A Course in Homological Algebra

Hilton, Peter; Stammbach, Urs

doi:10.1007/978-1-4419-8566-8

Cited by 293 publications

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“…Here we proceed in a more foundational manner. We show that solving linear systems is, as expected, the key to the complete computability of the category of finitely presented modules, merely viewed as an Abelian category [HS97,Rot09,Wei94].…”

Section: Introductionsupporting

confidence: 63%

An Axiomatic Setup for Algorithmic Homological Algebra and an Alternative Approach to Localization

Barakat

Lange-Hegermann

2011

J. Algebra Appl.

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Abstract. In this paper we develop an axiomatic setup for algorithmic homological algebra of Abelian categories. This is done by exhibiting all existential quantifiers entering the definition of an Abelian category, which for the sake of computability need to be turned into constructive ones. We do this explicitly for the often-studied example Abelian category of finitely presented modules over a so-called computable ring R, i.e., a ring with an explicit algorithm to solve one-sided (in)homogeneous linear systems over R. For a finitely generated maximal ideal m in a commutative ring R we show how solving (in)homogeneous linear systems over R m can be reduced to solving associated systems over R. Hence, the computability of R implies that of R m . As a corollary we obtain the computability of the category of finitely presented R m -modules as an Abelian category, without the need of a Mora-like algorithm. The reduction also yields, as a by-product, a complexity estimation for the ideal membership problem over local polynomial rings. Finally, in the case of localized polynomial rings we demonstrate the computational advantage of our homologically motivated alternative approach in comparison to an existing implementation of Mora's algorithm.

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Section: Introductionsupporting

confidence: 63%

An Axiomatic Setup for Algorithmic Homological Algebra and an Alternative Approach to Localization

Barakat

Lange-Hegermann

2011

J. Algebra Appl.

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“…e.g. [HS97,§6]) and by Lemma 2.1 both groups are free abelian groups. Now note that without loss of generality we can assume that A ⊂ B and that ϕ is the inclusion map.…”

Section: 2mentioning

confidence: 96%

Twisted Alexander polynomials detect fibered 3-manifolds

Friedl

Vidussi

2011

Ann. Math.

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A classical result in knot theory says that for a fibered knot the Alexander polynomial is monic and that the degree equals twice the genus of the knot. This result has been generalized by various authors to twisted Alexander polynomials and fibered 3-manifolds. In this paper we show that the conditions on twisted Alexander polynomials are not only necessary but also sufficient for a 3-manifold to be fibered. By previous work of the authors this result implies that if a manifold of the form S 1 ×N 3 admits a symplectic structure, then N fibers over S 1 . In fact we will completely determine the symplectic cone of S 1 × N in terms of the fibered faces of the Thurston norm ball of N .

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“…Conversely, from a tower as in (16), we can construct a sequence of finite Hirsch extensions M i as in (14). Furthermore, the dga M i , with differential defined by (15), coincides with the Chevalley-Eilenberg complex ( (L * i ), d) associated to the finite-dimensional Lie algebra L i = L(M i ), as in [23,§II] and [35,§VII]. In particular,…”

Section: 2mentioning

confidence: 99%

Formality properties of finitely generated groups and Lie algebras

Suciu

Wang

2019

Forum Mathematicum

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We explore the graded-formality and filtered-formality properties of finitely generated groups by studying the various Lie algebras over a field of characteristic 0 attached to such groups, including the Malcev Lie algebra, the associated graded Lie algebra, the holonomy Lie algebra, and the Chen Lie algebra. We explain how these notions behave with respect to split injections, coproducts, direct products, as well as field extensions, and how they are inherited by solvable and nilpotent quotients. A key tool in this analysis is the 1-minimal model of the group, and the way this model relates to the aforementioned Lie algebras. We illustrate our approach with examples drawn from a variety of group-theoretic and topological contexts, such as finitely generated torsion-free nilpotent groups, link groups, and fundamental groups of Seifert fibered manifolds.

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A Course in Homological Algebra

Cited by 293 publications

References 0 publications

An Axiomatic Setup for Algorithmic Homological Algebra and an Alternative Approach to Localization

An Axiomatic Setup for Algorithmic Homological Algebra and an Alternative Approach to Localization

Twisted Alexander polynomials detect fibered 3-manifolds

Formality properties of finitely generated groups and Lie algebras

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