The Integral Closure of Subrings Associated to Graphs

Simis, Aron; Vasconcelos, Wolmer V.; Villarreal, Rafael H.

doi:10.1006/jabr.1997.7171

Cited by 78 publications

(62 citation statements)

References 4 publications

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“…For the notion of an H-configuration, and the related notion of a bow tie configuration, we refer to [9,Chapter 8,p. 314] and [7]. Since rank(A(F G )) = n for a non-bipartite graph, the result follows immediately from Corollary 3.1 by using [7, Theorem 1.1 and Corollary 2.8] which together assert that the edge ideal (F G ) is normal if and only if the graph G admits no H-configurations.…”

Section: Normality Of the Rees Algebra Versus Birationalitymentioning

confidence: 65%

Constraints for the normality of monomial subrings and birationality

Simis

Villarreal

2002

Proc. Amer. Math. Soc.

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Section: Normality Of the Rees Algebra Versus Birationalitymentioning

confidence: 65%

Constraints for the normality of monomial subrings and birationality

Simis

Villarreal

2002

Proc. Amer. Math. Soc.

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“…The following configuration was introduced in [13]. (1) A bow tie of G(f) is the (connected) subgraph B of G(f) consisting of two odd cycles whose sets of edges are disjoint, connected by a unique non-empty path.…”

Section: Bow Tiesmentioning

confidence: 99%

“…Remark 3.7. These configurations were introduced in [13] in order to build the integral closure of the algebra k[f] in case G(f) had no loops. The Hochster monomial associated to a bow tie is the product of the variables corresponding to the totality of the vertices of the two cycles.…”

Section: Bow Tiesmentioning

confidence: 99%

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Polar syzygies in characteristic zero: The monomial case

Bermejo

Gimenez

Simis

2009

Journal of Pure and Applied Algebra

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a b s t r a c tGiven a set of forms f = {f 1 , . . . , f m } ⊂ R = k[x 1 , . . . , x n ], where k is a field of characteristic zero, we focus on the first syzygy module Z of the transposed Jacobian module D(f), whose elements are called differential syzygies of f. There is a distinct submodule P ⊂ Z coming from the polynomial relations of f through its transposed Jacobian matrix, the elements of which are called polar syzygies of f. We say that f is polarizable if equality P = Z holds.This paper is concerned with the situation where f are monomials of degree 2, in which case one can naturally associate to them a graph G(f) with loops and translate the problem into a combinatorial one. The main result is a complete combinatorial characterization of polarizability in terms of special configurations in this graph. As a consequence, we show that polarizability implies normality of the subalgebra k[f] ⊂ R and that the converse holds provided the graph G(f) is free of certain degenerate configurations. One main combinatorial class of polarizability is the class of polymatroidal sets. We also prove that if the edge graph of G(f) has diameter at most 2 then f is polarizable. We establish a curious connection with birationality of rational maps defined by monomial quadrics.

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“…Furthermore, Gröbner bases of I G have been studied among others by Ohsugi and Hibi [21] and Tatakis and Thoma [26]. A necessary condition for I G to have a squarefree initial ideal is the normality of K [G], which was characterized combinatorially by Ohsugi and Hibi [19] and Simis, Vasconcelos and Villarreal [23]. Normality, though, is not sufficient: Ohsugi and Hibi [16] gave an example of a graph G such that K [G] is normal but all possible initial ideals of I G are not squarefree.…”

Section: Introductionmentioning

confidence: 99%

Toric ideals associated with gap-free graphs

D’Alì

2015

Journal of Pure and Applied Algebra

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A b s t r ac t . In this paper we prove that every toric ideal associated with a gap-free graph G has a squarefree lexicographic initial ideal. Moreover, in the particular case when the complementary graph of G is chordal (i.e. when the edge ideal of G has a linear resolution), we show that there exists a reduced Gröbner basis G of the toric ideal of G such that all the monomials in the support of G are squarefree. Finally, we show (using work by Herzog and Hibi) that if I is a monomial ideal generated in degree 2, then I has a linear resolution if and only if all powers of I have linear quotients, thus extending a result by Herzog, Hibi and Zheng. I n t ro d u c t i o nAlgebraic objects depending on combinatorial data have attracted a lot of interest among both algebraists and combinatorialists: some valuable sources to learn about this research area are the books by Stanley [24], Villarreal [27], Miller and Sturmfels [12], and Herzog and Hibi [7]. It is often a challenge to establish relationships between algebraic and combinatorial properties of these objects.Let G be a simple graph and consider its vertices as variables of a polynomial ring over a field K. We can associate with each edge e of G the squarefree monomial M e of degree 2 obtained by multiplying the variables corresponding to the vertices of the edge. With this correspondence in mind, we can now introduce some algebraic objects associated with the graph G:• the edge ideal I(G) is the monomial ideal generated by {M e | e is an edge of G};• the toric ideal I G is the kernel of the presentation of the K-algebra K [G] generated by {M e | e is an edge of G}.An important result by Fröberg [5] gives a combinatorial characterization of those graphs G whose edge ideal I(G) admits a linear resolution: they are exactly the ones whose complementary graph G c is chordal. Another strong connection between the realms of commutative algebra and combinatorics is the one which links initial ideals of the toric ideal I G to triangulations of the edge polytope of G, see Sturmfels's book [25] and the recent article by Haase, Paffenholz, Piechnik and Santos [6]. Furthermore, Gröbner bases of I G have been studied among others by Ohsugi and Hibi [21] and Tatakis and Thoma [26]. A necessary condition for I G to have a squarefree initial

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The Integral Closure of Subrings Associated to Graphs

Cited by 78 publications

References 4 publications

Constraints for the normality of monomial subrings and birationality

Constraints for the normality of monomial subrings and birationality

Polar syzygies in characteristic zero: The monomial case

Toric ideals associated with gap-free graphs

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