Algebraic $K$-theory and algebraic cobordism of almost mathematics

TL;DR

This paper develops an almost mathematics version of algebraic $K$-theory and cobordism, connecting these theories to bilocalization in abelian categories and proving properties like the Gersten property in the almost setting.

Contribution

It introduces the almost algebraic $K$-theory and cobordism, extending classical theories via Quillen's bilocalization, and establishes their key properties and equivalences.

Findings

Almost $K$-theory is a direct factor of the $K$-theory of the base field.

Almost algebraic cobordism satisfies tilting equivalence.

Almost $K$-theory holds the Gersten property.

Abstract

Faltings; Gabber and Ramero introduced almost mathematics. In another way, almost mathematics can be characterized bilocalization abelian category of modules mentioned in Quillen's unpublished note. Applying the concept of Quillen's bilocalization to Gabber and Ramero's work, this paper establishes the almost version of algebraic $K$-theory and cobordism. As a result of almost $K$-theory, we prove that in the case an almost algebra containing a field, the almost $K$-theory of the almost algebra is a direct factor of the $K$-theory of the field, implying that almost $K$-theory holds the Gersten property. We clarify that an almost $K$-theory is a $K$-theory spectrum of non-unital firm algebras in the sense of Quillen. Furthermore, we obtain that almost algebraic cobordism holds tilting equivalence on the category of zero-section stable integral perfectoid algebras with finite syntomic topology.