41 results on '"Mathematics - Commutative Algebra"'
Search Results
2. Use DG-methods to build a matrix factorization
- Author
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Kustin, Andrew R.
- Subjects
Mathematics::Commutative Algebra ,FOS: Mathematics ,High Energy Physics::Experiment ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,13D02, 16E45 - Abstract
Let 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 resolution N of Pbar/K Pbar by free Pbar-modules. The resolution N is built from a Differential Graded Algebra resolution of P/(K:f) by free P-modules, together with one homotopy map. In particular, we give an explicit form for the matrix factorization which is the infinite tail of the resolution N.
- Published
- 2022
3. The quantitative behavior of asymptotic syzygies for Hirzebruch surfaces
- Author
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Bruce, Juliette
- Subjects
Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,Mathematics::Commutative Algebra ,FOS: Mathematics ,Computer Science::Symbolic Computation ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,Algebraic Geometry (math.AG) - Abstract
The goal of this note is to quantitatively study the behavior of asymptotic syzygies for certain toric surfaces, including Hirzebruch surfaces. In particular, we show that the asymptotic linear syzygies of Hirzebruch surfaces embedded by $\mathcal{O}(d,2)$ conform to Ein, Erman, and Lazarsfeld's normality heuristic. We also show that the higher degree asymptotic syzygies are not asymptotically normally distributed., Comment: 6 pages. Corrected minor typos
- Published
- 2022
4. Structure theory for a class of grade 3 homogeneous ideals defining type 2 compressed rings
- Author
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VandeBogert, Keller
- Subjects
Mathematics::Commutative Algebra ,FOS: Mathematics ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) - Abstract
Let $R=k[x,y,z]$ be a standard graded $3$-variable polynomial ring, where $k$ denotes any field. We study grade $3$ homogeneous ideals $I \subseteq R$ defining compressed rings with socle $k(-s) \oplus k(-2s+1)$, where $s \geq3$ is some integer. We prove that all such ideals are obtained by a trimming process introduced by Christensen, Veliche, and Weyman. We also construct a general resolution for all such ideals which is minimal in sufficiently generic cases. Using this resolution, we can give bounds on the minimal number of generators $\mu(I)$ of $I$ depending only on $s$; moreover, we show these bounds are sharp by constructing ideals attaining the upper and lower bounds for all $s\geq 3$. Finally, we study the Tor-algebra structure of $R/I$. It is shown that these rings have Tor algebra class $G(r)$ for $s \leq r \leq 2s-1$. Furthermore, we produce ideals $I$ for all $s \geq 3$ and all $r$ with $s \leq r \leq 2s-1$ such that $\textrm{Soc} (R/I ) = k(-s) \oplus k(-2s+1)$ and $R/I$ has Tor-algebra class $G(r)$, partially answering a question of realizability posed by Avramov., Comment: 24 pages
- Published
- 2022
5. Totally Reflexive Modules Over Connected Sums with m3=0
- Author
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Vraciu, Adela
- Subjects
Mathematics::Commutative Algebra ,13D02 ,FOS: Mathematics ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) - Abstract
We 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.
- Published
- 2021
6. On the containment problem for fat points
- Author
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Jafarloo, Iman Bahmani and Zito, Giuseppe
- Subjects
Mathematics - Algebraic Geometry ,Mathematics::Commutative Algebra ,FOS: Mathematics ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,Algebraic Geometry (math.AG) - Abstract
Given an ideal $I$, the containment problem is concerned about finding the values $m$ and $n$ such that the $m$-th symbolic power of $I$ is contained in its $n$-th ordinary power. In this paper we consider this problem focusing on two classes of ideals of fat points. In particular we study the ideal of $n$ points on a line in $\mathbb{P}^N$ and the ideal of three nonlinear points in $\mathbb{P}^N$ for $N\geq2$.
- Published
- 2021
7. The Lefschetz question for ideals generated by powers of linear forms in few variables
- Author
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Migliore, Juan and Nagel, Uwe
- Subjects
Mathematics - Algebraic Geometry ,Mathematics::Commutative Algebra ,13D40, 13E10, 14N05 ,FOS: Mathematics ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,Algebraic Geometry (math.AG) - Abstract
The Lefschetz question asks if multiplication by a power of a general linear form, $L$, on a graded algebra has maximal rank (in every degree). We consider a quotient by an ideal that is generated by powers of linear forms. Then the Lefschetz question is, for example, related to the problem whether a set of fat points imposes the expected number of conditions on a linear system of hypersurfaces of fixed degree. Our starting point is a result that relates Lefschetz properties in different rings. It suggests to use induction on the number of variables, $n$. If $n = 3$, then it is known that multiplication by $L$ always has maximal rank. We show that the same is true for multiplication by $L^2$ if all linear forms are general. Furthermore, we give a complete description of when multiplication by $L^3$ has maximal rank (and its failure when it does not). As a consequence, for such ideals that contain a quadratic or cubic generator, we establish results on the so-called Strong Lefschetz Property for ideals in $n=3$ variables, and the Weak Lefschetz Property for ideals in $n=4$ variables., Comment: 24 pages. Substantial revision from first version, with new results
- Published
- 2021
8. Regularity of the vanishing ideal over a parallel composition of paths
- Author
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Antonio Macchia, Jorge Alexandre Barbosa Neves, Maria Vaz Pinto, and Rafael H. Villarreal
- Subjects
13F20 ,Polynomial ring ,14G15 ,MathematicsofComputing_GENERAL ,Structure (category theory) ,Parameterized complexity ,0102 computer and information sciences ,parallel composition of paths ,Commutative Algebra (math.AC) ,binomial ideals ,01 natural sciences ,Castelnuovo–Mumford regularity ,Combinatorics ,Mathematics - Algebraic Geometry ,11T55 ,FOS: Mathematics ,0101 mathematics ,Invariant (mathematics) ,Algebraic Geometry (math.AG) ,Quotient ,Mathematics ,Mathematics::Commutative Algebra ,010102 general mathematics ,13F20 (primary), 14G15, 11T55 ,Composition (combinatorics) ,Mathematics - Commutative Algebra ,010201 computation theory & mathematics ,Bipartite graph ,MathematicsofComputing_DISCRETEMATHEMATICS - Abstract
Let [math] be a graph obtained by taking [math] paths and identifying all first vertices and identifying all last vertices. We compute the Castelnuovo–Mumford regularity of the quotient [math] , where [math] is the polynomial ring on the edges of [math] and [math] is the vanishing ideal of the projective toric subset parameterized by [math] . This invariant is known for several special families of graphs such as trees, cycles, complete graphs and complete bipartite graphs. For bipartite graphs, it is also known that the computation of the regularity can be reduced to the [math] -connected case. Thus, we focused on the first case of a bipartite graph where the regularity was unknown. We also prove new inequalities relating the Castelnuovo–Mumford regularity of [math] with the combinatorial structure of [math] , for a general graph.
