  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 when M=N^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 when 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 Beilinson-Soule and Farrell-Jones are also
given.
