Bivariant algebraic cobordism with bundles

TL;DR

This paper extends bivariant algebraic cobordism to include vector bundles, explores a restricted rank 1 case, and develops foundational properties over arbitrary Noetherian rings, advancing towards a base ring cobordism theory.

Contribution

It introduces an extended bivariant cobordism theory with bundles, proves a projective bundle formula in a restricted case, and develops foundational properties over general base rings.

Findings

Extension of cobordism with vector bundles over characteristic 0 fields.

Proof of a weak projective bundle formula for rank 1 bundles.

Development of precobordism theories over arbitrary Noetherian rings.

Abstract

The purpose of this paper is to study an extended version of bivariant derived algebraic cobordism where the cycles carry a vector bundle on the source as additional data. We show that, over a field of characteristic 0, this extends the analogous homological theory of Lee and Pandharipande constructed earlier. We then proceed to study in detail the restricted theory where only rank 1 vector bundles are allowed, and prove a weak version of projective bundle formula for bivariant cobordism. Since the proof of this theorem works very generally, we introduce precobordism theories over arbitrary Noetherian rings of finite Krull dimension as a reasonable class of theories where the proof can be carried out, and prove some of their basic properties. These results can be considered as the first steps towards a Levine-Morel style algebraic cobordism over a base ring that is not a field of characteristic 0.