Finitely generated cohomology Hopf algebras and torsion
type: Item
instance of: scholarly article
title: Finitely generated cohomology Hopf algebras and torsion
zbMATH Open document ID: 0849.57034
author: James P. Lin
published in: Pacific Journal of Mathematics
publication date: 1996-08-29T00:00:00Z
review text: Let \(p\) be a prime. We begin a study of H-spaces whose cohomology mod \(p\) is finitely generated as an algebra. Such cohomology algebras are Hopf algebras and hence have Borel decompositions into tensor products of exterior algebras, truncated polynomial algebras and free polynomial algebras. The existence of \(p\)-torsion in the integral cohomology \(H^*(X; Z)\) plays a role in determining if the mod \(p\) cohomology \(H^*(X; Z_p)\) is finite-dimensional. In the following theorems, let \(X\) be a simply connected H-space whose mod \(p\) cohomology is finitely generated as an algebra. We prove: Theorem A. Generators of infinite height in the Borel decomposition lie in degrees \(2p^j\), for \(j \geq 0\). Theorem B. If \(X\) is two-connected and \(p^rH^*(X;Z)\) is torsion free for some \(r \geq 1\), then \(H^*(X; Z_p)\) is finite dimensional. Theorem C. Let \(X\) be two-connected. If \(H^* (X;Z_p)\) is not finite dimensional, then \(H^*(X;Z)\) has \(p\)-torsion of all orders. Theorem D. Let \(X\) be homotopy associative with \(H^*(X; Z_p)\) primitively generated. If \(p\) is an odd prime, then all even generators lie in degrees \(2p^j\), \(j \geq 0\) and for every even generator \(x\) of degree greater than 2, \(\beta_1 x \neq 0\).
Mathematics Subject Classification ID: 57T05
Mathematics Subject Classification ID: 55P45
zbMATH DE Number: 910253
zbMATH Keywords: torsion
zbMATH Keywords: H-spaces
zbMATH Keywords: cohomology
zbMATH Keywords: Hopf algebras
MaRDI profile type: Publication
full work available at URL: pjm.1996.172.215
OpenAlex ID: W1559723760
DOI: 10.2140/PJM.1996.172.215
Recommended article: Finite generation of Hochschild homology algebras
Recommended article: Q4206909
Recommended article: Cohomology of finite-dimensional connected cocommutative Hopf algebras
Recommended article: On the hochschild cohomology of finite dimensional algebras
Recommended article: Tate and Tate-Hochschild cohomology for finite dimensional Hopf algebras.
Recommended article: Finite generation of André–Quillen (co-)homology of F-finite algebras
Recommended article: TORSION IN THE COHOMOLOGY OF FINITE H-SPACES
Recommended article: Cohomology of finite-dimensional pointed Hopf algebras
Recommended article: Finite generation of the invariants of finite dimensional Hopf algebras
Recommended article: Q4288699
description: scientific article; zbMATH DE number 910253