A structure theorem for a class of grade three perfect ideals

Brown, Anne

doi:10.1016/0021-8693(87)90196-7

Cited by 33 publications

(52 citation statements)

References 7 publications

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“…Now we review the structure theorems for a class of perfect ideals I of grade 3 with type 2, λ(I) > 0 and for a class of perfect ideals I of grade 3 with type 3, λ(I) ≥ 2 given by Brown [2] and Sanchez [14]. Kustin and Miller introduced the numerical invariant λ(I) defined in [12] to distinguish Gorenstein ideals I of grade 4 in terms of a resolution of R/I.…”

Section: Preliminariesmentioning

confidence: 99%

“…Brown provided a structure theorem for a class of perfect ideals I of grade 3 with type 2, λ(I) > 0. The minimal free resolution F of R/I is described in [2].…”

Section: Preliminariesmentioning

confidence: 99%

“…Let I be a homogeneous Gorenstein ideal of grade 3. For this purpose, we use the explicit description of minimal free resolution for R/I in [2] to describe the Hilbert function of R/I. Consider the minimal free resolution F in the form (4.1)…”

Section: Hilbert Functions Of Some Classes Of Perfect Ideals Of Gradementioning

confidence: 99%

“…They also showed that every perfect ideal of grade 3 has a differential, graded commutative algebra structure. In 1987, Brown [2] provided a structure theorem for a class of perfect ideals of grade 3 with type 2 and λ(I) > 0, where λ(I) is the numerical invariant introduced by Kustin and Miller [12] to distinguish classes of Gorenstein ideals I of grade 4 in terms of free resolutions of R/I. In 1989, Sanchez [14] provided a structure theorem for a class of type 3 perfect ideals of grade 3 and λ(I) ≥ 2.…”

Section: Introductionmentioning

confidence: 99%

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Perfect Ideals of Grade Three Defined by Skew-Symmetrizable Matrices

Cho¹,

Kang²,

Ko³

2012

Bulletin of the Korean Mathematical Society

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Abstract. Brown provided a structure theorem for a class of perfect ideals of grade 3 with type 2 and λ > 0. We introduced a skew-symmetrizable matrix to describe a structure theorem for complete intersections of grade 4 in a Noetherian local ring. We construct a class of perfect ideals I of grade 3 with type 2 defined by a certain skew-symmetrizable matrix. We present the Hilbert function of the standard k-algebras R/I, where R is the polynomial ring R = k[v 0 , v 1 , . . . , vm] over a field k with indeterminates v i and deg v i = 1.

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Section: Preliminariesmentioning

confidence: 99%

“…Brown provided a structure theorem for a class of perfect ideals I of grade 3 with type 2, λ(I) > 0. The minimal free resolution F of R/I is described in [2].…”

Section: Preliminariesmentioning

confidence: 99%

Section: Hilbert Functions Of Some Classes Of Perfect Ideals Of Gradementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Perfect Ideals of Grade Three Defined by Skew-Symmetrizable Matrices

Cho¹,

Kang²,

Ko³

2012

Bulletin of the Korean Mathematical Society

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“…They employed the multilinear algebra and algebra structure on finite free resolution to describe the structure theorems. Brown ( [1]) and Sanchez ([8]) provided structure theorems for a class of perfect ideals of grade 3 with type 2 and λ > 0 and for a class of perfect ideals of grade 3 with type 3 and λ ≥ 2, respectively. Kang, Cho and Ko ( [5]) described a structure theorem for some classes of perfect ideals of grade 3 which are algebraically linked to an almost complete intersection of grade 3 by a regular sequence.…”

Section: Introductionmentioning

confidence: 99%

A Class of Grade Three Determinantal Ideals

Kang¹,

Kim²

2012

Honam Mathematical Journal

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Abstract. Let k be a field containing the field Q of rational numbers and let R = k[xij|1 ≤ i ≤ m, 1 ≤ j ≤ n] be the polynomial ring over a field k with indeterminates xij. Let It(X) be the determinantal ideal generated by the t-minors of an m × n matrix X = (xij). Eagon and Hochster proved that It(X) is a perfect ideal of grade (m − t + 1)(n − t + 1). We give a structure theorem for a class of determinantal ideals of grade 3. This gives us a characterization that It(X) has grade 3 if and only if n = m + 2 and It(X) has the minimal free resolution F such that the second differential map of F is a matrix defined by complete matrices of grade n + 2.

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Planar Graphs and the Koszul Algebra Structure for Trivariate Monomial Ideals

Painter

2014

Connections Between Algebra, Combinatorics, and Geometry

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We will describe how we can identify the structure of the Koszul algebra for trivariate monomial ideals from minimal free resolutions. We use recent work of L. Avramov, where he classifies the behavior of Bass numbers of embedding codepth 3 commutative local rings. His classification relies on a corresponding classification of their respective Koszul algebras, which is comprised of 5 categories. Using Avramov's classification of the Koszul algebra, along with their respective Bass series we will learn how to identify the Koszul algebra structure by inspecting the minimal free resolution of the quotient ring. We give a complete classification of the Koszul algebra for generic monomial ideals and offer several examples. In addition we will describe a class of ideals with a specific Koszul algebra structure which was previously unknown.

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A structure theorem for a class of grade three perfect ideals

Cited by 33 publications

References 7 publications

Perfect Ideals of Grade Three Defined by Skew-Symmetrizable Matrices

Perfect Ideals of Grade Three Defined by Skew-Symmetrizable Matrices

A Class of Grade Three Determinantal Ideals

Planar Graphs and the Koszul Algebra Structure for Trivariate Monomial Ideals

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