Hochschild homology, Frobenius homomorphism and mac lane homology

Abstract

We prove that H i(A,Φ(A))=0, i>0. Here A is a commutative algebra over the prime field Fp 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 Fp. This result has implications in Mac Lane homology theory. Among other results, we prove that HML •(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 < Card(K). © 2007 Mathematical Sciences Publishers.