Mostrando recursos 1 - 20 de 225

  1. Index

  2. Associated primes of local cohomology after adjoining indeterminates. Part 2: The general case

    Robbins, Hannah
    Let $A$ be a domain finitely generated as an algebra over a field, $k$ of characteristic zero, $ R=A[t_1,\dots ,t_\ell ]\quad \mbox {or}\quad A[[t_1,\dots ,t_\ell ]]$, and $I$ an ideal of $R$. If $A$ has a resolution of singularities, $Y_0$, which is the blowup of $A$ along an ideal of depth at least 2 and is covered by a finite number of open affines with $H^j(Y_0,\mathcal {O}_{Y_0})$ of finite length over $A$ for $j>0$, we prove that $\rm{Ass} _RH^i_I(R) $ is finite for every $i$. In particular, this holds when $A$ is a finite-dimensional normal domain with an isolated singularity...

  3. The Buchberger resolution

    Olteanu, Anda; Welker, Volkmar
    We define the Buchberger resolution, which is a graded free resolution of a monomial ideal in a polynomial ring. Its construction uses a generalization of the Buchberger graph and encodes much of the combinatorics of the Buchberger algorithm. The Buchberger resolution is a cellular resolution that, when it is minimal, coincides with the Scarf resolution. The simplicial complex underlying the Buchberger resolution is of interest for its own sake, and its combinatorics is not fully understood. We close with a conjecture on the clique complex of the Buchberger graph.

  4. Sperner property and finite-dimensional Gorenstein algebras associated to matroids

    Maeno, Toshiaki; Numata, Yasuhide
    We prove the Lefschetz property for a certain class of finite-dimensional Gorenstein algebras associated to matroids. Our result implies the Sperner property of the vector space lattice. More generally, it is shown that the modular geometric lattice has the Sperner property. We also discuss the Gr\"obner fan of the defining ideal of our Gorenstein algebra.

  5. The cone of Betti tables over three non-collinear points in the plane

    Gheorghita, Iulia; Sam, Steven V
    We describe the cone of Betti tables of all finitely generated graded modules over the homogeneous coordinate ring of three non-collinear points in the projective plane. We also describe the cone of Betti tables of all finite length modules.

  6. On a topological characterization of Prüfer $v$-multiplication domains among essential domains

    Finocchiaro, Carmelo Antonio; Tartarone, Francesca
    In this paper, we characterize the Pr\"ufer $v$-multiplication domain as a class of essential domains verifying an additional property on the closure of some families of prime ideals, with respect to the constructible topology.

  7. A Northcott type inequality for Buchsbaum-Rim coefficients

    Balakrishnan, R.; Jayanthan, A.V.
    In 1960, Northcott \cite {DGN} proved that, if $e_0(I)$ and $e_1(I)$ denote the 0th and first Hilbert-Samuel coefficients of an $\mathfrak m$-primary ideal $I$ in a Cohen-Macaulay local ring $(R,\mathfrak m)$, then $e_0(I)-e_1(I)\le \ell (R/I)$. In this article, we study an analogue of this inequality for Buchsbaum-Rim coefficients. We prove that, if $(R,\mathfrak m)$ is a two dimensional Cohen-Macaulay local ring and $M$ is a finitely generated $R$-module contained in a free module $F$ with finite co-length, then $\rm{br} _0(M)-\rm{br} _1(M)\le \ell (F/M)$, where $\rm{br} _0(M)$ and $\rm{br} _1(M$) denote 0th and 1st Buchsbaum-Rim coefficients, respectively.

  8. Representations of finite posets over the ring of integers modulo a prime power

    Arnold, David; Mader, Adolf; Mutzbauer, Otto; Solak, Ebru
    The classical category Rep$(S,\mathbb {Z}_{p})$ of representations of a finite poset $S$ over the field $\mathbb {Z}_{p}$ is extended to two categories, Rep$(S,\mathbb {Z}_{p^{m}})$ and uRep$(S,\mathbb {Z}_{p^{m}})$, of representations of $S$ over the ring $\mathbb {Z}_{p^{m}}$. A list of values of $S$ and $m$ for which Rep$(S,\mathbb {Z}_{p^{m}})$ or uRep$(S,\mathbb {Z}_{p^{m}})$ has infinite representation type is given for the case that $S$ is a forest. Applications include a computation of the representation type for certain classes of abelian groups, as the category of sincere representations in (uRep$(S,\mathbb {Z}_{p^{m}})$) Rep$(S,Z_{p^{m}})$ has the same representation type as (homocyclic) $(S,p^{m})$-groups, a class of...

  9. Multigraded modules of nested type

    Sabzrou, Hossein; Tousi, Massoud
    Extending the notion of monomial ideals of nested type, we introduce multigraded modules of nested type. We characterize them algebraically, resulting in the explicit construction of their dimension filtration. We compute several important invariants, including their Castelnuovo-Mumford regularity and projective dimension without {us\nobreak {ing}} the construction of their minimal graded free resolution. We show that they are pretty clean and hence sequentially Cohen-Macaulay.

