Hochschild cohomology algebra of abelian groups

1997 ◽  
Vol 68 (1) ◽  
pp. 17-21 ◽  
Author(s):  
Claude Cibils ◽  
Andrea Solotar
Keyword(s):  
Hochschild Cohomology ◽  
Abelian Groups ◽  
Cohomology Algebra ◽  
Hochschild Cohomology Algebra

scholarly journals Extreme nonuniqueness of end-sum

2020 ◽  
pp. 1-43
Author(s):  
Jack S. Calcut ◽  
Craig R. Guilbault ◽  
Patrick V. Haggerty
Keyword(s):  
Abelian Groups ◽  
Cohomology Algebra ◽  
Homotopy Types ◽  
Proper Homotopy ◽  
Open Formula

We give explicit examples of pairs of one-ended, open [Formula: see text]-manifolds whose end-sums yield uncountably many manifolds with distinct proper homotopy types. This answers strongly in the affirmative a conjecture of Siebenmann regarding nonuniqueness of end-sums. In addition to the construction of these examples, we provide a detailed discussion of the tools used to distinguish them; most importantly, the end-cohomology algebra. Key to our Main Theorem is an understanding of this algebra for an end-sum in terms of the algebras of summands together with ray-fundamental classes determined by the rays used to perform the end-sum. Differing ray-fundamental classes allow us to distinguish the various examples, but only through the subtle theory of infinitely generated abelian groups. An appendix is included which contains the necessary background from that area.

scholarly journals Hochschild Cohomology, Monoid Objects and Monoidal Categories

2020 ◽  
Author(s):  
Magnus Hellstrøm-Finnsen
Keyword(s):  
Hochschild Cohomology ◽  
Abelian Groups ◽  
Cochain Complex ◽  
Monoidal Categories ◽  
Equivalent Formulation ◽  
Theoretical Formulation ◽  
Cohomology Monoid

Abstract This paper expands further on a category theoretical formulation of Hochschild cohomology for monoid objects in monoidal categories enriched over abelian groups, which has been studied in Hellstrøm-Finnsen (Commun Algebra 46(12):5202–5233, 2018). This topic was also presented at ISCRA, Isfahan, Iran, April 2019. The present paper aims to provide a more intuitive formulation of the Hochschild cochain complex and extend the definition to Hochschild cohomology with values in a bimodule object. In addition, an equivalent formulation of the Hochschild cochain complex in terms of a cosimplicial object in the category of abelian groups is provided.

scholarly journals DEFINABLY COMPACT ABELIAN GROUPS

2004 ◽  
Vol 04 (02) ◽  
pp. 163-180 ◽  
Author(s):  
MÁRIO J. EDMUNDO ◽  
MARGARITA OTERO
Keyword(s):  
Fundamental Group ◽  
Abelian Groups ◽  
Exterior Algebra ◽  
Real Closed Field ◽  
Cohomology Algebra ◽  
Torsion Subgroup ◽  
Closed Field ◽  
Dimensional Group ◽  
Compact Abelian Groups

Let M be an o-minimal expansion of a real closed field. Let G be a definably compact definably connected abelian n-dimensional group definable in M. We show the following: the o-minimal fundamental group of G is isomorphic to ℤn; for each k>0, the k-torsion subgroup of G is isomorphic to (ℤ/kℤ)n, and the o-minimal cohomology algebra over ℚ of G is isomorphic to the exterior algebra over ℚ with n generators of degree one.

scholarly journals On inverse categories and transfer in cohomology

2012 ◽  
Vol 56 (1) ◽  
pp. 187-210 ◽  
Author(s):  
Markus Linckelmann
Keyword(s):  
Hochschild Cohomology ◽  
Abelian Groups ◽  
Mackey Functor ◽  
Symmetric Algebras ◽  
Transfer Maps ◽  
Category Algebras

AbstractIt follows from methods of B. Steinberg, extended to inverse categories, that finite inverse category algebras are isomorphic to their associated groupoid algebras; in particular, they are symmetric algebras with canonical symmetrizing forms.We deduce the existence of transfer maps in cohomology and Hochschild cohomology from certain inverse subcategories. This is in part motivated by the observation that, for certain categories $\mathcal{C}$, being a Mackey functor on $\mathcal{C}$ is equivalent to being extendible to a suitable inverse category containing $\mathcal{C}$. We further show that extensions of inverse categories by abelian groups are again inverse categories.

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