Algebraic Spivak’s theorem and applications

Annala, Toni

doi:10.2140/gt.2023.27.351

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Algebraic Spivak's theorem and applications

Toni Annala

Geometry & Topology 27 (2023) 351–396

DOI: 10.2140/gt.2023.27.351

Abstract

We prove an analogue of Lowrey and Schürg’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 quasiprojective 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 quasismooth 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 quasiprojective derived $A$ –schemes.

Keywords

algebraic bordism, algebraic cobordism, derived algebraic geometry

Mathematical Subject Classification

Primary: 14F43

Secondary: 14C99, 14J99

References

Publication

Received: 1 February 2021

Revised: 8 August 2021

Accepted: 25 September 2021

Published: 1 May 2023

Proposed: Haynes R Miller

Seconded: Mark Behrens, Marc Levine

Authors

Toni Annala
Department of Mathematics University of British Columbia Vancouver BC Canada

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Volume 27, issue 1 (2023)