Algebraic geometry of topological spaces. I. (English) Zbl 1266.19003

Cortiñas, Guillermo; Thom, Andreas

Acta Math. 209, No. 1, 83-131 (2012).

The paper deals with homotopy invariance of functors on algebras over the field \(\mathbb C\) of complex numbers. Let \(M\) be a countable cancellative torsion-free seminormal abelian monoid, \(X\) a compact Hausdorff space. The first main result of the paper is local triviality of the bundles of finitely generated free \(\mathbb C[M]\)-modules over \(X\) that are direct summands of trivial bundles. In particular, if \(X\) is contractible then every finitely generated projective module over \(C(X)[M]\) is free. The case \(M=\mathbb N_0^n\) gives a version of the Quillen-Suslin theorem on freeness of finitely generated projective modules over the Laurent polynomial ring, and the case \(M=\mathbb Z^n\) allows the authors to prove the J. Rosenberg’s conjecture on homotopy invariance of the negative algebraic \(K\)-theory groups of \(C(X)\). The second main result of the paper is the following criterion for homotopy invariance. Let \(F\) be a functor on the category \(\mathfrak C\mathfrak o\mathfrak m\mathfrak m/\mathbb C\) of commutative \(\mathbb C\)-algebras with values in the category \(\mathfrak A\mathfrak b\) of abelian groups. If \(F\) is split-exact on \(C^*\)-algebras, vanishes on coordinate rings of smooth affine varieties, and commutes with filtering colimits then the functor \(X\mapsto F(C(X))\) is homotopy invariant. This result is used to derive a vanishing theorem and some other useful results for homology theories, in particular, for Hochschild and cyclic homology.

Reviewer: Vladimir M. Manuilov (Moskva)

MSC:

19D10	Algebraic \(K\)-theory of spaces
14P10	Semialgebraic sets and related spaces
19D35	Negative \(K\)-theory, NK and Nil
19D55	\(K\)-theory and homology; cyclic homology and cohomology

Keywords:

homotopy invariance; algebraic \(K\)-theory; semialgebraic set; homology theories

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