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Cyclic homology and nonsingularity. (English) Zbl 0838.19002
This is the second of a series of two articles [together with “Algebraic extensions and nonsingularity”, ibid. 8, No. 2, 251-289 (1995; see the review above)] describing a new approach to cyclic homology. (Periodic) cyclic homology is developed here as noncommutative analogue of algebraic de Rham cohomology (for not necessarily smooth varieties). It is known that the algebraic de Rham complex is well behaved only for smooth varieties. In the general case one uses the following strategy. Any variety \(X\) can be imbedded into a nonsingular variety \(Y\) and it turns out that the cohomology of the de Rham complex of the formal neighbourhood \(\widehat {X}_Y\) of \(X\) in \(Y\) is independent of the choice of \(Y\) and the imbedding. This cohomology is taken by definition as the de Rham cohomology of \(X\) and coincides for nonsingular varieties with the usual definition. For an affine variety \(X\) with coordinate ring \(A\) this means: Realise \(A\) as quotient of a smooth algebra \(R: 0\to I\to R\to A\to 0\) and consider the algebraic de Rham cohomology of the \(I\)-adic completion \(\widehat {R}:= \varprojlim R/I^n\) as the de Rham cohomology of \(A\).
Thus the program of defining a noncommutative de Rham cohomology is divided into the following steps: 1. Find the notion of nonsingularity for noncommutative algebras. 2. Define the correct de Rham complex for smooth algebras. 3. Proceed then as in the commutative case.
In the preceding article the authors showed that the notion of a (formally) smooth algebraic geometry corresponds to the notion of a quasifree algebra in the category of (not necessarily commutative) algebras.
The correct noncommutative de Rham cohomology is provided by Connes’ periodic cyclic (co)homology \(HP\) so that one is tempted to define the “de Rham complex” of a smooth (= quasifree) algebra \(R\) as the cyclic bicomplex \(CC_* (R)\) of Connes. As quasifree algebras are precisely the algebras of Hochschild cohomological dimension at most one, however (see the review above) the cyclic complex of a quasifree algebra contains a large acyclic subcomplex and becomes quasiisomorphic to a very small quotient, the authors’ \(X\)-complex: \[ \begin{aligned} X_* R&:=\qquad \to R@>d\;>>\Omega^1 R/[\Omega^1 R,R]@>b\;>>R\to\\ d(x) &:= dx;\qquad b(\overline{x dy}):= [x, y].\end{aligned} \] This finally leads to the following definition: If \(A\) is an algebra, let \(0\to I\to R\to A\to 0\) be any extension with \(R\) quasifree. Observe that the \(I\)-adic completion \(\widehat {R}\) is still quasifree (see the review above) and put \[ H_* (A):= H_* (X_* (\widehat {R})). \] In this article the authors show that if one takes the universal quasifree resolution \[ 0\to IA\to RA\to A\to 0 \] of \(A\) (with \(RA\) the reduced tensor algebra of \(A\)) then the cyclic bicomplex \(CC_* (A)\) becomes a natural deformation retract of the \(X\)-complex \(X_* (\widehat {RA})\) of the \(I\)-adic completion of \(RA\). Thus \[ HP_* (A)\simeq H_* (X_* (\widehat {RA})). \] Moreover, the cohomology is independent of the chosen quasifree resolution. This approach to cyclic (co)homology has the merit of replacing the rather complicated cyclic bicomplex \(CC_*\) by the much simpler \(X\)-complex at the cost of replacing the original algebra by the \(I\)-adic completion of its tensor algebra which is quite big but not very difficult to understand. At the same time one obtains a new definition of bivariant cyclic (co)homology as the cohomology of the complex of continuous maps between the individual \(X\)-complexes with their adic topologies: \[ HP^* (A,B):= H_* (\operatorname{Hom}^*_{\text{cont}} (X_* (\widehat {RA}), X_* (\widehat {RB})). \] Note that this differs from the bivariant theory of Jones–Kassel [cf. J. D. S. Jones and C. Kassel, \(K\)-Theory 3, No. 4, 339-365 (1989; Zbl 0755.18008)]. An immediate consequence of the definition is the generalized Goodwillie theorem that a surjection \(A\to B\) with nilpotent kernel induces isomorphisms on bivariant periodic cyclic cohomology. Such a result necessarily has to hold if the bivariant theory should satisfy excision but fails in the Jones–Kassel theory. Besides this the article contains a simple treatment of Nistor’s bivariant Chern character of a finitely summable Fredholm module. The Cuntz-Quillen approach allows a very intriguing interpretation of the Chern character in \(K\)-theory. The article also deals with the Cartan homotopy formula in periodic cyclic theory for polynomial (smooth) homotopies and with Künneth type results on the periodic cyclic (co)homology of a tensor product.

MSC:
19D55 \(K\)-theory and homology; cyclic homology and cohomology
14A22 Noncommutative algebraic geometry
46L85 Noncommutative topology
46L87 Noncommutative differential geometry
19L10 Riemann-Roch theorems, Chern characters
14F40 de Rham cohomology and algebraic geometry
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