Differentiation and integration between Hopf algebroids and Lie algebroidsMay 24 2019In this paper we investigate the formal notions of differentiation and integration in the context of commutative Hopf algebroids and Lie algebroid, or more precisely Lie-Rinehart algebras. Specifically, we construct a contravariant functor from the category ... More
2-irreducible and strongly 2-irreducible submodules of a moduleMay 24 2019Let R be a commutative ring with identity and M be an R-module. In this paper, we will introduce the concept of 2-irreducible (resp., strongly 2- irreducible) submodules of M as a generalization of irreducible (resp., strongly irreducible) submodules ... More
Tate Resolutions on Products of Projective Spaces: Cohomology and Direct Image ComplexesMay 24 2019We describe the Macaulay2 package TateOnProducts and its capabilities, which include computing cohomology tables and Beilinson monads of sheaves on products of projective spaces and the derived category pushForward of a sheaf under a morphism from a projective ... More
Higher derivations of Jacobian type in positive characteristicMay 24 2019In this paper, we study higher derivations of Jacobian type in positive characteristic. We give a necessary and sufficient condition for $(n-1)$-tuples of polynomials to be extendable in the polynomial ring in $n$ variables over an integral domain $R$ ... More
Higher derivations of Jacobian type in positive characteristicMay 24 2019Jun 02 2019In this paper, we study higher derivations of Jacobian type in positive characteristic. We give a necessary and sufficient condition for $(n-1)$-tuples of polynomials to be extendable in the polynomial ring in $n$ variables over an integral domain $R$ ... More
Virtual Complete Intersections in $\mathbb{P}^1 \times \mathbb{P}^1$May 24 2019The minimal free resolution of the coordinate ring of a complete intersection in projective space is a Koszul complex on a regular sequence. In the product of projective spaces $\mathbb{P}^1 \times \mathbb{P}^1$, we investigate which sets of points have ... More
Vanishing of Tor over fiber productsMay 23 2019Let $(S,\mathfrak{m},k)$ and $(T,\mathfrak{n},k)$ be local rings, and let $R$ denote their fiber product over their common residue field $k$. We explore consequences of vanishing of ${\rm Tor}^R_m(M,N)$ for small values of $m$, where $M$ and $N$ are finitely ... More
Vanishing of Tor over fiber productsMay 23 2019Jun 13 2019Let $(S,\mathfrak{m},k)$ and $(T,\mathfrak{n},k)$ be local rings, and let $R$ denote their fiber product over their common residue field $k$. We explore consequences of vanishing of ${\rm Tor}^R_m(M,N)$ for small values of $m$, where $M$ and $N$ are finitely ... More
An explicit matrix factorization of cubic hypersurfaces of small dimensionMay 23 2019In this paper, we compute an explicit matrix factorization of a rank 9 Ulrich sheaf on a general cubic hypersurface of dimension at most 7, whose existence was proved by Manivel. Instead of using the invariant theory, we use Shamash's construction with ... More
Bass numbers of local cohomology of cover ideals of graphsMay 23 2019We develop splitting techniques to study Lyubeznik numbers of cover ideals of graphs which allow us to describe them for large families of graphs including forests, cycles, wheels and cactus graphs. More generally we are able to compute all the Bass numbers ... More
Tutte Short Exact Sequences of GraphsMay 22 2019We associate two modules, the $G$-parking critical module and the toppling critical module, to an undirected connected graph $G$. We establish a Tutte-like short exact sequence relating the modules associated to $G$, an edge contraction $G/e$ and edge ... More
Subspace arrangements and Cherednik algebrasMay 21 2019The purpose of this article is to study the relationship between numerical invariants of certain subspace arrangements coming from reflection groups and numerical invariants arising in the representation theory of Cherednik algebras. For instance, we ... More
Bivariate Semialgebraic SplinesMay 21 2019Semialgebraic splines are bivariate splines over meshes whose edges are arcs of algebraic curves. They were first considered by Wang, Chui, and Stiller. We compute the dimension of the space of semialgebraic splines in two extreme cases. If the polynomials ... More
