Conservative descent for semi-orthogonal decompositions

Abstract

Motivated by the local flavor of several well-known semi-orthogonal decompositions in algebraic geometry, we introduce a technique called conservative descent, which shows that it is enough to establish these decompositions locally. The decompositions we have in mind are those for projectivized vector bundles and blow-ups, due to Orlov, and root stacks, due to Ishii and Ueda. Our technique simplifies the proofs of these decompositions and establishes them in greater generality for arbitrary algebraic stacks.

Gluing semi-orthogonal decompositions

“…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:(1) 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], (2) 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].…”

“…Gluing psod-s and conservative descent. A theory for gluing semi-orthogonal decompositions across faithfully-flat covers was proposed in [4,Theorem B]. In this section we explain the relationship between that result and our work.…”

“…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:…”

“…We will study limits of dg-categories equipped with compatible psod-s. As an application of our theory, we will obtain gluing results for semi-orthogonal decompositions along appropriate faithfully-flat covers. With different methods, and in the setting of classical triangulated categories, related results have been obtained in the recent paper [4]. We remark that the concept of preordered semi-orthogonal decomposition was also considered in [3].…”

“…In this section we explain the relationship between that result and our work. We will show that our Proposition 3.9 and Theorem 3.13 imply Theorem B from [4]. We start by briefly recalling the setting of [4], referring the reader to the original reference for full details.…”