Gluing semi-orthogonal decompositions

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 the main result 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 divisor, 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 Niziol in the simple normal crossing case [15], [23]. (C) 2020 Elsevier Inc. All rights reserved.