  10. Pseudo-convergent sequences and Prüfer domains of integer-valued polynomials

    Loper, K. Alan; Werner, Nicholas J.
    Let $K$ be a field with rank one valuation and $V$ the valuation domain of $K$. For a subset $E$ of $V$, the ring of integer-valued polynomials on $E$ is \[ \Int (E, V) = \{f \in K[x] \mid f(E) \subseteq V \}. \] A question of interest regarding $\Int (E, V)$ is: for which $E$ is $\Int (E, V)$ a Pr\"{u}fer domain? In this paper, we contribute a partial answer to this question. We classify exactly when $\Int (E, V)$ is Pr\"{u}fer in the case where the elements of $E$ comprise a pseudo-convergent sequence in $V$. Our work expands...

  11. On the functoriality of marked families

    Lella, Paolo; Roggero, Margherita
    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 associating to each ideal $J$ of this type a scheme $\MFScheme {J}$, called a $J$-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...

  12. On formal local cohomology modules with respect to a pair of ideals

    Freitas, T.H.; Pérez, V.H. Jorge
    We introduce a generalization of the formal local cohomology module, which we call a formal local cohomology module with respect to a pair of ideals, and study its various properties. We analyze their structure, upper and lower vanishing and non-vanishing properties. There are various exact sequences concerning formal cohomology modules, among them we have a Mayer-Vietoris sequence with respect to pair ideals. Also, we give another proof for a generalized version of the local duality theorems for Gorenstein, Cohen-Macaulay rings, and a generalization of the Grothendieck duality theorem for Gorenstein rings. We discuss the concept of formal grade with respect...

  13. A structure theorem for $2$-stretched Gorenstein algebras

    Casnati, Gianfranco; Notari, Roberto
    In this paper, we study isomorphism classes of local, Artinian, Gorenstein $k$-algebras $A$ whose maximal ideal $\frak M$ satisfies $\dim _k(\fM ^3/\fM ^4)=1$ by means of Macaulay's inverse system generalizing a recent result by Elias and Rossi. Then we use such results in order to complete the description of the singular locus of the Gorenstein locus of $\Hilb _{11}(\p n)$.

  14. Homological dimensions with respect to a semidualizing complex

    Totushek, Jonathan
    In this paper, we build off of Takahashi and White's $\catpc $-projective dimension and $\catic $-injective dimension to define these dimensions for when $C$ is a semidaulizing complex. We develop the framework for these homological dimensions by establishing base change results and local-global behavior. Furthermore, we investigate how these dimensions interact with other invariants.

  15. $s$-Hankel hypermatrices and $2\times 2$ determinantal ideals

    Sammartano, Alessio
    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.

  16. Reduced symmetric algebras and linear syzygies

    Johnson, Mark R.

  17. On complete monomial ideals

    Gimenez, Philippe; Simis, Aron; Vasconcelos, Wolmer V.; Villarreal, Rafael H.
    In dimension 2, we study complete monomial ideals combinatorially, their Rees algebras and develop effective means of finding their defining equations.

  18. Polynomial foldings and rank of tensors

    Diaz, Steven P.; Lutoborski, Adam
    We review facts about rank, multilinear rank, multiplex rank and generic rank of tensors as well as folding of a tensor into a matrix of multihomogeneous polynomials. We define the new concept of folding rank of tensors and compare its properties to other ranks. We review the concept of determinantal schemes associated to a tensor. Then we define the new concept of a folding generic tensor meaning that all its determinantal schemes behave generically. Our main theorem states that for ``small'' 3-tensors, any folding generic tensor has generic rank, and the reverse does not always hold.

  19. Cut structures in zero-divisor graphs of commutative rings

    Axtell, M.; Baeth, N.; Stickles, J.
    Zero-divisor graphs, and more recently, compressed zero-divisor graphs are well represented in the commutative ring literature. In this work, we consider various cut structures, sets of edges or vertices whose removal disconnects the graph, in both compressed and non-compressed zero-divisor graphs. In doing so, we connect these graph-theoretic concepts with algebraic notions and provide realization theorems of zero-divisor graphs for commutative rings with identity.

  20. The ring of polynomials integral-valued over a finite set of integral elements

    Peruginelli, Giulio
    Let $D$ be an integral domain with quotient field $K$ and $\Omega $ a finite subset of $D$. McQuillan proved that the ring $\Int (\Omega ,D)$ of polynomials in $K[X]$ which are integer-valued over $\Omega $, that is, $f\in K[X]$ such that $f(\Omega )\subset D$, is a Pr\"ufer domain if and only if $D$ is Pr\"ufer. Under the further assumption that $D$ is integrally closed, we generalize his result by considering a finite set $S$ of a $D$-algebra $A$ which is finitely generated and torsion-free as a $D$-module, and the ring $\Int _K(S,A)$ of integer-valued polynomials over $S$, that is,...

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