Twisted Hochschild homology and MacLane homology

We prove that Hi.A; ˆ.A// D 0, i > 0. Here A is a commutative algebra over the prime field Fp of characteristic p > 0 and ˆ.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 / D 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