On the cyclic homology of ringed spaces and schemes.

Keller, Bernhard

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We extend the definition of Hochschild and cyclic homologies of a scheme over a commutative ring k to define the Hochschild homologies HH⁎(X/S) and cyclic homologies HC⁎(X/S) of a scheme X with respect to an arbitrary base scheme S. Our main purpose is to study product structures on the Hochschild homology groups HH⁎(X/S). In particular, we show that $H H ⁎ (X / S) = ⨁_{n \in ℤ} H H ₙ (X / S)$ carries the structure of a graded algebra.