A non-Golod ring with a trivial product on its Koszul homology

Katthän, Lukas

doi:10.1016/j.jalgebra.2017.01.042

Cited by 32 publications

(36 citation statements)

References 24 publications

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“…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%

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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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Section: Polyhedral Products and The Golodness Of Monomial Ideal Ringsmentioning

confidence: 99%

“…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%

Polyhedral products and features of their homotopy theory

Bahri¹,

Bendersky²,

Cohen³

2020

Handbook of Homotopy Theory

View full text Add to dashboard Cite

show abstract

“…It is well-known that, for homogeneous ideals inside polynomial rings in three variables, being Golod is equivalent to requiring that the product on the Koszul homology is trivial (for instance, see [15,Theorem 6.3]). In the same article, it is shown that this is not the case more generally, even for monomial ideals.…”

Section: Characterization Of Monomial Golod Ideals In Three Variablesmentioning

confidence: 99%

“…The notion is defined and studied extensively in the local setting, but in this paper we shall restrict ourselves to the graded situation. Golod rings and ideals have attracted increasing attention recently (see [6,8,9,12,15]), but they remain mysterious even when n = 3. For instance, we do not know if the product of any two homogeneous ideals in Q = k[x, y, z] is Golod.…”

Section: Introductionmentioning

confidence: 99%