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2012 Algebraic geometry of topological spaces I
Guillermo Cortiñas, Andreas Thom
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Acta Math. 209(1): 83-131 (2012). DOI: 10.1007/s11511-012-0082-6

Abstract

We use techniques from both real and complex algebraic geometry to study K-theoretic and related invariants of the algebra C(X) of continuous complex-valued functions on a compact Hausdorff topological space X. For example, we prove a parameterized version of a theorem by Joseph Gubeladze; we show that if M is a countable, abelian, cancellative, torsion-free, semi-normal monoid, and X is contractible, then every finitely generated projective module over C(X)[M] is free. The particular case $ M = \mathbb{N}_0^n $ gives a parameterized version of the celebrated theorem proved independently by Daniel Quillen and Andrei Suslin that finitely generated projective modules over a polynomial ring over a field are free. The conjecture of Jonathan Rosenberg which predicts the homotopy invariance of the negative algebraic K-theory of C(X) follows from the particular case $ M = {\mathbb{Z}^n} $. We also give algebraic conditions for a functor from commutative algebras to abelian groups to be homotopy invariant on C*-algebras, and for a homology theory of commutative algebras to vanish on C*-algebras. These criteria have numerous applications. For example, the vanishing criterion applied to nil K-theory implies that commutative C*-algebras are K-regular. As another application, we show that the familiar formulas of Hochschild–Kostant–Rosenberg and Loday–Quillen for the algebraic Hochschild and cyclic homology of the coordinate ring of a smooth algebraic variety remain valid for the algebraic Hochschild and cyclic homology of C(X). Applications to the conjectures of Beĭlinson-Soulé and Farrell–Jones are also given.

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Guillermo Cortiñas. Andreas Thom. "Algebraic geometry of topological spaces I." Acta Math. 209 (1) 83 - 131, 2012. https://doi.org/10.1007/s11511-012-0082-6

Information

Received: 18 March 2010; Revised: 19 August 2010; Published: 2012
First available in Project Euclid: 31 January 2017

zbMATH: 1266.19003
MathSciNet: MR2979510
Digital Object Identifier: 10.1007/s11511-012-0082-6

Subjects:
Primary: 13D15
Secondary: 13C10 , 46J10

Keywords: algebraic $K$-theory , algebraic approximation , projective modules , rings of continuous functions , Serre’s conjecture

Rights: 2012 © Institut Mittag-Leffler

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Vol.209 • No. 1 • 2012
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