- Published
- 2020
9. Trimming a Gorenstein ideal
- Author
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Lars Winther Christensen, Jerzy Weyman, and Oana Veliche
- Subjects
Koszul homology ,Pure mathematics ,Gorenstein ring ,Dimension (graph theory) ,13C99, 13H10 ,0102 computer and information sciences ,Homology (mathematics) ,Commutative Algebra (math.AC) ,01 natural sciences ,symbols.namesake ,FOS: Mathematics ,Ideal (ring theory) ,0101 mathematics ,Poincaré duality ,Mathematics ,13C99 ,Graded vector space ,Mathematics::Commutative Algebra ,13H10 ,Mathematics::Rings and Algebras ,010102 general mathematics ,Poincare duality algebra ,Regular local ring ,Mathematics - Commutative Algebra ,16. Peace & justice ,010201 computation theory & mathematics ,symbols ,Quotient ring - Abstract
Let Q be a regular local ring of dimension 3. We show how to trim a Gorenstein ideal in Q to obtain an ideal that defines a quotient ring that is close to Gorenstein in the sense that its Koszul homology algebra is a Poincare duality algebra P padded with a non-zero graded vector space on which P_{\ge 1} acts trivially. We explicitly construct an infinite family of such rings., Corrected statement of Lemma 2.3 and updated proof of Theorem 2.4. Final version, to appear in J. Commut. Algebra; 11 pp
- Published
- 2019
10. Toric representations of algebras defined by certain nonsimple polyominoes
- Author
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Akihiro Shikama
- Subjects
Polyomino ,13C05 ,Polyominoes ,0102 computer and information sciences ,toric rings ,Computer Science::Computational Geometry ,Commutative Algebra (math.AC) ,01 natural sciences ,Combinatorics ,Computer Science::Discrete Mathematics ,FOS: Mathematics ,05E40 ,Rectangle ,0101 mathematics ,Mathematics ,Ring (mathematics) ,Mathematics::Combinatorics ,Mathematics::Commutative Algebra ,010102 general mathematics ,Representation (systemics) ,Regular polygon ,toric ideals ,Mathematics - Commutative Algebra ,13C05, 05E40 ,010201 computation theory & mathematics ,Computer Science::Formal Languages and Automata Theory - Abstract
In this paper, we give a toric representation of the associated ring of a polyomino which is obtained by removing a convex polyomino from its ambient rectangle.
- Published
- 2018
11. Topics on sequentially Cohen-Macaulay modules
- Author
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Tran Thi Phuong, Naoki Taniguchi, Tran Nguyen An, and Nguyen Thi Dung
- Subjects
Discrete mathematics ,Property (philosophy) ,Mathematics::Commutative Algebra ,13H10 ,010102 general mathematics ,13E05 ,0102 computer and information sciences ,Extension (predicate logic) ,13E05, 13H10 ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,01 natural sciences ,localization ,sequentially Cohen-Macaulay module ,010201 computation theory & mathematics ,FOS: Mathematics ,Dimension filtration ,0101 mathematics ,Mathematics - Abstract
In this paper, we study the two different topics related to sequentially Cohen-Macaulay modules. The questions are when the sequentially Cohen-Macaulay property preserve the localization and the module-finite extension of rings., 7 pages. arXiv admin note: substantial text overlap with arXiv:1406.3423
- Published
- 2018
12. Valuative and geometric characterizations of Cox sheaves
- Author
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Benjamin Bechtold
- Subjects
13A02 ,Algebraic properties ,Pure mathematics ,Strategy and Management ,media_common.quotation_subject ,Algebraic geometry ,Characterization (mathematics) ,Commutative Algebra (math.AC) ,01 natural sciences ,Industrial and Manufacturing Engineering ,Mathematics - Algebraic Geometry ,0103 physical sciences ,FOS: Mathematics ,Cox rings ,0101 mathematics ,Algebraic Geometry (math.AG) ,Normality ,Quotient ,Mathematics ,media_common ,Mathematics::Commutative Algebra ,13A18 ,Mechanical Engineering ,Mathematics::Rings and Algebras ,010102 general mathematics ,Metals and Alloys ,graded schemes ,14A20 ,Mathematics - Commutative Algebra ,010307 mathematical physics ,Krull schemes - Abstract
We give an intrinsic characterization of Cox sheaves on Krull schemes in terms of their valuative algebraic properties. We also provide a geometric characterization of their graded relative spectra in terms of good quotients of graded schemes, extending the work of Arzhantsev, Derenthal, Hausen and Laface on relative spectra of Cox sheaves on normal varieties. Moreover, we obtain an irredundant characterization of Cox rings which in turn produces a normality criterion for certain graded rings., Comment: 25 pages
- Published
- 2018
13. Systems of parameters and the Cohen-Macaulay property
- Author
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Katharine Shultis
- Subjects
Noetherian ,Cohen-Macaulay property ,13H10, 13C05 ,Mathematics::Commutative Algebra ,13H10 ,13C05 ,010102 general mathematics ,Structure (category theory) ,Local ring ,Free module ,0102 computer and information sciences ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,01 natural sciences ,Combinatorics ,System of parameters ,010201 computation theory & mathematics ,FOS: Mathematics ,Rank (graph theory) ,Homomorphism ,system of parameters ,0101 mathematics ,Mathematics::Representation Theory ,Commutative property ,Mathematics - Abstract
Let $R$ be a commutative, Noetherian, local ring and $M$ a finitely generated $R$-module. Consider the module of homomorphisms $Hom _R(R/\mathfrak{a} ,M/\mathfrak{b} M)$ where $\mathfrak{b} \subseteq \mathfrak{a} $ are parameter ideals of $M$. When $M=R$ and $R$ is Cohen-Macaulay, Rees showed that this module of homomorphisms is isomorphic to $R/\mathfrak{a} $, and in particular, a free module over $R/\mathfrak{a} $ of rank one. In this work, we study the structure of such modules of homomorphisms for a not necessarily Cohen-Macaulay $R$-module $M$.
