Gabber's Presentation Lemma

Abstract

Gabber’s presentation lemma is a foundational result in A 1 -homotopy theory. This result can be thought of as an algebro-geometric analog of the tubular neighborhood theorem in differential geometry. Similar to tubular neighbourhood theorem, this lemma gives the local model of the inclusion of a closed subscheme into a smooth scheme. The lemma was proved in 1994 by O. Gabber in the case where the base is a spectrum of an infinite field. We present a proof when the base is a finite field. Further in 2018, S. Schmidt and F. Strunck proved Gabber’s presentation lemma over the Henslian discrete valuation rings. We further generalize this result over any noetherian domain with all its residue fields infinite. We also discuss various applications of this lemma in A 1 -homotopy theory, which includes a connectivity result.