  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].

Algebraic Geometry