The Dedekind-Mertens formula and determinantal rings

Bruns, Winfried; Guerrieri, Anna

doi:10.1090/s0002-9939-99-04535-9

Cited by 20 publications

(7 citation statements)

References 13 publications

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“…. , y m ); using a variety of techniques including determinantal ideals, residual intersections and Gröbner basis, [15] and [6] describe additional features of this and other related subideals in connection with the so-called Dedekind-Mertens Lemma. A hook-shaped tableau is the other extremal case; in this situation I λ = x 1 (y 1 , .…”

Section: Betti Numbers and Primary Decompositions Of Ferrers Idealsmentioning

confidence: 99%

Monomial and toric ideals associated to Ferrers graphs

Corso

Nagel

2008

Trans. Amer. Math. Soc.

106

121

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Abstract. Each partition λ = (λ 1 , λ 2 , . . . , λ n ) determines a so-called Ferrers tableau or, equivalently, a Ferrers bipartite graph. Its edge ideal, dubbed a Ferrers ideal, is a squarefree monomial ideal that is generated by quadrics. We show that such an ideal has a 2-linear minimal free resolution; i.e. it defines a small subscheme. In fact, we prove that this property characterizes Ferrers graphs among bipartite graphs. Furthermore, using a method of Bayer and Sturmfels, we provide an explicit description of the maps in its minimal free resolution. This is obtained by associating a suitable polyhedral cell complex to the ideal/graph. Along the way, we also determine the irredundant primary decomposition of any Ferrers ideal. We conclude our analysis by studying several features of toric rings of Ferrers graphs. In particular we recover/establish formulae for the Hilbert series, the Castelnuovo-Mumford regularity, and the multiplicity of these rings. While most of the previous works in this highly investigated area of research involve path counting arguments, we offer here a new and self-contained approach based on results from Gorenstein liaison theory.

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Section: Betti Numbers and Primary Decompositions Of Ferrers Idealsmentioning

confidence: 99%

Monomial and toric ideals associated to Ferrers graphs

Corso

Nagel

2008

Trans. Amer. Math. Soc.

106

121

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“…. , m) ∈ Z n , the ideal I λ is the edge ideal of a complete bipartite graph, and a distinguished minimal reduction of I λ is given by the Dedekind-Mertens content formula (see [54,15,5]). Here we extend this result to arbitrary Ferrers ideals.…”

Section: Minimal Reductionsmentioning

confidence: 99%

“…That is, c(f g) is a minimal reduction of c(f ) • c(g) with reduction number min{n, m} − 1 (see Figure 1). Subsequently, a combinatorial proof of the Dedekind-Mertens formula was given by Bruns and Guerrieri [5] via a study of the Gröbner basis of the ideal c(f g). The boxes in Figure 1 are naturally associated…”

Section: Introductionmentioning

confidence: 99%

Blow-up algebras, determinantal ideals, and Dedekind–Mertens-like formulas

et al. 2016

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We give a combinatorial proof of the Dedekind-Mertens formula by computing the initial ideal of the content ideal of the product of two generic polynomials. As a side effect we obtain a complete classification of the rank 1 Cohen-Macaulay modules over the determinantal rings K[X]/I 2 (X). Let f, g be polynomials in one indeterminate over a commutative ring A. The Dedekind-Mertens formula relates the content ideals of f , g, and their product f g: one has c(f g)c(f) d = c(g)c(f) d+1 , d= deg g. It is the best universally valid variant of Gauß' classical formula c(f g) = c(f)c(g) for polynomials over a principal ideal domain. (The content ideal of f ∈ A[T ] is the ideal generated by the coefficients of f in A.) Content ideals and the Dedekind-Mertens formula have recently received much attention; see Glaz and Vasconcelos [8], Corso, Vasconcelos, and Villarreal [6] and Heinzer and Huneke [9], [10]. For detailed historical information about the Dedekind-Mertens formula, see [9]. The main objective of this paper is a combinatorial proof of the formula based on a Gröbner basis approach to the ideal c(f g) for polynomials with indeterminate coefficients; in fact we will determine the initial ideal of c(f g) with respect to a suitable term order. (For information on term orders and Gröbner bases we refer the reader to Eisenbud [7].) A side effect of our approach is very precise numerical information about the rank one Cohen-Macaulay modules over the determinantal ring S = K[X]/I 2 (X) where X is an m × n matrix of indeterminates and I 2 (X) the ideal generated by its 2-minors. This connection extends the ideas of [6] and was in fact suggested by them. The actual motive for our work was the need for some explicit computation modulo c(f g) in Boffi, Bruns, and Guerrieri [2], or, more precisely, modulo an ideal generalizing c(f g) slightly. Theorem 1. Let K be a field, R = K[

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“…Proof. Let R be the ring k[X]/I 2 (X) and d be the Krull dimension of R. It is well-known; see, for example, [7,Cor. 4], [12, Cor.1], [23], or [1], that…”

Section: The Valuementioning

confidence: 99%

Exact pairs of homogeneous zero divisors

2016

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Abstract. Let S be a standard graded Artinian algebra over a field k. We identify constraints on the Hilbert function of S which are imposed by the hypothesis that S contains an exact pair of homogeneous zero divisors. As a consequence, we prove that if S is a compressed level algebra, then S does not contain any homogeneous zero divisors.In [17], Henriques and Şega defined the pair of elements (a, b) in a commutative ring S to be an exact pair of zero divisors if (0 : S a) = (b) and (0 : S b) = (a). We take S be a standard graded Artinian algebra over a field and we identify constraints on the Hilbert function of S which are imposed by the hypothesis that S contains an exact pair (θ 1 , θ 2 ) of homogeneous zero divisors. In Theorem 2.9 we prove that the main numerical constraint depends on the sum deg θ 1 + deg θ 2 , but not on the individual numbers deg θ 1 or deg θ 2 . In other words, the numerical constraint imposed on S by having an exact pair of homogeneous zero divisors of degrees d 1 and d 2 is the same as the constraint imposed by having an exact pair of homogeneous zero divisors of degrees 1 and d 1 + d 2 − 1. This result is especially curious because it is possible for S to have an exact pair of homogeneous zero divisors of degrees 2 and 2 without having any homogeneous exact zero divisors of degree 1; see Example 3.1. Our main result is Theorem 2.9.Theorem 2.9. Let S be a standard graded Artinian k-algebra. Suppose that (θ 1 , θ 2 ) is an exact pair of homogeneous zero divisors in S. If D = deg θ 1 + deg θ 2 , then the Hilbert series of S is divisible by2010 Mathematics Subject Classification. 13D02, 13A02.

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The Dedekind-Mertens formula and determinantal rings

Cited by 20 publications

References 13 publications

Monomial and toric ideals associated to Ferrers graphs

Monomial and toric ideals associated to Ferrers graphs

Blow-up algebras, determinantal ideals, and Dedekind–Mertens-like formulas

Exact pairs of homogeneous zero divisors

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