- Published
- 2018
14. Tensor product of dualizing complexes over a field
- Author
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Liran Shaul
- Subjects
Noetherian ,Pure mathematics ,0211 other engineering and technologies ,Field (mathematics) ,02 engineering and technology ,010103 numerical & computational mathematics ,Commutative Algebra (math.AC) ,adic completion ,01 natural sciences ,Mathematics - Algebraic Geometry ,FOS: Mathematics ,Dualizing complex ,0101 mathematics ,Algebraic Geometry (math.AG) ,Mathematics ,Discrete mathematics ,Mathematics::Commutative Algebra ,13H10 ,Mathematics::Rings and Algebras ,13J10 ,010102 general mathematics ,021107 urban & regional planning ,13D09, 13H10, 18G20, 13J10 ,Mathematics - Commutative Algebra ,18G20 ,Tensor product ,13D09 ,Krull dimension - Abstract
Let $k$ be a field, and let $X,Y$ be two locally noetherian $k$-schemes (respectively $k$-formal schemes) with dualizing complexes $R_X$ and $R_Y$ respectively. We show that $R_X \boxtimes_{k} R_Y$ (respectively its derived completion) is a dualizing complex over $X\times_{k} Y$ if and only if $X\times_{k} Y$ is locally noetherian of finite Krull dimension., 13 pages, final version, to appear in the Journal of Commutative Algebra
- Published
- 2018
15. Powers of edge ideals of regularity three bipartite graphs
- Author
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Arindam Banerjee and Ali Alilooee
- Subjects
Ideal (set theory) ,13F55 ,010102 general mathematics ,0102 computer and information sciences ,Edge (geometry) ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,even-connected vertices ,01 natural sciences ,bipartite complement ,Combinatorics ,Castelnuovo–Mumford regularity ,010201 computation theory & mathematics ,FOS: Mathematics ,Bipartite graph ,0101 mathematics ,Castelnuovo-Mumford regularity ,Bipartite graphs ,05C10 ,Mathematics - Abstract
In this paper, we prove that, if $I(G)$ is the edge ideal of a connected bipartite graph with regularity 3, then, for all $s\geq 2$, the regularity of $I(G)^s$ is exactly $2s+1$.
- Published
- 2017
16. The realization problem for delta sets of numerical semigroups
- Author
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Stefan Colton and Nathan Kaplan
- Subjects
010103 numerical & computational mathematics ,20M14 ,Commutative Algebra (math.AC) ,20M13 ,01 natural sciences ,Set (abstract data type) ,Combinatorics ,Factorization ,20M14, 20M13, 11B75 ,Numerical semigroup ,FOS: Mathematics ,Mathematics - Combinatorics ,Delta set ,0101 mathematics ,Invariant (mathematics) ,Finite set ,Mathematics ,11B75 ,010102 general mathematics ,Mathematics - Commutative Algebra ,delta set ,Embedding ,non-unique factorization ,Combinatorics (math.CO) ,Realization (systems) ,factorization theory - Abstract
The delta set of a numerical semigroup $S$, denoted $\Delta(S)$, is a factorization invariant that measures the complexity of the sets of lengths of elements in $S$. We study the following problem: Which finite sets occur as the delta set of a numerical semigroup $S$? It is known that $\min \Delta(S) = \gcd \Delta(S)$ is a necessary condition. For any two-element set $\{d,td\}$ we produce a semigroup $S$ with this delta set. We then show that for $t\ge 2$, the set $\{d,td\}$ occurs as the delta set of some numerical semigroup of embedding dimension three if and only if $t=2$., Comment: 13 pages
- Published
- 2017
17. Lattice-ordered abelian groups finitely generated as semirings
- Author
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Vítězslav Kala
- Subjects
finitely generated ,Lattice (group) ,Structure (category theory) ,MV-algebra ,semiring ,01 natural sciences ,Semiring ,Combinatorics ,Lattice-ordered abelian group ,06D35 ,0101 mathematics ,Abelian group ,Commutative property ,16Y60 ,Mathematics ,parasemifield ,Group (mathematics) ,010102 general mathematics ,52B20 ,Mathematics - Commutative Algebra ,010101 applied mathematics ,Primary 06F20, 12K10, Secondary 06D35, 16Y60, 52B20 ,Idempotence ,12K10 ,06F20 ,order-unit ,Mathematics - Group Theory - Abstract
A lattice-ordered group (an $\ell$-group) $G(\oplus, \vee, \wedge)$ can be naturally viewed as a semiring $G(\vee,\oplus)$. We give a full classification of (abelian) $\ell$-groups which are finitely generated as semirings, by first showing that each such $\ell$-group has an order-unit so that we can use the results of Busaniche, Cabrer and Mundici [8]. Then we carefully analyze their construction in our setting to obtain the classification in terms of certain $\ell$-groups associated to rooted trees (Theorem 4.1). This classification result has a number of important applications: for example it implies a classification of finitely generated ideal-simple (commutative) semirings $S(+, \cdot)$ with idempotent addition and provides important information concerning the structure of general finitely generated ideal-simple (commutative) semirings, useful in obtaining further progress towards Conjecture 1.1 discussed in [2], [15]., Comment: 16 pages; revised and slightly extended version
- Published
- 2017
18. On the functoriality of marked families
- Author
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Paolo Lella and Margherita Roggero
- Subjects
Pure mathematics ,Borel-fixed ideal ,Hilbert scheme ,marked family ,open subfunctor ,Structure (category theory) ,010103 numerical & computational mathematics ,Type (model theory) ,Commutative Algebra (math.AC) ,01 natural sciences ,Mathematics - Algebraic Geometry ,14C05, 13P99 ,FOS: Mathematics ,0101 mathematics ,13P99 ,Computational algebra ,Algebraic Geometry (math.AG) ,Mathematics ,Ring (mathematics) ,Ideal (set theory) ,Mathematics::Commutative Algebra ,14C05 ,010102 general mathematics ,Mathematics - Commutative Algebra ,Scheme (mathematics) - Abstract
