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.

τ -rigid modules are essential in the τ -tilting theory introduced by Adachi, Iyama and Reiten. In this paper, we give equivalent conditions for Iwanaga-Gorenstein algebras with selfinjective dimension at most one in terms of τ -rigid modules. We show that every indecomposable module over iterated tilted algebras of Dynkin type is τ -rigid. Finally, we give a τ -tilting theorem on homological dimension which is an analog to that of classical tilting modules.

In this paper, we introduce the notions of Gorenstein projective τ -rigid modules, Gorenstein projective support τ -tilting modules and Gorenstein torsion pairs and give a Gorenstein analog to Adachi-Iyama-Reiten's bijection theorem on support τ -tilting modules. More precisely, for an algebra Λ, we show that there is a bijection between the set of Gorenstein projective τ -rigid pairs in mod Λ and the set of Gorenstein injective τ −1 -rigid pairs in mod Λ op . We prove that there is a bijection between the set of Gorenstein projective support τ -tilting modules and the set of functorially finite Gorenstein projective torsion classes. As an application, we introduce the notion of CM-τ -tilting finite algebras and show that Λ is CM-τ -tilting finite if and only if Λ op is CM-τ -tilting finite. MSC2020: 16G10, 18G25.

Let B be the one-point extension algebra of A by an A-module X. We proved that every support τ -tilting A-module can be extended to be a support τ -tilting B-module by two different ways. As a consequence, it is shown that there is an inequality | sτ -tilt B| 2| sτ -tilt A|.

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