# Gluing semi-orthogonal decompositions

## TL;DR

This paper develops a method to glue semi-orthogonal decompositions of dg-categories using psod-s, enabling new constructions for root stacks and computations of Kummer flat K-theory in complex log pairs.

## Contribution

It introduces preordered semi-orthogonal decompositions (psod-s) for dg-categories and demonstrates how to glue these structures along covers, extending existing results.

## Key findings

- Constructed semi-orthogonal decompositions for root stacks of log pairs.
- Computed Kummer flat K-theory for general log pairs.
- Extended semi-orthogonal decomposition techniques to broader contexts.

## Abstract

We introduce preordered semi-orthogonal decompositions (psod-s) of dg-categories. We show that homotopy limits of dg-categories equipped with compatible psod-s carry a natural psod. This gives a way to glue semi-orthogonal decompositions along faithfully-flat covers, extending some results of [4]. As applications we will construct semi-orthogonal decompositions for root stacks of log pairs (X,D) where D is a (not necessarily simple) normal crossing divisors, generalizing results from [17] and [3]. Further we will compute the Kummer flat K-theory of general log pairs (X,D), generalizing earlier results of Hagihara and Nizio{\l} in the simple normal crossing case [15], [23].