Support $τ$-tilting modules and recollements

Abstract

Let (mod Λ ′ , mod Λ, mod Λ ′′ ) be a recollement of abelian categories for artin algebras Λ ′ , Λ and Λ ′′ . Under certain conditions, we present an explicit construction of gluing of (support) τ -tilting modules in mod Λ with respect to (support) τ -tilting modules in mod Λ ′ and mod Λ ′′ respectively. On the other hand, we study the construction of (support) τ -tilting modules in mod Λ ′ and mod Λ ′′ obtained from (support) τ -tilting modules in mod Λ.

Recollements and tilting modules

“…Since j ! is exact, we get that j * preserves injective objects by[15, Proposition 2.5].It follows that j * (I) is injective in C and j * (I) ∈ T ′′ . Since i !…”

“…injective) objects, we have that A has enough projective (resp. injective) objects by [15,Proposition 2.5].…”

“…injective) objects, then A has enough projective (resp. injective) objects by[15, Proposition 2.5]. (1) Since (T ′ , F ′ ) and (T ′′ , F ′′ ) are tilting torsion pairs in A and C respectively by assumption, we have that T ′ and T ′′ contain all injective objects in A and C respectively by Remark 3.2.…”

“…([15, Lemma 2.4] and[22, Lemma 5.3]) Let (A, B, C) be a recollement of abelian categories as (2.1). Then we have (1) If i * is exact, then j !…”