On triviality of the Euler class group of a deleted neighbourhood of a smooth local scheme

Abstract

Let k be an infinite perfect field and let X = SpecR, where R is the localization of a k-algebra B of finite type with respect to a regular prime ideal p ∈ SpecB. LetD be a divisor onX with a decompositionD = ∪i=1Di into its irreducible components Di such that each Di is nonsingular and they intersect transversally. Consider Y = X − D, a deleted neighbourhood of the closed point x0 of X . Let E be a vector bundle on Y . In this set up, E is trivial in the following cases : (i) dim(X) ≤ 3 (Gabber [Ga1]) ; (ii) t = 1 (Bhatwadekar-Rao [BR]) ; (iii) rankE ≥ min{[dimR 2 ], t} (Rao [Ra1]). The above exposition is borrowed from a paper of Nisnevich [Ni], which contains some beautiful applications of these ‘purity theorems’. In the same set up, now let y ∈ Y be a closed point and let my be the corresponding maximal ideal of OY,y. It is natural to ask whether my is a complete intersection. Observing that the conormal bundle my/my is trivial, we consider a more general situation : Let I be the defining ideal of a closed subscheme Z of Y such that the conormal bundle I/I2 (defined on Z) is trivial. Is I a complete intersection? The questions we are addressing in this paper are a little stronger in nature : Can a given basis of I/I2 be lifted to a minimal set of generators of I? We remark that with the hypothesis on X , each component Di is defined by a single equation fi which vanishes at the closed point x0 of X . Let m be the maximal ideal at x0. The condition of transversality of intersection of Di’s is equivalent to saying that the images of f1, · · · , ft in the A/m-vector space m/m2 are linearly independent. Keeping this discussion in mind, we now give an excerpt of our main results in commutative algebraic terms (see 4.2, 4.3, 4.4).