Algebraic cobordism via spans

TL;DR

This paper introduces algebraic cobordism for $ abla$-categories using spans, recovering known motivic models and extending to perfectoid geometry with new tilting and local theory comparisons.

Contribution

It defines a new algebraic cobordism framework via spans, recovers classical motivic models, and extends to perfectoid geometry with tilting and local theory results.

Findings

Recovering the Thom spectrum model in the motivic framework.

Proving the projective bundle formula and Chern-class identities under certain conditions.

Constructing perfectoid cobordism and establishing tilting equivalences.

Abstract

We define the algebraic cobordism of $\infty$-categories equipped with universal line bundle data as an initial oriented functor in the associated span category. In the standard motivic framework, this recovers the Thom spectrum model established by Voevodsky, Gepner, and Snaith. Furthermore, assuming that the $\infty$-category contains Grassmann objects of all ranks, we prove that the projective bundle formula and the corresponding Chern-class and Whitney-sum identities hold for any oriented functor satisfying the splitting principle property. We apply the span formalism to perfectoid geometry. For perfectoid algebras $R$ with tilt $R^\flat$, we construct perfectoid cobordism, prove tilting equivalences, and compare the arc-local and $v$-local $p$-adic theories.