Hochschild and cyclic homology are far from being homotopy functors Academic Article

Given a homology theory H ( A ) {H_*}(A) on rings, based on a natural chain complex, one can form a new theory H h ( A ) H_*^h(A) which is universal with respect to the homotopy property H ( A ) H ( A [ t ] ) {H_*}(A) simeq {H_*}(A[t]) . We show that the homotopy theories H H h HH_*^h and H C h HC_*^h associated to Hochschild and cyclic homology are both zero. On the other hand, if H C HC_*^ - denotes Goodwillies variant of cyclic homology, and A A contains a field of characteristic 0, we show that ( H C ) h A (H{C^ - })_*^hA is Connes periodic cyclic homology H P ( A )