Oriented cohomology, Borel-Moore homology and algebraic cobordism

TL;DR

This paper develops a unified framework for oriented cohomology and Borel-Moore homology theories in algebraic geometry, introducing an oriented duality theory that extends previous work and relates algebraic cobordism to Voevodsky's MGL theory.

Contribution

It constructs a Borel-Moore homology version of algebraic cobordism and proposes a conjecture linking it to Voevodsky's MGL theory, extending duality theories in algebraic geometry.

Findings

Defined an oriented duality theory generalizing Bloch-Ogus duality.

Constructed a Borel-Moore homology version of MGL-theory.

Proposed a conjecture that the natural map is an isomorphism.

Abstract

We examine various versions of oriented cohomology and Borel-Moore homology theories in algebraic geometry and put these two together in the setting of an "oriented duality theory", a generalization of Bloch-Ogus twisted duality theory. This combines and exends work of Panin and Mocanasu. We apply this to give a Borel-Moore homology version $MGL'_{*,*}$ of Voevodsky's $MGL^{*,*}$-theory, and a natural map $\vartheta:\Omega_*\to MGL'_{2*,*}$, where $\Omega_*$ is the algebraic cobordism theory defined by Levine-Morel. We conjecture that $\vartheta$ is an isomorphism and describe a program for proving this conjecture.