Bivariant algebraic cobordism with bundles.

The purpose of this paper is to study an extended version of bivariant derived algebraic cobordism in which 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. We then proceed to study in detail the restricted theory where only rank 1 vector bundles are allowed, and employ the obtained structural results to prove a weak version of the projective bundle formula for bivariant cobordism. Since the proof of this theorem works very generally, we introduce the notion of precobordism theories for quasi-projective derived schemes over an arbitrary Noetherian ring 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. [ABSTRACT FROM AUTHOR]