The application of methods of computational algebra has recently introduced new tools for the study of Hilbert schemes. The key idea is to define flat families of ideals endowed with a scheme structure whose defining equations can be determined by algorithmic procedures. For this reason, several authors developed new methods, based on the combinatorial properties of Borel-fixed ideals, that allow to associate to each ideal $J$ of this type a scheme $\mathbf{Mf}_{J}$, called $J$-marked scheme. In this paper we provide a solid functorial foundation to marked schemes and show that the algorithmic procedures introduced in previous papers do not depend on the ring of coefficients. We prove that for all strongly stable ideals $J$, the marked schemes $\mathbf{Mf}_{J}$ can be embedded in a Hilbert scheme as locally closed subschemes, and that they are open under suitable conditions on $J$. Finally, we generalize Lederer's result about Gr\"obner strata of zero-dimensional ideals, proving that Gr\"obner strata of any ideals are locally closed subschemes of Hilbert schemes., Comment: final version, accepted for publication on Journal of Commutative Algebra
- Published
- 2016
19. $s$-Hankel hypermatrices and $2\times 2$ determinantal ideals
- Author
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Alessio Sammartano
- Subjects
Pure mathematics ,Ideal (set theory) ,Mathematics::Commutative Algebra ,primary decomposition ,Structure (category theory) ,13C40 ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,Primary decomposition ,Mathematics - Algebraic Geometry ,hypermatrix ,switchable set ,FOS: Mathematics ,13C40, 13P10 ,Hankel ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,binomial ideal ,13P10 ,Algebraic Geometry (math.AG) ,$2\times 2$ minor ,Mathematics - Abstract
We introduce the concept of an $s$-Hankel hypermatrix, which generalizes both Hankel matrices and generic hypermatrices. We study two determinantal ideals associated to an $s$-Hankel hypermatrix: the ideal $\I {s}{t}$ generated by certain $2 \times 2$ slice minors, and the ideal $\It {s}{t}$ generated by certain $2 \times 2$ generalized minors. We describe the structure of these two ideals, with particular attention to the primary decomposition of $\I {s}{t}$, and provide the explicit list of minimal primes for large values of $s$. Finally we give some geometrical interpretations and generalize a theorem of Watanabe.
- Published
- 2016
20. The degree of the algebra of covariants of a binary form
- Author
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Leonid Bedratyuk and Nadia Ilash
- Subjects
Integral representation ,Degree (graph theory) ,Binary number ,algebra of covariants of binary form ,13A50 ,Mathematics - Rings and Algebras ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,algebra of invariants of binary form ,Poincaré series ,degree of algebra ,Algebra ,Mathematics - Algebraic Geometry ,Binary form ,Rings and Algebras (math.RA) ,FOS: Mathematics ,13N15, 13A50, 13F20 ,Classical invariant theory ,Algebra over a field ,Algebraic Geometry (math.AG) ,13N15 ,Mathematics - Abstract
We calculate the degree of the algebra of covariants $\mathcal{C}_d$ for binary $d$-form. Also, for the degree we obtain its integral representation and asymptotic behavior., 9 pages
- Published
- 2015
21. Monomial valuations, cusp singularities, and continued fractions
- Author
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Molly Logue, David J. Bruce, and Robert Walker
- Subjects
13F30 ,Cusp (singularity) ,Computer Science::Computer Science and Game Theory ,Monomial ,Pure mathematics ,resolution of singularities ,Mathematics::Commutative Algebra ,Coprime integers ,14E15 ,Plane curve ,Zero (complex analysis) ,continued fractions ,Resolution of singularities ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,Mathematics - Algebraic Geometry ,Monomial valuations ,FOS: Mathematics ,16W60 ,Fraction (mathematics) ,Algebraic Geometry (math.AG) ,Mathematics ,Real number - Abstract
This paper explores the relationship between real valued monomial valuations on $k(x,y)$, the resolution of cusp singularities, and continued fractions. It is shown that up to equivalence there is a one to one correspondence between real valued monomial valuations on $k(x,y)$ and continued fraction expansions of real numbers between zero and one. This relationship with continued fractions is then used to provide a characterization of the valuation rings for real valued monomial valuations on $k(x,y)$. In the case when the monomial valuation is equivalent to an integral monomial valuation, we exhibit explicit generators of the valuation rings. Finally, we demonstrate that if $\nu$ is a monomial valuation such that $\nu(x)=a$ and $\nu(y)=b$, where $a$ and $b$ are relatively prime positive integers larger than one, then $\nu$ governs a resolution of the singularities of the plane curve $x^{b}=y^{a}$ in a way we make explicit. Further, we provide an exact bound on the number of blow ups needed to resolve singularities in terms of the continued fraction of $a/b$, Comment: 20 pages, Corrected Author Information
- Published
- 2015
22. A Baer-Kaplansky theorem for modules over principal ideal domains
- Author
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Simion Breaz
- Subjects
Discrete mathematics ,Pure mathematics ,Endomorphism ,Endomorphism ring ,20K30 ,Cancellation property ,Dedekind domain ,Principal ideal domain ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,Principal ideal ,16D70 ,principal ideal domain ,FOS: Mathematics ,13G05 ,16S50 ,cancellation property ,Mathematics - Abstract
