Homotopy obstructions for projective modules

Homotopy obstructions for projective modules

Satya Mandal, University of Kansas, Lawrence, KS 66045

Wash U., 14 September 2018

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)). This theory emanates out of some germs of ideas given by Madhav V. Nori (around 1990).

Host: Mohan Kumar