Hochschild cohomology and p-adic Lie groups

We carry out a detailed comparison of group cohomology and Lie algebra cohomology in the context of a compact p-adic Lie group G admitting a p-valuation omega, using Hochschild cohomology as an intermediary. As a result we provide a new spectral sequence for F-p[[G]]-bimodules W which computes H*(G, W-ad) from an E-1-page of Lie algebra cohomology. This generalizes the May spectral sequence for (one-sided) F-p[[G]]-modules. We believe our results are new even in the case where W is the trivial bimodule, in which case we can quantify at which stage the spectral sequence collapses in terms of the amplitude of omega. When G is equi-p-valued we recover the Lazard isomorphism with Lambda Hom(G, F-p) as an edge map. We include various applications, such as the computation of the Hochschild cohomology of the mod p Iwasawa algebra F-p[[G]] with coefficients in a discrete quotient F-p[[G]]/I. The mod p cohomology of the p-adic quaternion group O-D(x) is worked out in detail for p > 3 as an example.