A family of complexes associated to an almost alternating map, with applications to residual intersections

Kustin, Andrew R.; Ulrich, Bernd

doi:10.1090/memo/0461

Cited by 41 publications

(71 citation statements)

References 36 publications

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“…For any integer q ≥ 1 Kustin and Ulrich [6] and Boffi and Sánchez [2] construct in a canonical way a finite complex D q (ξ) = D q R (G, G) of free modules over R of the form…”

Section: Complexes Associated With An Alternating Mapmentioning

confidence: 99%

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Torsion freeness of symmetric powers of ideals

Tchernev

2007

Trans. Amer. Math. Soc.

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Abstract. Let I be an ideal in a Noetherian commutative ring R with unit, let k ≥ 2 be an integer, and let α k : S k I −→ I k be the canonical surjective R-module homomorphism from the kth symmetric power of I to the kth power of I. When pd R I ≤ 1 or when I is a perfect Gorenstein ideal of grade 3, we provide a necessary and sufficient condition for α k to be an isomorphism in terms of upper bounds for the minimal number of generators of the localisations of I. When I = m is a maximal ideal of R we show that α k is an isomorphism if and only if R m is a regular local ring. In all three cases for I our results yield that if α k is an isomorphism, then α t is also an isomorphism for each 1 ≤ t ≤ k.

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“…For any integer q ≥ 1 Kustin and Ulrich [6] and Boffi and Sánchez [2] construct in a canonical way a finite complex D q (ξ) = D q R (G, G) of free modules over R of the form…”

Section: Complexes Associated With An Alternating Mapmentioning

confidence: 99%

“…For details on the construction and properties of the complexes D q R (G, G) we refer to [6] and [2], and also to [12], where a concise summary is given. As the presentation in [12, Section 1] appears to be best suited to our purpose, for the rest of this section we follow the notation introduced there.…”

Section: Complexes Associated With An Alternating Mapmentioning

confidence: 99%

Torsion freeness of symmetric powers of ideals

Tchernev

2007

Trans. Amer. Math. Soc.

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“…A version of this argument, which contains more details, may be found in [3, Theorem 2.10]. The proof of (b) also follows a standard argument; see, for example, [9,Theorem 9.4]. Let P be a prime of R with H ⊆ P and depth R P ≤ k. For (i) it suffices to show that R P is Cohen-Macaulay; for (ii) it suffices to show that R P is regular.…”

Section: Further Applications and Questionsmentioning

confidence: 99%

Ideals associated to two sequences and a matrix

Kustin

1995

Communications in Algebra

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Abstract. Let u 1×n , X n×n , and v n×1 be matrices of indeterminates, Adj X be the classical adjoint of X, and H(n) be the ideal I 1 (uX) + I 1 (Xv) + I 1 (vu − Adj X). Vasconcelos has conjectured that H(n) is a perfect Gorenstein ideal of grade 2n. In this paper, we obtain the minimal free resolution of H(n); and thereby establish Vasconcelos' conjecture.Let u 1×n , X n×n , and v n×1 be matrices of indeterminates over a commutative noetherian ring R 0 , and let H(n) be the ideal I 1 (uX) + I 1 (Xv) + I 1 (vu − Adj X) of the polynomial ring R = R 0 [{u i , v i

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“…In their influential work [5], Buchsbaum and Eisenbud show that when R is local, grade 3 ideals such as I classify the grade 3 Gorenstein ideals of R. It is therefore of considerable interest to obtain information on the homological properties of the ideal I and its powers I q , q ≥ 1. One approach to this problem was pursued by Boffi and Sánchez [2] and Kustin and Ulrich [6]. They construct a family of free complexes {D q (ξ)} q≥0 associated with the alternating matrix ξ such that each complex D q (ξ) approximates a resolution of the qth symmetric power S q M of the cokernel M of ξ; and they show that if ξ is a generic alternating matrix, then each complex D q (ξ) is in fact a resolution of the ideal I q .…”

Section: Introductionmentioning

confidence: 99%

“…For the convenience of the reader, and in order to establish notation, we recall in this section, after [6], the definition and some properties of the complexes D q (ξ). Let R be a ring, let G be a free R-module of rank g, and let G * = Hom R (G, R) be the dual of G. For the rest of this paper ξ : G * → G is an alternating map.…”

Section: Introductionmentioning

confidence: 99%

Acyclicity criteria for complexes associated with an alternating map

Tchernev

2001

Proc. Amer. Math. Soc.

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Abstract. When I is a Gorenstein ideal of grade 3 in a local ring R, results of Boffi and Sánchez, and of Kustin and Ulrich show that for each t ≥ 1 one can construct in a canonical way a finite free complex D t that is "approximately" a resolution for the ideal I t . Kustin and Ulrich also provide a sufficient condition that D t is acyclic, and a sufficient condition that D t is a resolution of I t . We complete these two acyclicity criteria by showing that the corresponding sufficient conditions are also necessary.

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A family of complexes associated to an almost alternating map, with applications to residual intersections

Cited by 41 publications

References 36 publications

Torsion freeness of symmetric powers of ideals

Torsion freeness of symmetric powers of ideals

Ideals associated to two sequences and a matrix

Acyclicity criteria for complexes associated with an alternating map

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