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Cyclic homology and nonsingularity

Joachim Cuntz, Daniel Quillen

1995
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Journal of The American Mathematical Society
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From the pioneering work of Connes [Col] one knows that periodic cyclic homology can be regarded as a natural extension of de Rham cohomology to the realm of noncommutative geometry. Our aim in this paper is to present the noncommutative analogue of the approach of Deligne [D] and Hartshorne [H] to de Rham cohomology in algebraic geometry. In this approach de Rham cohomology is first obtained for a nonsingular algebraic variety by means of the de Rham complex of differential forms. An arbitrary
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... variety is then treated by embedding it in a nonsingular variety and completing the de Rham complex of the latter along the subvariety. In our noncommutative version algebraic varieties are replaced by associative unital algebras over the complex numbers, and nonsingular varieties become algebras which are quasi-free [CQ1]. Indeed, nonsingular varieties are described locally by commutative algebras which behave like free commutative algebras with respect to nilpotent extensions of commutative algebras, while quasi-free algebras are those algebras behaving like free algebras relative to nilpotent algebra extensions. Like a free algebra, a quasi-free algebra R has cohomological dimension::; I with respect to Hochschild cohomology, and this implies that its periodic cyclic homology H PIJR, v E Z/2 , is calculated by the supercomplex (1) discussed for free algebras and co algebras in [Ql]. This means that X(R) for R quasi-free plays the role in the noncommutative setting of the de Rham complex of a nonsingular variety. Our version of the way de Rham cohomology can be obtained by embedding into a nonsingular variety says that for any algebra extension A = R/ I with R quasi-free we have a canonical isomorphism (2) HP*A = H* (~X(R/ln)) . As an immediate consequence we deduce Goodwillie's theorem [G] that a nilpotent extension A' -+ A gives rise to an isomorphism on periodic cyclic homology. In fact, as we show in § 10, our methods yield a refinement of this theorem in which an inverse with respect to cup product for the homomorphism A' -+ A is constructed in bivariant periodic cyclic cohomology. For this result to be valid, it is necessary to define bivariant periodic cyclic cohomology in a

doi:10.1090/s0894-0347-1995-1303030-7
fatcat:rbobsdxi7jc4lepediwhq73ih4