Serre-Swan theorem for non-commutative C∗-algebras. Revised edition

Serre-Swan Theorem for non commutative C∗-algebras

The Standard Model – the Commutative Case: Spinors, Dirac Operator and De Rham Algebra

The present paper is a short survey on the mathematical basics of Classical Field Theory including the Serre-Swan’ theorem, Clifford algebra bundles and spinor bundles over smooth Riemannian manifolds, Spin-structures, Dirac operators, exterior algebra bundles and Connes’ differential algebras in the commutative case, among other elements. We avoid the introduction of principal bundles and put ...

On Simplicial Commutative Algebras with Vanishing André-quillen Homology

In this paper, we study the André-Quillen homology of simplicial commutative l-algebras, l a field, having certain vanishing properties. When l has non-zero characteristic, we obtain an algebraic version of a theorem of J.-P. Serre and Y. Umeda that characterizes such simplicial algebras having bounded homotopy groups. We further discuss how this theorem fails in the rational case and, as an ap...

Serre Finiteness and Serre Vanishing for Non-commutative P-bundles

Suppose X is a smooth projective scheme of finite type over a field K, E is a locally free OX -bimodule of rank 2, A is the non-commutative symmetric algebra generated by E and ProjA is the corresponding non-commutative P -bundle. We use the properties of the internal Hom functor HomGrA(−,−) to prove versions of Serre finiteness and Serre vanishing for ProjA. As a corollary to Serre finiteness,...

Poincaré Duality and Spinc Structures for Complete Noncommutative Manifolds

We prove a noncompact Serre-Swan theorem characterising modules which are sections of vector bundles not necessarily trivial at infinity. We then identify the endomorphism algebras of the resulting modules. The endomorphism results continue to hold for the generalisation of these modules to noncommutative, nonunital algebras. Finally, we apply these results to not necessarily compact noncommuta...