Algebraic geometry of topological spaces I |
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Authors: | Guillermo Cortiñas Andreas Thom |
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Institution: | 1.Departamento de Matemática,Universidad de Buenos Aires,Buenos Aires,Argentina;2.Mathematisches Institut,Universit?t Leipzig,Leipzig,Germany |
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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 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 Beĭlinson-Soulé and Farrell–Jones are also given. |
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Keywords: | |
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