Algebraic Spivak's theorem and applications

TL;DR

This paper extends algebraic Spivak's theorem to certain base rings, demonstrating that algebraic bordism groups can be generated by classical cycles and establishing homotopy invariance for inverted bordism groups.

Contribution

It proves an analogue of Lowrey--Schürg's algebraic Spivak's theorem over specific base rings and inverts residual characteristic exponent, enabling new generation and invariance results.

Findings

Algebraic bordism groups are generated by classical cycles.

Vanishing results for low degree e-inverted bordism classes.

Homotopy invariance of e-inverted bordism groups.

Abstract

We prove an analogue of Lowrey--Sch\"urg's algebraic Spivak's theorem when working over a base ring $A$ that is either a field or a nice enough discrete valuation ring, and after inverting the residual characteristic exponent $e$ in the coefficients. By this result algebraic bordism groups of quasi-projective derived $A$-schemes can be generated by classical cycles, leading to vanishing results for low degree $e$-inverted bordism classes, as well as to the classification of quasi-smooth projective $A$-schemes of low virtual dimension up to $e$-inverted cobordism. As another application, we prove that $e$-inverted bordism classes can be extended from an open subset, leading to the proof of homotopy invariance of $e$-inverted bordism groups for quasi-projective derived $A$-schemes.