Recollements of derived categories I: Exact contexts

Abstract

Recollements were introduced originally by Beilinson, Bernstein and Deligne to study the derived categories of perverse sheaves, and nowadays become very powerful in understanding relationship among three algebraic, geometric or topological objects. The purpose of this series of papers is to study recollements in terms of derived module categories and homological ring epimorphisms, and then to apply our results to both representation theory and algebraic K-theory. In this paper we present a new and systematic method to construct recollements of derived module categories. For this aim, we introduce a new ring structure, called the noncommutative tensor product, and give necessary and sufficient conditions for noncommutative localizations which appears often in representation theory, topology and K-theory, to be homological. The input of our machinery is an exact context which can be easily obtained from a rigid morphism that exists in very general circumstances. The output is a recollement of derived module categories of rings in which the noncommutative tensor product of an exact context plays a crucial role. Thus we obtain a large variety of new recollements from commutative and noncommutative localizations, ring epimorphisms and extensions.