The Homotopy Obstructions in Complete Intersections

Abstract

Let A be a regular ring over a field k, with 1/2 Є k, and dim A = d. We discuss the Homotopy Obstruction Program, in the complete intersection case. Fix an integer n ≥ 2. A local orientation is a pair (I,ω), where I is an ideal and ω - A^n ↠ I/I^2 is a surjective map. The goal is to define and detect homotopy obstructions, for ω to lift to a surjective map A^n ↠ I . Denote the set of all local orientations by LO(A, n). A homotopy relations on LO(A, n) is induced by the maps LO(A, n)←T=0 LO(A[T ], n) →T=1 LO(A, n). The homotopy obstruction set π0(LO(A, n)) is defined to be the set of all equivalence classes. Assume 2n ≥d+2. We prove that π0(LO(A, n)) is an abelian group. We also establish a surjective map ρ - E^n(A)a↠ π0(LO(A, n)), where E^n(A) denotes the Euler class group. When 2n ≥ d + 3, and A is essentially smooth, we prove ρ is an isomorphism. This settles a conjecture of Morel.