We will prove that if $G$ and $H$ are modules over a principal ideal domain $R$ such that the endomorphism rings $\mathrm{End}_R(R\oplus G)$ and $\mathrm{End}_R(R\oplus H)$ are isomorphic then $G\cong H$. Conversely, if $R$ is a Dedekind domain such that two $R$-modules $G$ and $H$ are isomorphic whenever the rings $\mathrm{End}_R(R\oplus G)$ and $\mathrm{End}_R(R\oplus H)$ are isomorphic then $R$ is a PID., preprint version; the final version is accepted by JCA
- Published
- 2015
23. The strong Lefschetz property in codimension two
- Author
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David Cook
- Subjects
13A35 ,13E10 ,Monomial ,Pure mathematics ,Property (philosophy) ,Mathematics::Commutative Algebra ,Zero (complex analysis) ,positive characteristic ,Field (mathematics) ,Monomial ideal ,Codimension ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,13A35, 13E10 ,FOS: Mathematics ,Ideal (ring theory) ,lexsegment ideals ,Quotient ,Strong Lefschetz property ,Mathematics - Abstract
Every artinian quotient of $K[x,y]$ has the strong Lefschetz property if $K$ is a field of characteristic zero or is an infinite field whose characteristic is greater than the regularity of the quotient. We improve this bound in the case of monomial ideals. Using this we classify when both bounds are sharp. Moreover, we prove that the artinian quotient of a monomial ideal in $K[x,y]$ always has the strong Lefschetz property, regardless of the characteristic of the field, exactly when the ideal is lexsegment. As a consequence we describe a family of non-monomial complete intersections that always have the strong Lefschetz property., Comment: 18 pages, 1 figure; v2: Updated history and references
- Published
- 2014
24. Derived supersymmetries of determinantal varieties
- Author
-
Steven V Sam
- Subjects
Statement (computer science) ,Pure mathematics ,13D02 ,Mathematics::Commutative Algebra ,Koszul duality ,17B10, 13D02, 15A69 ,Mathematics::Rings and Algebras ,Structure (category theory) ,17B10 ,Lie superalgebra ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,15A69 ,Irreducible representation ,Lie algebra ,FOS: Mathematics ,Representation Theory (math.RT) ,Mathematics::Representation Theory ,Realization (systems) ,Mathematics - Representation Theory ,Mathematics - Abstract
We show that the linear strands of the Tor of determinantal varieties in spaces of symmetric and skew-symmetric matrices are irreducible representations for the periplectic (strange) Lie superalgebra. The structure of these linear strands is explicitly known, so this gives an explicit realization of some representations of the periplectic Lie superalgebra. This complements results of Pragacz and Weyman, who showed an analogous statement for the generic determinantal varieties and the general linear Lie superalgebra. We also give a simpler proof of their result. Via Koszul duality, this is an odd analogue of the fact that the coordinate rings of these determinantal varieties are irreducible representations for a certain classical Lie algebra., 17 pages
- Published
- 2014
25. A remark on a conjecture of Derksen
- Author
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Andrew Snowden
- Subjects
Graded vector space ,Finite group ,Conjecture ,Mathematics::Commutative Algebra ,Graded ring ,Cyclic group ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,Combinatorics ,Conjugacy class ,Symmetric group ,Irreducible representation ,FOS: Mathematics ,Representation Theory (math.RT) ,Mathematics - Representation Theory ,Mathematics - Abstract
Let V be a complex representation of a finite group G of order g. Derksen conjectured that the pth syzygies of the invariant ring Sym(V)^G are generated in degrees at most (p+1)g. We point out that a simple application of the theory of twisted commutative algebras -- using an idea due to Weyl -- gives the bound pg^3., Comment: 2 pages
- Published
- 2014
26. Examples of non-Noetherian domains inside power series rings
- Author
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Christel Rotthaus, William Heinzer, and Sylvia Wiegand
- Subjects
Noetherian ,13B35 ,Pure mathematics ,Noetherian and non-Noetherian integral domains ,13A15 ,13A15, 13B35, 13J10 ,Mathematics::Commutative Algebra ,Prime ideal ,Polynomial ring ,13J10 ,Unique factorization domain ,Power series ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,Prime (order theory) ,Integral domain ,FOS: Mathematics ,Maximal ideal ,Quotient ring ,Mathematics - Abstract
Let R* be an ideal-adic completion of a Noetherian integral domain R and let L be a subfield of the total quotient ring of R* such that L contains R. Let A denote the intersection of L with R*. The integral domain A sometimes inherits nice properties from R* such as the Noetherian property. For certain fields L it is possible to approximate A using a localzation B of a nested union of polynomial rings over R associated to A; if B is Noetherian, then B = A. If B is not Noetherian, we can sometimes identify the prime ideals of B that are not finitely generated. We have obtained in this way, for each positive integer s, a 3-dimensional local unique factorization domain B such that the maximal ideal of B is 2-generated, B has precisely s prime ideals of height 2, each prime ideal of B of height 2 is not finitely generated and all the other prime ideals of B are finitely generated. We examine the map Spec A to Spec B for this example. We also present a generalization of this example to dimension 4. We describe a 4-dimensional local non-Noetherian UFD B such that the maximal ideal of B is 3-generated, there exists precisely one prime ideal Q of B of height 3, the prime ideal Q is not finitely generated. We consider the question of whether Q is the only prime ideal of B that is not finitely generated, but have not answered this question., Comment: 32 pages to appear in JCA
- Published
- 2014
27. Rees algebras of diagonal ideals
- Author
-
Kuei-Nuan Lin
- Subjects
