Precobordism and cobordism

Abstract

The purpose of this paper is to compare three versions of bivariant algebraic cobordism: the bivariant algebraic cobordism, the universal precobordism, and the operational algebraic cobordism. We show that there is a very close relationship between universal precobordism and bivariant algebraic cobordism, and that, over a base field of characteristic 0, the former can be used to give a new presentation of the algebraic bordism groups of Levine-Morel, which simplifies slightly the presentation achieved by Lowrey-Schürg. We also strengthen a result of Vezzosi on operational derived K-theory. In the appendix, we fill the gaps in Lowrey-Schürg's construction of virtual pullbacks in algebraic bordism.

K-theory and G-theory of derived algebraic stacks

“…On the other hand Annala has recently constructed good theories of algebraic cobordism and Chow cohomology of singular schemes in characteristic zero [An1]. After further simplification and partial extension to general bases in [AnYo,An2], the generators are similar to those of A * der but a simple description of the relations needed to pass from cobordism to Chow does not yet seem to be known. Any relation with filtrations on K-theory has not been studied beyond the remarks above as far as I know and comparisons with the Levine-Weibel Chow groups [LW] also have yet to be investigated.…”