Mathematics

To Be Announced

Pierre Will (Institut Fourier)

Abstract: TBA

To Be Announced

Alastair Fletcher (Northern Illinois University)

Splitting property of projective modules, by Homotopy obstructions

Satya Mandal (University of Kansas)

Abstract: The theory of vector bundles on compact hausdorff spaces X, guided the research on projective modules over noetherian commutative rings A. There has been a steady stream of results on projective modules over A, that were formulated by imitating existing results on vector bundles on X. The first part of this talk would be a review of this aspects of results on projective modules, leading up to some results on splitting projective Amodules P, as direct sum P ∼= Q ⊕ A. Our main interest in this talk is to define an obstruction class ε(P) in a suitable obstruction set (preferably a group), to be denoted by π0 (LO(P)). Under suitable smoothness and other conditions, we prove that ε(P) is trivial ⇐⇒ P ∼= Q ⊕ A Under similar conditions, we prove π0 (LO(P)) has an additive structure, which is associative, commutative and has n unit (a "monoid"). In deed, LO(P) =  (I, ω) : I ⊆ A is an ideal, and ω : P  I I 2 is a surjective map.  The two maps LO(P) LO(P ⊗ A[T]) T =1 / T =0 o LO(P) induce a chain homotopy equivalence on LO(P), and the set of equivalence classes is defined to be π0 (LO(P)). I