Algebraic geometry of topological spaces I

Guillermo Cortiñas Departamento de Matemática, Universidad de Buenos Aires Andreas Thom Mathematisches Institut, Universität Leipzig

K-Theory and Homology mathscidoc:1701.01002

Acta Mathematica, 209, (1), 83-131, 2010.3
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.
algebraic; Serre’s conjecture; projective modules; rings of continuous functions; algebraic approximation
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  title={Algebraic geometry of topological spaces I},
  author={Guillermo Cortiñas, and Andreas Thom},
  booktitle={Acta Mathematica},
Guillermo Cortiñas, and Andreas Thom. Algebraic geometry of topological spaces I. 2010. Vol. 209. In Acta Mathematica. pp.83-131.
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