Recollements and tilting modules

Abstract

Let (mod Λ ′ , mod Λ, mod Λ ′′) be a recollement of module categories for artin algebras Λ ′ , Λ and Λ ′′. We provide a sufficient condition such that a glued torsion pair in mod Λ is tilting when the given two torsion pairs are tilting in mod Λ ′ and mod Λ ′′ respectively. Using this result, we give a construction of gluing of tilting modules in mod Λ with respect to tilting modules in mod Λ ′ and mod Λ ′′ respectively.

Gluing n-tilting and n-cotilting subcategories

“…Note that T 1 , T 3 are just T ′ and T ′′ given in [MZ,Example 4.1(1)] respectively. They are also tilting modules, and the tilting cotorsion pairs they induce are still (U 1 , V 1 ) and (U 3 , V 3 ) respectively.…”

Recollements of extriangulated categories

“…Recollements of abelian categories and triangulated categories play an important role in algebraic geometry and representation theory, see for instance [10], [1], [9]. In recollements of abelian categories, the relations with tilting modules and torsion pairs have been studied in [6] and [7], respectively. Chen [5] studied the relationship of cotorsion pairs among three triangulated categories in a recollement of triangulated categories.…”

Recollements of extriangulated categories

“…Ma, Xie and Zhao [10] study the gluing of support τ -tilting modules in a recollement of module categories; see also [8,7] for support τ -tilting modules. In recollements of abelian categories, the relations with tilting modules and torsion pairs have been studied in [11] and [9], respectively. Chen [5] studied the relationship of cotorsion pairs among three triangulated categories in a recollement of triangulated categories.…”