Homology of Maximal Orders in Central Simple Algebras

and moreover that HH2(O/R) = 0. These two facts greatly facilitate the computation of the cyclic homology HC∗(O/R) ([3]). This paper presents non-commutative analogues of the main results of [3]. In particular, L is to be replaced by a central simple algebra D over K, while O is taken to be a maximal R-order in D. Of course, O is not, in general, uniquely defined by R and D, but it turns out that the homology of O/R is independent of the choice of maximal order. We prove that (0.1) remains valid in the non-commutative context but that the odd Hochschild homology groups vanish. The first section is devoted to generalities on Hochschild homology. We do not claim to prove any new result but only to establish notation and give self-contained proofs of various facts for which we could not find convenient references. The main results, the periodicity theorem and the vanishing theorem, are proved in §2 when R is a complete discrete valuation ring with perfect residue field. We construct an element in Hochschild cohomology such that Yoneda product gives the desired periodicity. The calculations needed to compute the low-dimensional Hochschild homology groups are laborious, but the resulting formulae are quite simple. The globalization is carried out in §3 and is completely standard. The vanishing of odd Hochschild homology guarantees that the Loday-Quillen spectral sequence degenerates at E, so it is easy to compute cyclic homology, in both the local and the global case, up to extension. This is explained in §4, where the extension problem is partially solved by an application of the universal coefficient theorem.