The syzygy theorem for Bézout ringsMay 20 2019We provide constructive versions of Hilbert's syzygy theorem for Z and Z/nZ following Schreyer's method. Moreover, we extend these results to arbitrary coherent strict B\'ezout rings with a divisibility test for the case of finitely generated modules ... More
On the zero-sum constant, the Davenport constant and their analoguesMay 18 2019Let $D(G)$ be the Davenport constant of a finite Abelian group $G$. Let $ZS_m(G)$ be the least $t\in\mathbb{N}\cup\{\infty\}$ such that every sequence of length $t$ in $G$ contains $m$ disjoint zero-sum sequences, each of length $|G|.$ The main result ... More
Degree bounds for Gröbner bases of modulesMay 18 2019May 30 2019Let $F$ be a non-negatively graded free module over a polynomial ring $\mathbb{K}[x_1,\dots,x_n]$ generated by $m$ basis elements. Let $M$ be a submodule of $F$ generated by elements in $F$ with degrees bounded by $D$ and dim $F/M$=$r$. We prove that ... More
Gröbner bases bounds for modulesMay 18 2019Let $F$ be a non-negatively graded free module over a polynomial ring $\mathbb{K}[x_1,\dots,x_n]$ generated by $m$ basis elements. Let $M$ be a submodule of $F$ generated by elements in $F$ with degrees bounded by $D$ and dim $F/M$=$r$. We prove that ... More
Reduced group schemes as iterative differential Galois groupsMay 17 2019This article is on the inverse Galois problem in Galois theory of linear iterative differential equations in positive characteristic. We show that it has an affirmative answer for reduced algebraic group schemes over any iterative differential field which ... More
Irreducibility and factorizations in monoid ringsMay 17 2019For an integral domain $R$ and a commutative cancellative monoid $M$, the ring consisting of all polynomial expressions with coefficients in $R$ and exponents in $M$ is called the monoid ring of $M$ over $R$. An integral domain is called atomic if every ... More
Simplicial generation of Chow rings of matroidsMay 17 2019We introduce a new presentation for the Chow ring of a matroid with far-reaching geometric and combinatorial implications that include recovering a central portion of the Hodge theory of matroids developed by Adiprasito, Huh, and Katz in \cite{AHK18}. ... More
The Virtual Resolutions Package for Macaulay2May 16 2019We introduce the VirtualResolution package for the computer algebra system Macaulay2. This package has tools to construct, display, and study virtual resolutions for products of projective spaces. The package also has tools for generating curves in $\mathbb{P}^1\times\mathbb{P}^2$, ... More
Gorenstein graphic matroidsMay 14 2019The toric variety of a matroid is projectively normal, and therefore it is Cohen-Macaulay. We provide a complete graph-theoretic classification when the toric variety of a graphic matroid is Gorenstein.
Minimal log discrepancies of determinantal varieties via jet schemesMay 14 2019We compute the minimal log discrepancies of determinantal varieties of square matrices, and more generally of pairs $\bigl(D^k,\sum \alpha_i D^{k_i}\bigr)$ consisting of a determinantal variety (of square matrices) and an $\mathbb R$-linear sum of determinantal ... More
Minimal log discrepancies of determinantal varieties via jet schemesMay 14 2019May 21 2019We compute the minimal log discrepancies of determinantal varieties of square matrices, and more generally of pairs $\bigl(D^k,\sum \alpha_i D^{k_i}\bigr)$ consisting of a determinantal variety (of square matrices) and an $\mathbb R$-linear sum of determinantal ... More
Minimal Cohen-Macaulay Simplicial ComplexesMay 13 2019We define and study the notion of a minimal Cohen-Macaulay simplicial complex. We prove that any Cohen-Macaulay complex is shelled over a minimal one in our sense, and we give sufficient conditions for a complex to be minimal Cohen-Macaulay. We show that ... More
Componentwise linearity of projective varieties with almost maximal degreeMay 13 2019The degree of a projective subscheme has an upper bound in term of the codimension and the reduction number. If a projective variety has an almost maximal degree, that is, the degree equals to the upper bound minus one, then its Betti table has been described ... More
Reducing invariants and total reflexivityMay 12 2019Motivated by a recent result of Yoshino, and the work of Bergh on reducible complexity, we introduce reducing versions of invariants of finitely generated modules over commutative Noetherian local rings. Our main result considers modules which have finite ... More
