Base independent algebraic cobordism

TL;DR

This paper proves that bivariant algebraic cobordism groups are independent of the base ring and extends the theory to derived schemes, establishing key formulas and theorems in this broader context.

Contribution

It demonstrates base independence of algebraic cobordism and extends fundamental results to divisorial Noetherian derived schemes.

Findings

Bivariant algebraic cobordism groups are base independent.

Extended projective bundle formula and Chern class theory.

Generalized Grothendieck--Riemann--Roch theorem to derived schemes.

Abstract

The purpose of this article is to show that the bivariant algebraic $A$-cobordism groups considered previously by the author are independent of the chosen base ring $A$. This result is proven by analyzing the bivariant ideal generated by the so called snc relations, and, while the alternative characterization we obtain for this ideal is interesting by itself because of its simplicity, perhaps more importantly it allows us to easily extend the definition of bivariant algebraic cobordism to divisorial Noetherian derived schemes of finite Krull dimension. As an interesting corollary, we define the corresponding homology theory called algebraic bordism. We also generalize projective bundle formula, the theory of Chern classes, the Conner--Floyd theorem and the Grothendieck--Riemann--Roch theorem to this setting. The general definitions of bivariant cobordism is based on the careful study of ample line bundles and quasi-projective morphisms of Noetherian derived schemes, also undertaken in this work.