Étale descent for hochschild and cyclic homology

Abstract

IfB is an étale extension of ak-algebraA, we prove for Hochschild homology thatHH *(B)≅HH*(A)⊗AB. For Galois descent with groupG there is a similar result for cyclic homology:HC *≅HC*(B)G if $$\mathbb{Q} \subseteq A$$ . In the process of proving these results we give a localization result for Hochschild homology without any flatness assumption. We then extend the definition of Hochschild homology to all schemes and show that Hochschild homology satisfies cohomological descent for the Zariski, Nisnevich and étale topologies. We extend the definition of cyclic homology to finite-dimensional noetherian schemes and show that cyclic homology satisfies cohomological descent for the Zariski and Nisnevich topologies, as well as for the étale topology overQ. Finally we apply these results to complete the computation of the algebraicK-theory of seminormal curves in characteristic zero.