The Homotopy Obstructions in Complete Intersections

Let A be a regular ring over a field k, with 1/2 ∈ 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 : Aⁿ ↠I/I 2 is a surjective map. The goal is to define and detect homotopy obstructions, for ω to lift to a surjective map Aⁿ ↠ 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 : Eⁿ(A)↠ π0(LO(A, n)), where Eⁿ(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.