Recollements induced from tilting modules over tame hereditary algebras

Abstract

Abstract In this paper, we consider the endomorphism algebra of an infinitely generated tilting module of the form R 𝒰 ⊕ R 𝒰/R over a tame hereditary k-algebra R with k an arbitrary field, where R 𝒰 is the universal localization of R at an arbitrary set 𝒰 of simple regular R-modules. We show that the derived module category of this endomorphism algebra is a recollement of the derived module category 𝒟(R) of R and the derived module category 𝒟(𝔸𝒰) of the adèle ring 𝔸𝒰 associated with 𝒰. When k is an algebraically closed field, the ring 𝔸𝒰 can be precisely described in terms of Laurent power series ring k((x)) over k. Moreover, if 𝒰 is a union of finitely many cliques, we give two different stratifications of the derived category of this endomorphism algebra by derived categories of rings such that the two stratifications are of different finite lengths.