Hochschild homology, Frobenius homomorphism and Mac Lane homology

Pirashvili, Teimuraz

doi:10.2140/agt.2007.7.1071

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Hochschild homology, Frobenius homomorphism and Mac Lane homology

Teimuraz Pirashvili

Algebraic & Geometric Topology 7 (2007) 1071–1079

DOI: 10.2140/agt.2007.7.1071

arXiv: math.KT/0703376

Abstract

We prove that $H_{i} (A, Φ (A)) = 0$ , $i > 0$ . Here $A$ is a commutative algebra over the prime field $F_{p}$ of characteristic $p > 0$ and $Φ (A)$ is $A$ considered as a bimodule, where the left multiplication is the usual one, while the right multiplication is given via Frobenius endomorphism and $H_{∙}$ denotes the Hochschild homology over $F_{p}$ . This result has implications in Mac Lane homology theory. Among other results, we prove that ${H M L}_{∙} (A, T) = 0$ , provided $A$ is an algebra over a field $K$ of characteristic $p > 0$ and $T$ is a strict homogeneous polynomial functor of degree $d$ with $1 < d < C a r d (K)$ .

Keywords

Hochschild Homology, Mac Lane homology

Mathematical Subject Classification 2000

Primary: 55P43, 16E40

Secondary: 19D55, 55U10

References

Publication

Received: 14 March 2007

Accepted: 26 March 2007

Published: 2 August 2007

Authors

Teimuraz Pirashvili
Department of Mathematics University of Leicester Leicester LE1 7RH UK

Volume 7, issue 2 (2007)