Formal completions and idempotent completions of triangulated categories of singularities

TL;DR

This paper proves that the idempotent completions of triangulated categories of singularities are equivalent when the formal completions of the schemes are isomorphic, linking to Thomason's theorem and negative K-theory.

Contribution

It establishes an equivalence of idempotent completions of singularity categories based on formal scheme completions, connecting to Thomason's theorem and negative K-theory.

Findings

Idempotent completions are equivalent under formal completion isomorphisms.

Relation established between singularity categories and negative K-theory.

Discussion of Thomason theorem on dense subcategories.

Abstract

The main goal of this paper is to prove that the idempotent completions of the triangulated categories of singularities of two schemes are equivalent if the formal completions of these schemes along singularities are isomorphic. We also discuss Thomason theorem on dense subcategories and a relation to the negative K-theory.