Standard Conjecture D for matrix factorizationsMay 12 2019We prove the non-commutative analogue of Grothendieck's Standard Conjecture D for the dg-category of matrix factorizations of an isolated hypersurface singularity in characteristic 0. Along the way, we show the Euler pairing for such dg-categories of ... More
A little more on the zero-divisor graph and the annihilating-ideal graph of a reduced ringMay 11 2019We have tried to translate some graph properties of AG(R) and Gamma(R) to the topological properties of Zariski topology. We prove that Rad(Gamma(R)) and Rad(AG(R)) are equal and they are equal to 3, if and only if the zero ideal of R is an anti fixed-place ... More
Seshadri Constants and Fujita's Conjecture via Positive Characteristic MethodsMay 09 2019In 1988, Fujita conjectured that there is an effective and uniform way to turn an ample line bundle on a smooth projective variety into a globally generated or very ample line bundle. We study Fujita's conjecture using Seshadri constants, which were first ... More
On the canonical ideal of the Ehrhart ring of the chain polytope of a posetMay 09 2019Let P be a poset, O(P) the order polytope of P and C(P) the chain polytope of P. In this paper, we study the canonical ideal of the Ehrhart ring K[C(P)] of C(P) over a field K and characterize the level (resp. anticanonical level) property of K[C(P)] ... More
Multiplicativity and nonrealizable equivariant chain complexesMay 08 2019Let $G$ be a finite $p$-group and $\mathbb{F}$ a field of characteristic $p$. We filter the cochain complex of a free $G$-space with coefficients in $\mathbb{F}$ by powers of the augmentation ideal of $\mathbb{F} G$. We show that the cup product induces ... More
The ascent-descent property for $2$-term silting complexesMay 08 2019We will prove that over commutative rings the silting property of $2$-term complexes induced by morphisms between projective modules is preserved and reflected by faithfully flat extensions.
Induced matchings in strongly biconvex graphs and some algebraic applicationsMay 07 2019In this paper, motivated by a question posed in \cite{AH}, we introduce strongly biconvex graphs as a subclass of weakly chordal and bipartite graphs. We give a linear time algorithm to find an induced matching for such graphs and we prove that this algorithm ... More
Extending valuations to the field of rational functions using pseudo-monotone sequencesMay 07 2019For a valuation domain $V$ of rank one and quotient field $K$, Ostrowski introduced in 1935 the notion of pseudo-convergent sequence and proved his Fundalmentalsatz, which describes all the possible rank one extensions of $V$ to $K(X)$. In this paper, ... More
Minimal set of binomial generators for certain Veronese 3-fold projectionsMay 07 2019The goal of this paper is to explicitly describe a minimal binomial generating set of a class of lattice ideals, namely the ideal of certain Veronese $3$-fold projections. More precisely, for any integer $d\ge 4$ and any $d$-th root $e$ of 1 we denote ... More
The (ir)regularity of Tor and ExtMay 07 2019We investigate the asymptotic behaviour of Castelnuovo-Mumford regularity of Ext and Tor, with respect to the homological degree, over complete intersection rings. We derive from a theorem of Gulliksen a linearity result for the regularity of Ext modules ... More
Burch ideals and Burch ringsMay 07 2019May 09 2019We introduce the notion of Burch ideals and Burch rings. They are easy to define, and can be viewed as generalization of many well-known concepts, for example integrally closed ideals of finite colength and Cohen--Macaulay rings of minimal multiplicity. ... More
Burch ideals and Burch ringsMay 07 2019We introduce the notion of Burch ideals and Burch rings. They are easy to define, and can be viewed as generalization of many well-known concepts, for example integrally closed ideals of finite colength and Cohen--Macaulay rings of minimal multiplicity. ... More
Burch ideals and Burch ringsMay 07 2019Jun 13 2019We introduce the notion of Burch ideals and Burch rings. They are easy to define, and can be viewed as generalization of many well-known concepts, for example integrally closed ideals of finite colength and Cohen--Macaulay rings of minimal multiplicity. ... More
Extensions of a valuation from $K$ to $K[x]$May 06 2019In this paper we give an introduction on how one can extend a valuation from a field $K$ to the polynomial ring $K[x]$ in one variable over $K$. This follows a similar line as the one presented by the author in his talk at ALaNT 5. We will discuss the ... More
