Bivariant derived algebraic cobordism

Abstract

We extend the derived algebraic bordism of Lowrey and Schürg to a bivariant theory in the sense of Fulton and MacPherson and establish some of its basic properties. As a special case, we obtain a completely new theory of cobordism rings of singular quasi-projective schemes. The extended cobordism is shown to specialize to algebraic K 0 K^0 analogously to the Conner-Floyd theorem in topology. We also give a candidate for the correct definition of Chow rings of singular schemes.

Enriched categories of correspondences and characteristic classes of singular varieties

“…In [24] P. Lowrey and T. Schürg have constructed a derived algebraic cobordism dΩ * (X) for derived schemes. In [1] T. Annala has obtained a bivariant-theoretic version Ω * (X → Y ) of Levine-Morel's algebraic cobordism, using the construction of Lowrey and Schürg and the construction of a universal bivariant theory of the author [33] (cf. [34]).…”

“…[34]). Furthermore, in [3] (see also [2]) T. Annala and the author have constructed a bivariant-theoretic version of Lee-Pandharipande's algebraic cobordism of vector bundles [20].…”

Precobordism and cobordism

“…In [3], the author extended the derived bordism groups of Lowrey-Schürg to a bivariant theory Ω * called bivariant algebraic bordism. As a special case, the bivariant theory yields a ring valued cohomology theory called algebraic cobordism, which can also be used to construct a candidate for Chow rings of singular varieties.…”

“…As a special case, the bivariant theory yields a ring valued cohomology theory called algebraic cobordism, which can also be used to construct a candidate for Chow rings of singular varieties. Many expected properties of these theories, were verified in [3], but unfortunately they too make heavy use of the comparison results with the classical bordism groups Ω * , and therefore the proofs work only over fields of characteristic 0. However, in [5,4] an alternative construction was studied, the universal precobordism theory Ω * , and most of the properties proven in [3] for Ω * (in characteristic 0) were proven directly for Ω * over an arbitrary Noetherian ring A (of finite Krull dimension).…”

“…Many expected properties of these theories, were verified in [3], but unfortunately they too make heavy use of the comparison results with the classical bordism groups Ω * , and therefore the proofs work only over fields of characteristic 0. However, in [5,4] an alternative construction was studied, the universal precobordism theory Ω * , and most of the properties proven in [3] for Ω * (in characteristic 0) were proven directly for Ω * over an arbitrary Noetherian ring A (of finite Krull dimension). However, it was not clear exactly how the two theories Ω * and Ω * relate to each other.…”