Hochschild homology, Frobenius homomorphism and Mac Lane homology

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 $1<d<Card\left(K\right)$.