Matching numbers and the regularity of the Rees algebra of an edge idealMay 06 2019The regularity of the Rees ring of the edge ideal of a finite simple graph is studied. We show that the matching number is a lower and matching number~$+1$ is an upper bound of the regularity, if the Rees algebra is normal. In general the induced matching ... More
Supertropical Monoids II: Lifts, Transmissions, and EqualizersMay 06 2019The category $\operatorname{STROP}$ of commutative semirings, whose morphisms are transmissions, is a full and reflective subcategory of the category $\operatorname{STROP}_m$ of supertropical monoids. Equivalence relations on supertropical monoids are ... More
Corps valués locauxMay 06 2019Many active mathematical research topics nowadays include the concepts of valued fields and local fields, especially the local field of p-adic numbers Qp and the field of formal Laurent series F((X)). Local fields are a notion situated in the boundary ... More
The Structure of Ulrich ideals in hypersurfacesMay 06 2019This paper studies Ulrich ideals in hypersurface rings. A characterization of Ulrich ideals is given. Using the characterization, we construct a minimal free resolution of an Ulrich ideal concretely. We also explore Ulrich ideals in a hypersurface ring ... More
Totally reflexive modules over connected sums with m^3=0May 06 2019We give a criterion for rings with $\m^3=0$ which are obtained as connected sums of two other rings to have non-trivial totally acyclic modules.
A note on Flenner's extension theoremMay 06 2019We show that any $p$-form on the smooth locus of a normal complex space extends to a resolution of singularities, possibly with logarithmic poles, as long as $p \le \mathrm{codim}_X (X_{\mathrm{sg}}) - 2$, where $c$ is the codimension of the singular ... More
Differential forms on log canonical spaces in positive characteristicMay 06 2019Given a logarithmic $1$-form on the snc locus of a log canonical surface pair $(X, D)$ over a perfect field of characteristic $p \ge 7$, we show that it extends with at worst logarithmic poles to any resolution of singularities. We also prove the analogous ... More
On the behavior of modules of $m$-integrable derivations in the sense of Hasse-Schmidt under base changeMay 05 2019We study the behavior of modules of $m$-integrable derivations of a commutative finitely generated algebra in the sense of Hasse-Schmidt under base change. We focus on the case of separable ring extensions over a field of positive characteristic and on ... More
Mixed multiplicities of Divisorial FiltrationsMay 04 2019Suppose 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 ... More
Positivity of Mixed Multiplicities of FiltrationsMay 04 2019The theory of mixed multiplicities of filtrations by $m$-primary ideals in a ring is introduced in a recent paper by Cutkosky, Sarkar and Srinivasan. In this paper, we consider the positivity of mixed multiplicities of filtrations. We show that the mixed ... More
Betti numbers of monomial ideals in four variablesMay 03 2019We express the multigraded Betti numbers of monomial ideals in 4 variables in terms of the multigraded Betti numbers of 66 squarefree monomial ideals, also in 4 variables. We use this class of 66 ideals to prove that monomial resolutions in 4 variables ... More
Solving Linear Systems over Idempotent Semifields through $LU$-factorizationApr 30 2019In this paper, we introduce and analyze a generalized $LU$-factorization method for square matrices over idempotent semifields. We use this $LU$-factorization to propose a technique for solving linear systems of equations as an extension of similar techniques ... More
Solving Linear Systems over Tropical Semirings through Normalization Method and its ApplicationsApr 30 2019In this paper, we introduce and analyze a normalization method for solving a system of linear equations over tropical semirings. We use a normalization method to construct an associated normalized matrix, which gives a technique for solving the system. ... More
On the Maximal Solution of A Linear System over Tropical SemiringsApr 30 2019In this paper, we present methods for solving a system of linear equations, $ AX=b $, over tropical semirings. To this end, if possible, we first reduce the order of the system through some row-column analysis, and obtain a new system with fewer equations ... More