Acyclicity over local rings with radical cube zero

Christensen, Lars Winther; Veliche, Oana

doi:10.1215/ijm/1258138553

Cited by 17 publications

(17 citation statements)

References 10 publications

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“…that over a Golod local ring that is not Gorenstein, every totally reflexive module is free. Another partial answer to Problem 1.30 is obtained by Yoshino [116], and by Christensen and Veliche [36]. The problem is also studied by Takahashi in [105].…”

Section: Local Ringsmentioning

confidence: 90%

Beyond totally reflexive modules and back

Christensen

Foxby

Holm

2010

Commutative Algebra

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Section: Local Ringsmentioning

confidence: 90%

Beyond totally reflexive modules and back

Christensen

Foxby

Holm

2010

Commutative Algebra

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“…In light of Proposition 4.4, we write the coefficients of the polynomial d(t) in terms of the invariants a i , q ij , b and a for some special cases and provide a uniform expression of the Poincaré series in these cases. (1 + t) n−1 1 − t − (a 1 − 1)t 2 − (a 3 − q 11 )t 3 + q 12 t 4 − bt 5 .…”

Section: Applications To the Constructionmentioning

confidence: 99%

“…It follows that the kernel of the multiplication map φ 1 : A 1 ⊗ A 1 → A 2 is generated by a 2 1 − q 11 = 42 elements. We record the indices of basis elements of A 1 • A 1 by the set S = {(1, 2), (1,5), (1,6), (1, 7), (2, 3), (3,5), (4, 5)}.…”

Section: A Basis Ofmentioning

confidence: 99%

A truncated minimal free resolution of the residue field

Nguyen¹,

Veliche²

2020

Preprint

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In a paper in 1962, Golod proved that the Betti sequence of the residue field of a local ring attains an upper bound given by Serre if and only if the homology algebra of the Koszul complex of the ring has trivial multiplications and trivial Massey operations. This is the origin of the notion of Golod ring. Using the Koszul complex components he also constructed a minimal free resolution of the residue field. In this article, we extend this construction up to degree five for any local ring. We describe how the multiplicative structure and the triple Massey products of the homology of the Koszul algebra are involved in this construction. As a consequence, we provide explicit formulas for the first six terms of a sequence that measures how far the ring is from being Golod.

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“…Define N = T/M and let F and G be free resolutions of M and N , respectively. Since R is local with 3 = 0 we can assume that F i = R n and G i = R for all i ≥ 0, [8]. By applying the Horseshoe Lemma, we get the…”

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confidence: 99%