Nice triples and a moving lemma for motivic spaces

TL;DR

The paper proves the strict homotopy invariance of certain cohomology sheaves for smooth semi-local schemes using nice triples, avoiding Gabber's lemma, and clarifies a known inaccuracy in Morel’s arguments.

Contribution

It introduces the use of nice triples to establish homotopy invariance of cohomology sheaves without relying on Gabber's presentation lemma.

Findings

Cousin complex is exact for cohomology theories on semi-local schemes.

Nisnevich sheaves associated with these theories are strictly homotopy invariant.

The approach recovers known results and corrects previous inaccuracies.

Abstract

It is proved that for any cohomology theory A in the sense of [PS] and any essentially k-smooth semi-local X the Cousin complex is exact. As a consequence we prove that for any integer n the Nisnevich sheaf A^n_Nis, associated with the presheaf U |--> A^n(U), is strictly homotopy invariant. Particularly, for any presheaf of S^1-spectra E on the category of k-smooth schemes its Nisnevich sheves of stable A1-homotopy groups are strictly homotopy invariant. The ground field k is arbitrary. We do not use Gabber's presentation lemma. Instead, we use the machinery of nice triples as invented in [PSV] and developed further in [P3]. This recovers a known inaccuracy in Morel's arguments in [M]. The machinery of nice triples is inspired by the Voevodsky machinery of standard triples.