Algebraic Geometry of Topological Spaces I

TL;DR

This paper explores the algebraic geometry of topological spaces to analyze K-theoretic invariants of continuous function algebras, proving new theorems and establishing criteria with broad applications in algebraic topology and K-theory.

Contribution

It introduces new algebraic conditions and parametrized theorems connecting algebraic geometry with topological invariants of C(X), extending classical results to a topological setting.

Findings

Finiteness of projective modules over C(X)[M] when X is contractible

Validation of classical homology formulas for C(X)

Criteria for homotopy invariance and K-regularity of C*-algebras

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 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.