Symmetric algebra ,Rees algebra ,determinantal ring ,Pure mathematics ,Ideal (set theory) ,Mathematics::Commutative Algebra ,Diagonal ,symmetric algebra ,Epimorphism ,Mathematics - Commutative Algebra ,13C40 ,Commutative Algebra (math.AC) ,join variety ,14Q15 ,Mathematics - Algebraic Geometry ,Gröbner basis ,Kernel (algebra) ,FOS: Mathematics ,Isomorphism ,14M12 ,13P10 ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
There is a natural epimorphism from the symmetric algebra to the Rees algebra of an ideal. When this epimorphism is an isomorphism, we say that the ideal is of linear type. Given two determinantal rings over a field, we consider the diagonal ideal, the kernel of the multiplication map. We prove that the diagonal ideal is of linear type and recover the defining ideal of the Rees algebra in some special cases. The special fiber ring of the diagonal ideal is the homogeneous coordinate ring of the join variety., Comment: This work is based on author's Ph. D. thesis from Purdue University under the direction of Professor Bernd Ulrich
- Published
- 2013
28. Gorenstein Hilbert coefficients
- Author
-
Sabine El Khoury and Hema Srinivasan
- Subjects
Discrete mathematics ,Mathematics::Commutative Algebra ,FOS: Mathematics ,Algebra over a field ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,Mathematics - Abstract
We prove upper and lower bounds for all the coefficients in the Hilbert Polynomial of a graded Gorenstein algebra $S=R/I$ with a quasi-pure resolution over $R$. The bounds are in terms of the minimal and the maximal shifts in the resolution of $R$ . These bounds are analogous to the bounds for the multiplicity found in \cite{S} and are stronger than the bounds for the Cohen Macaulay algebras found in \cite{HZ}., Comment: 20 pages
- Published
- 2013
29. Triplets of pure free squarefree complexes
- Author
-
Gunnar Fløystad
- Subjects
pure complex ,Pure mathematics ,Rational number ,Functor ,Conjecture ,13F55 ,13D02 ,Mathematics::Commutative Algebra ,squarefree module ,Alexander duality ,Betti number ,Polynomial ring ,homology ,Square-free integer ,Homology (mathematics) ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,Pure resolution ,Mathematics::Category Theory ,FOS: Mathematics ,Betti numbers ,Mathematics - Abstract
On the category of bounded complexes of finitely generated free squarefree modules over the polynomial ring S, there is the standard duality functor D = Hom_S(-, omega_S) and the Alexander duality functor A. The composition AD is an endofunctor on this category, of order three up to translation. We consider complexes F of free squarefree modules such that both F, AD(F) and (AD)^2(F) are pure, when considered as singly graded complexes. We conjecture i) the existence of such triplets of complexes for given triplets of degree sequences, and ii) the uniqueness of their Betti numbers, up to scalar multiple. We show that this uniqueness follows from the existence, and we construct such triplets if two of them are linear., 25 pages, minor improvements
- Published
- 2013
30. New free divisors from old
- Author
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Aldo Conca and Ragnar-Olaf Buchweitz
- Subjects
Tangent bundle ,14H51 ,Pure mathematics ,discriminant ,Binomial (polynomial) ,Divisor ,Mathematics::Number Theory ,Divisor function ,Commutative Algebra (math.AC) ,01 natural sciences ,14J70 ,binomial ,Mathematics::Algebraic Geometry ,0103 physical sciences ,FOS: Mathematics ,Saito matrix ,Euler vector field ,0101 mathematics ,Mathematics ,Discrete mathematics ,Practical number ,14B05 ,010102 general mathematics ,Chain rule ,Mathematics - Commutative Algebra ,Table of divisors ,Free divisors ,32S25 ,010307 mathematical physics ,Free divisor ,Refactorable number ,14J17 - Abstract
We present several methods to construct or identify families of free divisors such as those annihilated by many Euler vector fields, including binomial free divisors, or divisors with triangular discriminant matrix. We show how to create families of quasihomogeneous free divisors through the chain rule or by extending them into the tangent bundle. We also discuss whether general divisors can be extended to free ones by adding components and show that adding a normal crossing divisor to a smooth one will not succeed.
- Published
- 2013
31. Bass numbers over local rings via stable cohomology
- Author
-
Luchezar L. Avramov and Srikanth B. Iyengar
- Subjects
Noetherian ,Pure mathematics ,stable cohomology ,13D02 ,Mathematics::Commutative Algebra ,010102 general mathematics ,Local ring ,0102 computer and information sciences ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,13D40 ,01 natural sciences ,Cohomology ,Bass (sound) ,13D07 (Primary) 13D02, 13D40 (Secondary) ,010201 computation theory & mathematics ,Residue field ,fiber products ,FOS: Mathematics ,Bass numbers ,13D07 ,Maximal ideal ,0101 mathematics ,Mathematics - Abstract
For any non-zero finite module M of finite projective dimension over a noetherian local ring R with maximal ideal m and residue field k, it is proved that the natural map Ext_R(k,M)-->Ext_R(k,M/mM) is non-zero when R is regular and is zero otherwise. A noteworthy aspect of the proof is the use of stable cohomology. Applications include computations of Bass series over certain local rings., 7 pages
- Published
- 2013
32. On the prime ideal structure of symbolic Rees algebras
- Author
-
Samir Bouchiba and Salah Kabbaj
- Subjects
Krull domain ,Noetherian ,13F20 ,Pure mathematics ,13B25 ,valuative dimension ,Prime ideal ,13E05 ,factorial domain ,Commutative Algebra (math.AC) ,Prime (order theory) ,Mathematics - Algebraic Geometry ,FOS: Mathematics ,fourteenth problem of Hilbert ,Algebraic Geometry (math.AG) ,Krull dimension ,Mathematics ,Mathematics::Commutative Algebra ,13C15 ,13F15 ,Jaffard domain ,13C15, 13F05, 13F15, 13E05, 13F20, 13G05, 13B25, 13B30 ,13B30 ,Mathematics - Commutative Algebra ,Domain (ring theory) ,associated graded ring ,Symbolic Rees algebra ,13F05 ,13G05 ,Maximal ideal ,Rees algebra ,subalgebra of an affine domain ,Dimension theory (algebra) - Abstract
