Algebraic geometry of topological spaces I

作者：Cortinas Guillermo^*; Thom Andreas

来源：Acta Mathematica, 2012, 209(1): 83-131.

DOI：10.1007/s11511-012-0082-6

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 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 . 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 BeAlinson-Soul, and Farrell-Jones are also given.

• 出版日期2012-9