#### Volume 7, issue 2 (2007)

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Hochschild homology, Frobenius homomorphism and Mac Lane homology

### Teimuraz Pirashvili

Algebraic & Geometric Topology 7 (2007) 1071–1079
 arXiv: math.KT/0703376
##### Abstract

We prove that ${H}_{i}\left(A,\Phi \left(A\right)\right)=0$, $i>0$. Here $A$ is a commutative algebra over the prime field ${\mathbb{F}}_{p}$ of characteristic $p>0$ and $\Phi \left(A\right)$ is $A$ considered as a bimodule, where the left multiplication is the usual one, while the right multiplication is given via Frobenius endomorphism and ${H}_{\bullet }$ denotes the Hochschild homology over ${\mathbb{F}}_{p}$. This result has implications in Mac Lane homology theory. Among other results, we prove that ${HML}_{\bullet }\left(A,T\right)=0$, 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.

##### Keywords
Hochschild Homology, Mac Lane homology
##### Mathematical Subject Classification 2000
Primary: 55P43, 16E40
Secondary: 19D55, 55U10