This paper contributes to the study of the prime spectrum and dimension theory of symbolic Rees algebra over Noetherian domains. We first establish some general results on the prime ideal structure of subalgebras of affine domains, which actually arise, in the Noetherian context, as domains between a domain $A$ and $A[a^{-1}]$. We then examine closely the special context of symbolic Rees algebras (which yielded the first counter-example to the Zariski-Hilbert problem). One of the results states that if $A$ is a Noetherian domain and $p$ a maximal ideal of $A$, then the Rees algebra of $p$ inherits the Noetherian-like behavior of being a stably strong S-domain. We also investigate graded rings associated with symbolic Rees algebras of prime ideals $p$ such that $A_{p}$ is a rank-one DVR and close with an application related to Hochster's result on the coincidence of the ordinary and symbolic powers of a prime ideal., 12 pages
- Published
- 2012
33. Existence of totally reflexive modules via Gorenstein homomorphisms
- Author
-
Kristen A. Beck
- Subjects
Class (set theory) ,Pure mathematics ,13C13 ,Gorenstein homomorphism ,Mathematics::Commutative Algebra ,13D05 ,Local ring ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,totally reflexive module ,Totally acyclic complex ,Reflexivity ,FOS: Mathematics ,13D07 ,Homomorphism ,embedded deformation ,Mathematics - Abstract
We define, via Gorenstein homomorphisms, a class of local rings over which there exist non-trivial totally reflexive modules. We also provide a general construction of such rings, which indicates their abundance., Comment: Redundancy problem fixed. Most recent version to appear in the Journal of Commutative Algebra
- Published
- 2012
34. Trivariate monomial complete intersections and plane partitions
- Author
-
Xin Jin, Charles Chen, Alan Guo, and Gaku Liu
- Subjects
Monomial ,0102 computer and information sciences ,Commutative Algebra (math.AC) ,01 natural sciences ,Bijective proof ,Primary: 05E40, Secondary: 05E05, 05E18, 13E10, 15A15 ,Combinatorics ,Identity (mathematics) ,05E05 ,FOS: Mathematics ,Mathematics - Combinatorics ,05E40 ,0101 mathematics ,Mathematics ,13E10 ,15A15 ,010102 general mathematics ,Block matrix ,Mathematics - Commutative Algebra ,16. Peace & justice ,Linear subspace ,Schur polynomial ,Toeplitz matrix ,010201 computation theory & mathematics ,05E18 ,Combinatorics (math.CO) ,Smith normal form - Abstract
We consider the homogeneous components U_r of the map on R = k[x,y,z]/(x^A, y^B, z^C) that multiplies by x + y + z. We prove a relationship between the Smith normal forms of submatrices of an arbitrary Toeplitz matrix using Schur polynomials, and use this to give a relationship between Smith normal form entries of U_r. We also give a bijective proof of an identity proven by J. Li and F. Zanello equating the determinant of the middle homogeneous component U_r when (A, B, C) = (a + b, a + c, b + c) to the number of plane partitions in an a by b by c box. Finally, we prove that, for certain vector subspaces of R, similar identities hold relating determinants to symmetry classes of plane partitions, in particular classes 3, 6, and 8., Comment: 21 pages, 15 figures
- Published
- 2011
35. Planes of the form $b(X,Y)Z^n-a(X,Y)$ over a DVR
- Author
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Prosenjit Das and Amartya Kumar Dutta
- Subjects
13F30 ,Primary 13F20, 14R25. Secondary: 13E15, 13B22, 13A30, 13B10 ,Discrete mathematics ,13F20 ,Polynomial ,13B25 ,Mathematics::Commutative Algebra ,Epimorphism ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,Valuation ring ,Discrete valuation ring ,epimorphism theorems ,residual variable ,Integer ,Residue field ,Domain (ring theory) ,FOS: Mathematics ,Discrete valuation ,14R10 ,Mathematics - Abstract
In this paper we extend an epimorphism theorem of D. Wright to the case of discrete valuation rings. We will show that if $(R, t)$ is a discrete valuation ring, $n \ge 2$ is an integer not divisible by the characteristic of the residue field $R/tR$, and $g \in R[X, Y, Z]$ is a polynomial of the form $g = b(X,Y)Z^n - a(X,Y)$ such that $R[X, Y, Z]/(g)$ is a polynomial algebra in two variables, then $g$ and $Z$ form a pair of variables in $R[X, Y, Z]$. We will also show that the result holds over any Noetherian domain containing $\mathbb{Q}$., Comment: 16 pages
- Published
- 2011
36. On the associated graded ring of a semigroup ring
- Author
-
Vincenzo Micale, Alessio Sammartano, and Marco D'Anna
- Subjects
Pure mathematics ,Ring (mathematics) ,Mathematics::Commutative Algebra ,13H10 ,Semigroup ,13A30, 13H10 ,13A30 ,Term (logic) ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,Associated graded ring ,Buchsbaum ring ,Numerical semigroup ,Apéry's constant ,Homogeneous ,FOS: Mathematics ,Maximal ideal ,Mathematics - Abstract
Let (R;m) be a numerical semigroup ring. In this paper we study the properties of its associated graded ring G(m). In particular, we describe the H^0_M for G(m) (where M is the homogeneous maximal ideal of G(m)) and we characterize when G(m) is Buchsbaum. Furthermore, we find the length of H^0_M as a G(m)-module, when G(m) is Buchsbaum. In the 3-generated numerical semigroup case, we describe the H^0_M in term of the Apery set of the numerical semigroup associated to R. Finally, we improve two characterizations of the Cohen-Macaulayness and Gorensteinness of G(m) given in [2] and [3], respectively., 20 pages
- Published
- 2011
37. A classification of one-dimensional local domains based on the invariant $(c-\delta)r-\delta$
- Author
-
Elsa Zatini and Anna Oneto
- Subjects
