Algebraic geometry of topological spaces I.

GUILLERMO CORTIÑAS; ANDREAS THOM

ACTA MATHEMATICA (DJURSHOLM)

SPRINGER

Año: 2012 vol. 209 p. 83 - 131

0001-5962

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 parametrized version of a theorem of Joseph Gubeladze; we show that if $M$ is a countable, abelian, cancellative, torsion-free, seminormal monoid, and $X$ is contractible, then every finitely generated projective module over $C(X)[M]$ is free. The particular case $M=N_{0}^n$ gives a parametrized 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 $M=Z^n$. 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 Beui linson-Soul´e and Farrell-Jones are also given.