Cohomology theories with supports

TL;DR

This paper proves a weak equivalence between two support conditions in cohomology theories with spectra, confirming cases of a conjecture by Marc Levine for affine and projective smooth schemes.

Contribution

It establishes the weak equivalence for cohomology theories with supports satisfying Nisnevich excision, advancing the understanding of support conditions in algebraic K-theory.

Findings

Weak equivalence between algebraic singular complex with quasi-finite supports and with proper face intersections.

Results hold for affine and projective smooth schemes.

Progress on Marc Levine's conjecture in specific cases.

Abstract

For $E$ a presheaf of spectra on the category of smooth $k$-schemes satisfying Nisnevich excision, we prove that the canonical map from the algebraic singular complex of the theory $E$ with quasi-finite supports to the theory $E$ with supports intersecting all the faces properly is a weak equivalence on smooth $k$-schemes that are affine or projective. This establishes some cases of a conjecture of Marc Levine.