Periodic Cyclic Homology as Sheaf Cohomology

0. Introduction. This paper continues the study of the noncommutative innn-itesimal cohomology we introduced in 3]. This is the cohomology of sheaves on a noncommutative version of the commutative innnitesimal site of Grothendieck ((8]). Grothendieck showed that, for a smooth scheme X of characteristic zero, the cohomology of the structure sheaf on the innnitesimal site gives de Rham cohomol-ogy: (1) H inf (X; O) = H dR (X) Here we prove that, for any associative, not necessarily unital algebra A over Q , the noncommutative innnitesimal cohomology of the structure sheaf modulo com-mutators gives periodic cyclic homology (Theorem 3.0): (2) H inf (A; O=O; O]) = HP (A) We view (2) as a noncommutative aane version of (1). Indeed, the noncommutative analogue of smoothness is quasi-freedom, and for quasi-free algebras HP agrees with (Karoubi's deenition of) noncommutative de Rham cohomology, i.e. we have: for R quasi-free. Here (R) is the DGA of noncommutative forms. Grothendieck's theorem includes also a description of the cohomology of the de Rham complex (E O X X ; r) associated with a bundle E with a at connection as the cohomology of a certain module on the innnitesimal site; in fact (1) is a obtained from this by setting E = O X , r = d. A noncommutative version of Grothendieck's theorem for the de Rham cohomology of at bundles is proved in Theorem 2.3. It expresses the cohomology of the complex H (E ~ R (R); r) associated with a right module