Delta ,Discrete mathematics ,Noetherian ,13H10 ,Local ring ,length ,Mathematics - Commutative Algebra ,Cohen Macaulay type ,Value set ,type-sequence ,Invariant (mathematics) ,Quotient ,Mathematics - Abstract
Let $R$ be a one-dimensional, local, Noetherian domain, $\R$ the integral closure of $R$ in its quotient field and $v(R)$ the value set defined by the usual valuation. The aim of the paper is to study the non-negative invariant $b:=(c-\delta)r- \delta $, where $c, \delta, r$ denote the conductor, the length of $\R/R$ and the Cohen Macaulay type, respectively. In particular, the classification of the semigroups $v(R)$ for rings having $b\leq 2(r-1)$ is realized. This method of classification might be successfully utilized with similar arguments but more boring computations in the cases $b\leq q(r-1), $ for reasonably low values of $q$. The main tools are type sequences and the invariant $k$ which estimates the number of elements in $v(R)$ belonging to the interval $[c-e,c), e$ being the multiplicity of $R$., Comment: Journal of Commutative Algebra (JCA), Rocky Mountain Consortium
- Published
- 2011
38. Non-vanishing forms in projective space over finite fields
- Author
-
Samuel Lundqvist
- Subjects
Mathematics - Algebraic Geometry ,Pure mathematics ,Finite field ,Degree (graph theory) ,FOS: Mathematics ,Projective space ,12E20, 12Y05, 14Q99 ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
We consider a subset of projective space over a finite field and give bounds on the minimal degree of a non-vanishing form with respect to this subset., 6 pages. Citations are updated and a section on how to compute a non-vanishing form is added
- Published
- 2010
39. On completely decomposable and separable modules over Prüfer domains
- Author
-
Jorge E. Macías-Díaz and László Fuchs
- Subjects
homogeneously decomposable ,continuous well-ordered ascending chain of modules ,Pure mathematics ,13C13 ,type of a rank 1 module ,finitely decomposable and separable module ,13C05 ,pure ,$H(\aleph_0)$ and $G(\aleph_0)$-family of submodules ,13C13, 13C05, 13F05 ,${\rm RD}$- and ${\rm RD*}$-submodule ,Mathematics - Commutative Algebra ,completely decomposable ,Separable space ,Mathematics::K-Theory and Homology ,Torsion-free module ,13F05 ,$h$-local Prüfer domain ,Mathematics - Group Theory ,Mathematics - Abstract
We generalize known results on summands of completely decomposable and separable torsion-free abelian groups to modules over h-local Pr\"ufer domains. Over such domains summands of completely decomposable torsion-free modules are again completely decomposable (Theorem 3.2) and summands of separable torsion-free modules are likewise separable (Theorem 4.2). In addition, a Pontryagin-Hill type theorem is established on countable chains of homogeneous completely decomposable modules over h-local Pr\"ufer domains.
- Published
- 2010
40. Maps on divisor class groups induced by ring homomorphisms of finite flat dimension
- Author
-
Sandra Spiroff and Sean Sather-Wagstaff
- Subjects
Discrete mathematics ,Noetherian ,Ring (mathematics) ,Pure mathematics ,normal integral domains ,13C20 (Primary) 13B22, 13F40 (Secondary) ,Ring homomorphism ,Divisor class group ,excellent rings ,Divisor (algebraic geometry) ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,13C20 ,13B22 ,13F40 ,Surjective function ,Kernel (algebra) ,FOS: Mathematics ,Homomorphism ,Element (category theory) ,ring homomorphisms of finite flat dimension ,Mathematics - Abstract
Let f: A\to B be a ring homomorphism between Noetherian normal integral domains. We establish a general criterion for f to induce a homomorphism Cl(f): Cl(A)\to Cl(B) on divisor class groups. For instance, this criterion applies whenever f has finite flat dimension; this special case generalizes the more classical situations where f is flat or is surjective with kernel generated by an A-regular element. We extend some of Spiroff's work on the kernels of induced maps to this more general setting., 17 pages, uses xypic; v4 contains minor changes; final version to appear in Journal of Commutative Algebra
- Published
- 2009
41. Relations between semidualizing complexes
- Author
-
Sean Sather-Wagstaff, Amelia Taylor, and Anders J. Frankild
- Subjects
Pure mathematics ,13D05, 13D07, 13D25, 13H10 ,13D25 ,Generalization ,13D05 ,Bass classes ,Picard group ,Commutative Algebra (math.AC) ,semidualizing complexes ,FOS: Mathematics ,Equivalence relation ,Gorenstein dimensions ,Commutative property ,Mathematics ,Discrete mathematics ,G-dimensions ,Noetherian ring ,Mathematics::Commutative Algebra ,13H10 ,Mathematics::Rings and Algebras ,complete intersection dimensions ,tilting complexes ,Mathematics - Commutative Algebra ,Auslander classes ,13D07 ,Isomorphism ,CI-dimensions - Abstract
We study the following question: Given two semidualizing complexes B and C over a commutative noetherian ring R, does the vanishing of Ext^n_R(B,C) for n>>0 imply that B is C-reflexive? This question is a natural generalization of one studied by Avramov, Buchweitz, and Sega. We begin by providing conditions equivalent to B being C-reflexive, each of which is slightly stronger than the condition Ext^n_R(B,C)=0 for all n>>0. We introduce and investigate an equivalence relation \approx on the set of isomorphism classes of semidualizing complexes. This relation is defined in terms of a natural action of the derived Picard group and is well-suited for the study of semidualizing complexes over nonlocal rings. We identify numerous alternate characterizations of this relation, each of which includes the condition Ext^n_R(B,C)=0 for all n>>0. Finally, we answer our original question in some special cases., final version, to appear in J. Commutative Algebra, 27 pages, uses XY-pic
- Published
- 2009
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