Publications (10)2.91 Total impact
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ABSTRACT: Let $R$ be a Noetherian local ring and let $I$ be an ideal in $R$. The ideal $I$ is called balanced if the colon ideal $J:I$ is independent of the choice of the minimal reduction $J$ of $I$. Under suitable assumptions, Ulrich showed that $I$ is balanced if and only if the reduction number, $r(I)$, of $I$ is at most the `expected' one, namely $\ell(I) \height I+1$, where $\ell(I)$ is the analytic spread of $I$. In this article we propose a generalization of balanced. We prove under suitable assumptions that if either $R$ is onedimensional or the associated graded ring of $I$ is CohenMacaulay, then $J^{n+1}:I^n$ is independent of the choice of the minimal reduction $J$ of $I$ if and only if $r(I) \leq \ell(I)\height I+n$.09/2012;  [Show abstract] [Hide abstract]
ABSTRACT: We study the defining equations of the Rees algebra of squarefree monomial ideals in a polynomial ring over a field. We determine that when an ideal $I$ is generated by $n$ squarefree monomials of the same degree then $I$ has relation type at most $n2$ as long as $n \leq 5$. In general, we establish the defining equations of the Rees algebra in this case. Furthermore, we give a class of examples with relation type at least $n2$, where $n=\mu(I)$. We also provide new classes of ideals of linear type. We propose the construction of a graph, namely the generator graph of an ideal, where the monomial generators serve as vertices for the graph. We show that when $I$ is a squarefree monomial ideal generated in the same degree that is at least 2 and the generator graph of $I$ is the graph of a disjoint union of trees and graphs with unique odd cycles then $I$ is an ideal of linear type.05/2012;  [Show abstract] [Hide abstract]
ABSTRACT: We study minimal reductions of edge ideals of graphs and determine restrictions on the coefficients of the generators of these minimal reductions. We prove that when $I$ is not basic, then $\core{I}\subset \m I$, where $I$ is an edge ideal in the corresponding localized polynomial ring and $\m$ is the maximal ideal of this ring. We show that the inclusion is an equality for the edge ideal of an even cycle with an arbitrary number of whiskers. Moreover, we show that the core is obtained as a finite intersection of homogeneous minimal reductions in the case of even cycles. The formula for the core does not hold in general for the edge ideal of any graph and we provide a counterexample. In particular, we show in this example that the core is not obtained as a finite intersection of general minimal reductions.12/2010; 
Article: The clcore of an ideal
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ABSTRACT: We expand the notion of core to $cl$core for Nakayama closures $cl$. In the characteristic $p>0$ setting, when $cl$ is the tight closure, denoted by *, we give some examples of ideals when the core and the *core differ. We note that *core$(I)=$ core$(I)$, if $I$ is an ideal in a onedimensional domain with infinite residue field or if $I$ is an ideal generated by a system of parameters in any Noetherian ring. More generally, we show the same result in a CohenMacaulay normal local domain with infinite perfect residue field, if the analytic spread, $\ell$, is equal to the *spread and $I$ is $G_{\ell}$ and weakly$(\ell1)$residually $S_2$. This last is dependent on our result that generalizes the notion of general minimal reductions to general minimal *reductions. We also determine that the *core of a tightly closed ideal in certain onedimensional semigroup rings is tightly closed and therefore integrally closed. Comment: Final version. Math. Proc. Camb. Phil. Soc 149 (2010) 247262Mathematical Proceedings of the Cambridge Philosophical Society 06/2010; 149(02):247262. · 0.68 Impact Factor 
Article: What is a system of parameters?
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ABSTRACT: In this paper we discuss various refinements and generalizations of a theorem of Sankar Dutta and Paul Roberts. Their theorem gives a criterion for $d$ elements in a $d$dimensional Noetherian CohenMacaulay local ring to be a system of parameters, i.e., to have height $d$. We chiefly remove the assumption that the ring be CohenMacaulay and discuss similar theorems. Comment: 15 pages, submitted for publication.03/2010; 
Article: A formula for the *core of an ideal
