Oriented bivariant theory II- Algebraic cobordism of $S$-schemes -

TL;DR

This paper extends algebraic cobordism theory to schemes over a fixed base scheme S, generalizing Levine-Morel's construction and providing a new geometric framework for cobordism relative to S.

Contribution

It constructs an algebraic cobordism theory for schemes over a base scheme S, generalizing previous work and aligning with Levine-Morel's cobordism when S is a point.

Findings

Defined algebraic cobordism $ ext{ extOmega}^*(X o S)$ for schemes over S.

Showed that it recovers Levine-Morel's cobordism when S is a point.

Provided a geometric construction consistent with previous theories.

Abstract

This is a sequel to our previous paper of oriented bivariant theory [14]. In 2001 M. Levine and F. Morel constructed algebraic cobordism $\Omega_*(X)$ for schemes $X$ over a field $k$ in an abstract way and later M. Levine and R. Pandhairpande reconstructed it more geometrically. In this paper in a similar manner we construct an algebraic cobordism $\Omega^*(X \xrightarrow {\pi_X} S)$ for a scheme $X$ over a fixed scheme $S$ in such a way that if the target scheme $S$ is the point $pt = \operatorname{Spec} k$, then $\Omega^{-i}(X \xrightarrow {\pi_X} pt)$ is isomorphic to Levine-Morel's algebraic cobordism $\Omega_i(X)$