On the Homology of Certain Local Rings

Golod, E. S.

doi:10.1070/rm1978v033n05abeh002521

Cited by 24 publications

(29 citation statements)

References 7 publications

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“…, x r ] and I is a stable monomial ideal in S, then S/I is a Golod ring. Hence, from a result of Golod [4] (see also [5]), it follows that…”

Section: Poincaré Seriesmentioning

confidence: 89%

Truncations of the ring of number-theoretic functions

Snellman

2000

Homology, Homotopy and Applications

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We study the ring Γ of all functions N + → K, endowed with the usual convolution product. Γ, which we call the ring of number-theoretic functions, is an inverse limit of the "truncations"Each Γ n is a zero-dimensional, finitely generated K-algebra, which may be expressed as the quotient of a finitely generated polynomial ring with a stable (after reversing the order of the variables) monomial ideal. Using the description of the free minimal resolution of stable ideals given by Eliahou-Kervaire, and some additional arguments by Aramova-Herzog and Peeva, we give the Poincaré-Betti series for Γ n .

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“…, x r ] and I is a stable monomial ideal in S, then S/I is a Golod ring. Hence, from a result of Golod [4] (see also [5]), it follows that…”

Section: Poincaré Seriesmentioning

confidence: 89%

Truncations of the ring of number-theoretic functions

Snellman

2000

Homology, Homotopy and Applications

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“…Formula (3.6.1) specializes to the following result of Herzog [9, Theorem 1]: If (R, m, k) and (S, n, k) are local rings, and α : S → R and β : R → S are homomorphisms of rings, such that βα = id S , then for every finite S-module N there is an equality of formal power series P R N (t) = P R S (t) P S N (t) . Among the original characterizations of Golod rings, which appear in the next result, is the property that Massey products are defined for every finite set of elements of H 1 (K R ): This is one direction of Golod's theorem in [8]. Remark.…”

Section: Local Ringsmentioning

confidence: 98%

Homology over trivial extensions of commutative DG algebras

Avramov

Iyengar

Nasseh

et al. 2019

Communications in Algebra

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Conditions on the Koszul complex of a noetherian local ring R guarantee that Tor R i (M, N ) is non-zero for infinitely many i, when M and N are finitely generated R-modules of infinite projective dimension. These conditions are obtained from results concerning Tor of differential graded modules over certain trivial extensions of commutative differential graded algebras.Date: August 5, 2015. 2010 Mathematics Subject Classification. 13D07 (primary); 16E45, 13D02, 13D40 (secondary).

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“…Remark. According to [89] and [66], Serre had observed that the coefficients on the left hand side in (13.1) are always less than or equal to the corresponding coefficients on the right.…”

Section: Polyhedral Products and The Golodness Of Monomial Ideal Ringsmentioning

confidence: 99%

“…In [66], Golod showed that (13.1) holds if and only if all products and all higher Massey products vanish in Tor S (R, K), (see also [89]). Higher Massey products in H * (X; Q) obstruct the rational formality of X.…”

Section: Polyhedral Products and The Golodness Of Monomial Ideal Ringsmentioning

confidence: 99%

Polyhedral products and features of their homotopy theory

Bahri¹,

Bendersky²,

Cohen³

2020

Handbook of Homotopy Theory

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A polyhedral product is a natural subspace of a Cartesian product that is specified by a simplicial complex. The modern formalism arose as a generalization of the spaces known as moment-angle complexes which were developed within the nascent subject of toric topology. This field, which began as a topological approach to toric geometry and aspects of symplectic geometry, has expanded rapidly in recent years. The investigation of polyhedral products and their homotopy theoretic properties has developed to the point where they are studied in various fields of mathematics far removed from their origin. In this survey, we provide a brief historical overview of the development of this subject, summarize many of the main results and describe applications.The paper is organized as follows. A brief technical discussion of the origin of polyhedral products as moment-angle complexes in Sections 2 and 3, includes the original topological construction of toric manifolds by M. Davis and T. Januszkiewicz [52] and the reformulation of one of their main spaces as a moment-angle complex by V. Buchstaber and T. Panov, [34,35].A discussion follows in Section 4 about the independent discovery of moment-angle complexes as intersections of certain quadrics. We describe briefly the work of S. López de Medrano [100], S. López de Medrano and A. Verjovsky [101], F. Bosio and L. Meersseman [30], followed by that of S. López de Medrano and S. Gitler [65]. This approach has proved effective in identifying classes of moment-angle manifolds that are connected sums of products of spheres. This is followed in Section 5 by a sketch of the computation of the cohomology ring of moment-angle complexes.We begin our focus on more general polyhedral products in Section 6, by describing their behaviour with respect to the exponentiation of CW-pairs. This has led to an application to the study of toric manifolds [58,128,129,124,70,72,45], and orbifolds [18]. An application to topological joins is included as an example of the utility of this property with respect to exponentiation.The behaviour of polyhedral products with respect to fibrations, due to G. Denham and A. Suciu [55], is surveyed briefly in Section 7. We take advantage of the formalism to describe briefly the early work in this area by G. Porter [117,118], T. Ganea [64] and A. Kurosh [92]. and G. W. Whitehead. Instances of polyhedral products appear also in the work of D. Anick [5].In Section 8, we review the various fundamental unstable and stable splitting theorems for the polyhedral product. These theorems give access to the homotopy type of the polyhedral product in many of the most important cases and drive a number of applications. The identification of the various wedge summands which appear in the stable decompositions, occupies the second part of this section.A fine theorem of A. Al-Raisi [2] shows that, in certain cases, the stable splitting of the polyhedral product can be chosen to be equivariant with respect to the action of Aut(K). This and other related results, described in Sec...

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On the Homology of Certain Local Rings

Cited by 24 publications

References 7 publications

Truncations of the ring of number-theoretic functions

Truncations of the ring of number-theoretic functions

Homology over trivial extensions of commutative DG algebras

Polyhedral products and features of their homotopy theory

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