A cohomological study of local rings of embedding codepth 3
TL;DR: In this paper, it was shown that there is a real number γ > 1, such that μ R d + i ≥ γ μ Rd + i − 1 holds for all i ≥ 0, except for i = 2 in two explicitly described cases.
About: This article is published in Journal of Pure and Applied Algebra.The article was published on 2012-11-01 and is currently open access. It has received 33 citations till now. The article focuses on the topics: Von Neumann regular ring & Residue field.
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TL;DR: In this article, the authors give an explicit presentation for the first syzygies of the Frobenius power F t (Q) of a positive field k in the polynomial ring k [x, y, z ] with positive characteristic.
17 citations
TL;DR: In this paper, a necessary and sufficient condition for the weak Lefschetz property for the polynomial ring k[x1,..., xn] was presented.
Abstract: Let A = k[x1, . . . , xn]/(x1, . . . , x d n), where k is an infinite field. If k has characteristic zero, then Stanley proved that A has the Weak Lefschetz Property (WLP). Henceforth, k has positive characteristic p. If n = 3, then Brenner and Kaid have identified all d, as a function of p, for which A has the WLP. In the present paper, the analogous project is carried out for 4 ≤ n. If 4 ≤ n and p = 2, then A has the WLP if and only if d = 1. If n = 4 and p is odd, then we prove that A has the WLP if and only if d = kq + r for integers k, q, r with 1 ≤ k ≤ p−1 2 , r ∈ { q−1 2 , q+1 2 } , and q = pe for some non-negative integer e. If 5 ≤ n, then we prove that A has the WLP if and only if ⌊ n(d−1)+3 2 ⌋ ≤ p. We first interpret the WLP for the ring k[x1, . . . , xn]/(x1 , . . . , x d n) in terms of the degrees of the non-Koszul relations on the elements x1 , . . . , x d n−1, (x1 + . . . + xn−1) d in the polynomial ring k[x1, . . . , xn−1]. We then exhibit a sufficient condition for k[x1, . . . , xn]/(x1, . . . , x d n) to have the WLP. This condition is expressed in terms of the non-vanishing in k of determinants of various Toeplitz matrices of binomial coefficients. Frobenius techniques are used to produce relations of low degree on x1 , . . ., x d n−1, (x1 + . . .+ xn−1) d. From this we obtain a necessary condition for A to have the WLP. We prove that the necessary condition is sufficient by showing that the relevant determinants are non-zero in k.
16 citations
TL;DR: This work gives examples of algebra structures that have been conjectured not to occur that carry a differential graded algebra structure, based on which one can classify local rings of embedding codepth 3.
Abstract: A complete local ring of embedding codepth 3 has a minimal free resolution of length 3 over a regular local ring. Such resolutions carry a differential graded algebra structure, based on which one can classify local rings of embedding codepth 3. We give examples of algebra structures that have been conjectured not to occur.
10 citations
TL;DR: In this article, a family of complexes called trimming complexes is proposed and used to deduce the Betti table for the minimal free resolution problem. But the complexity of trimming complex is not fixed.
Abstract: We produce a family of complexes called trimming complexes and explore its applications. We demonstrate how trimming complexes can be used to deduce the Betti table for the minimal free resolution ...
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TL;DR: In this paper, it was shown that every grade 3 perfect ideal in a regular local ring is linked to a complete intersection or a Golod ideal, which is known not to be the case for ideals of grade 3.
9 citations
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01 Jan 1998
TL;DR: The notes for a series of five lectures to the Barcelona Summer School in Commutative Algebra at the Centre de Recerca Matematica, Institut d'Estudis Catalans, July 15-26, 1996.
Abstract: This text is based on the notes for a series of five lectures to the Barcelona Summer School in Commutative Algebra at the Centre de Recerca Matematica, Institut d’Estudis Catalans, July 15–26, 1996
378 citations
TL;DR: In this article, a homological invariant for a finite module over a commutative noetherian ring, called its CI-dimension, is introduced, which provides a rich structure theory of free resolutions.
Abstract: A new homological invariant is introduced for a finite module over a commutative noetherian ring: its CI-dimension. In the local case, sharp quantitative and structural data are obtained for modules of finite CI-dimension, providing the first class of modules of (possibly) infinite projective dimension with a rich structure theory of free resolutions.
285 citations
139 citations