For recollements of module categories, semibricks on the two outer algebras glue via the intermediate extension functor to semibricks on the middle algebra, yielding a construction of support tau-tilting modules over tau-tilting finite algebras.
Properties of abelian categories via recollements
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abstract
A recollement is a decomposition of a given category (abelian or triangulated) into two subcategories with functorial data that enables the glueing of structural information. This paper is dedicated to investigating the behaviour under glueing of some basic properties of abelian categories (well-poweredness, Grothendieck's axioms AB3, AB4 and AB5, existence of a generator) in the presence of a recollement. In particular, we observe that in a recollement of a Grothendieck abelian category the other two categories involved are also Grothendieck abelian and, more significantly, we provide an example where the converse does not hold and explore multiple sufficient conditions for it to hold.
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A construction of support $\tau$-tilting modules over $\tau$-tilting finite algebras
For recollements of module categories, semibricks on the two outer algebras glue via the intermediate extension functor to semibricks on the middle algebra, yielding a construction of support tau-tilting modules over tau-tilting finite algebras.