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ABSTRACT: Expanding on the work of Fouli and Vassilev \cite{FV}, we determine a formula for the *$\rm{core}$ of an ideal in two different settings: (1) in a CohenMacaulay local ring of characteristic $p>0$, perfect residue field and test ideal of depth at least two, where the ideal has a minimal *reduction that is a parameter ideal and (2) in a normal local domain of characteristic $p>0$, perfect residue field and $\m$primary test ideal, where the ideal is a sufficiently high Frobenius power of an ideal. We also exhibit some examples where our formula fails if our hypotheses are not met. Comment: 11 pages, submitted for publicationProceedings of the American Mathematical Society 10/2009; · 0.61 Impact Factor 
Article: Annihilators of Graded Components of the Canonical Module, and the Core of Standard Graded Algebras
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ABSTRACT: We relate the annihilators of graded components of the canonical module of a graded CohenMacaulay ring to colon ideals of powers of the homogeneous maximal ideal. In particular, we connect them to the core of the maximal ideal. An application of our results characterizes CayleyBacharach sets of points in terms of the structure of the core of the maximal ideal of their homogeneous coordinate ring. In particular, we show that a scheme is CayleyBacharach if and only if the core is a power of the maximal ideal.Transactions of the American Mathematical Society 04/2009; · 1.02 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We prove formulas for the core of ideals that apply in arbitrary characteristic.The Michigan Mathematical Journal 05/2008; · 0.60 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: Let $R$ be a local Gorenstein ring with infinite residue field of arbitrary characteristic. Let $I$ be an $R$ideal with $g=\height I >0$, analytic spread $\ell$, and let $J$ be a minimal reduction of $I$. We further assume that $I$ satisfies $G_{\ell}$ and ${\depth} R/I^j \geq \dim R/I j+1$ for $1 \leq j \leq \ellg$. The question we are interested in is whether $\core{I}=J^{n+1}:\ds \sum_{b \in I} (J,b)^n$ for $n \gg 0$. In the case of analytic spread one Polini and Ulrich show that this is true with even weaker assumptions (\cite[Theorem 3.4]{PU}). We give a negative answer to this question for higher analytic spreads and suggest a formula for the core of such ideals.06/2007; 
Article: A study on the core of ideals
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ABSTRACT: Let R be a local Gorenstein ring with infinite residue field k and let I be an R ideal. The core of I , core(I ), is defined to be the intersection of all (minimal) reductions. We will usually assume that I satisfies G[cursive l] and depth R/Ij > or = dim R/I  j + 1 for 1 < or = j < or = [cursive l]  g , where [cursive l] = [cursive l](I ) is the analytic spread of I and g = ht I > 0. Under these conditions Polini and Ulrich show that if char k = 0 or char k > rJ (I )  [cursive l] + g then core(I ) = Jn+1 : In = Jn +1 : [Special characters omitted.] (J,y )n for n > or = max{rJ (I )  [cursive l] + g , 0} and any minimal reduction J of I ([27]). This formula for the core depends on the characteristic of the residue field k . We propose a conjecture for the core of such an ideal that should hold in any characteristic. We exhibit a series of examples that support this conjecture. Using the computer algebra program Macaulay 2 ([8]) we provide examples where Jn +1 : In [Special characters omitted.] core (I ) [Special characters omitted.]HASH(0x35aafbb4)Jn+1 : [Special characters omitted.] (J,y )n , and we prove several theoretical results that support the validity of these computations. We then provide new classes of ideals I for which core( I ) = Jn+1 : In for n >> 0 and any minimal reduction J of I . Our main theorem states that if in addition k is perfect and the special fiber ring [Special characters omitted.] (I ) of I has embedding dimension at most 1 locally at every minimal prime of maximal dimension, then core ( I ) = Jn+1 : In for n > or = max{r J (I )  [cursive l] + g , 0} and every minimal reduction J of I . We give several applications of our main theorem that show different instances where the above formula holds. In addition we obtain a description for the core of a power of the homogeneous maximal ideal of a standard graded CohenMacaulay k algebra. Finally we investigate the connection between the core and the reduction number. We prove that if the associated graded ring grI (R ) of I is CohenMacaulay then core( I ) = Jn +1 : In for every minimal reduction J of I if and only if n > or = max{ r (I )  [cursive l](I ) + g , 0}.ETD Collection for Purdue University.
Publication Stats
26  Citations  
2.91  Total Impact Points  
Top Journals
Institutions

2010

New Mexico State University
 Department of Mathematical Sciences
Las Cruces, New Mexico, United States


2007

University of Texas at Austin
Austin, Texas, United States
