Acta Mathematica

Algebraic geometry of topological spaces I

Guillermo Cortiñas and Andreas Thom

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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.

Article information

Source
Acta Math., Volume 209, Number 1 (2012), 83-131.

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

Permanent link to this document
https://projecteuclid.org/euclid.acta/1485892647

Digital Object Identifier
doi:10.1007/s11511-012-0082-6

Mathematical Reviews number (MathSciNet)
MR2979510

Zentralblatt MATH identifier
1266.19003

Subjects
Primary: 13D15: Grothendieck groups, $K$-theory [See also 14C35, 18F30, 19Axx, 19D50]
Secondary: 13C10: Projective and free modules and ideals [See also 19A13] 46J10: Banach algebras of continuous functions, function algebras [See also 46E25]

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

Rights
2012 © Institut Mittag-Leffler

Citation

Cortiñas, Guillermo; Thom, Andreas. Algebraic geometry of topological spaces I. Acta Math. 209 (2012), no. 1, 83--131. doi:10.1007/s11511-012-0082-6. https://projecteuclid.org/euclid.acta/1485892647


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