By E. M. Friedlander, M. R. Stein

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**Example text**

The following is a geometric example. 5 (Picard groups of algebras of functions) Let A = C ∞ (M ) be the algebra of smooth complex-valued functions on a manifold M . Using the Serre-Swan identiﬁcation of smooth complex vector bundles over M with projective modules over A, one can check that SPic(A) coincides with Pic(M ), the group of isomorphism classes of complex line bundles on M , which is isomorphic to H 2 (M, Z) via the Chern class map. We then have a purely geometric description of Pic(A) as Pic(C ∞ (M )) = Diﬀ(M ) H 2 (M, Z), (37) where the action of Diﬀ(M ) on H 2 (M, Z) is given by pull back.

This path lifting property suggests that complete Poisson maps play the role of “coverings” in Poisson geometry. This idea is borne out by some of the examples below. 23 (Complete functions) Let us regard R as a Poisson manifold, equipped with the zero Poisson bracket. ) Then any map f : P → R is a Poisson map, which is complete if and only if Xf is a complete vector ﬁeld. Observe that the notion of completeness singles out the subset of C ∞ (P ) consisting of complete functions, which is preserved under complete Poisson maps.

15 holds for n = 2 [75], but not in general. In fact, for n = 3, one can ﬁnd Π and Π , not in the same SO(n, n|Z)orbit, for which AΠ and AΠ are isomorphic (hence Morita equivalent) [77]. 15 and its converse hold with respect to a reﬁned notion of Morita equivalence, called “complete Morita equivalence” [78], in which bimodules carry connections of constant curvature. For the algebraic Morita equivalence of smooth quantum tori, see [36]. 15 was proven under an additional hypothesis. Rieﬀel and Schwarz consider three types of generators of SO(n, n|Z), and prove that their action preserves Morita equivalence.

### Algebraic K-Theory by E. M. Friedlander, M. R. Stein

by Brian

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