Tilting pairs and Wakamatsu tilting subcategories over triangular matrix algebras

Abstract

Abstract Let A and B be Artin algebras and let M be an ( A , B ) (A,B) -bimodule with M A {}_{A}M and M B M_{B} finitely generated. In this paper, we construct tilting pairs of subcategories and Wakamatsu tilting subcategories over an upper triangular matrix Artin algebra Λ = ( A M 0 B ) {\Lambda=\bigl{(}\begin{smallmatrix}A&M\\ 0&B\\ \end{smallmatrix}\bigr{)}} using tilting pairs and Wakamatsu tiling subcategories over A and B. Let 𝒞 {\mathcal{C}} be a subcategory of A ⁢ -mod {A\mbox{-mod}} and let 𝒟 {\mathcal{D}} be a subcategory of B ⁢ -mod {B\mbox{-mod}} . Consider the subcategory of left Λ-modules 𝔅 𝒟 𝒞 = { ( X Y ) f : f  is a monomorphism,  Y ∈ 𝒟  and  Coker ⁡ f ∈ 𝒞 } {\mathfrak{B}^{\mathcal{C}}_{\mathcal{D}}=\{{\bigl{(}\begin{smallmatrix}{X}\\ {Y}\\ \end{smallmatrix}\bigr{)}_{f}}:\text{$f$ is a monomorphism, $Y\in\mathcal{D}$ % and $\operatorname{Coker}f\in\mathcal{C}$}\}} . We prove the following results: (1) Assume that M ⊗ B 𝒯 ′ ⊆ 𝒯 {M\otimes_{B}\mathcal{T}^{\prime}\subseteq\mathcal{T}} , M ⊗ B 𝒞 ′ ⊆ 𝒞 {M\otimes_{B}\mathcal{C}^{\prime}\subseteq\mathcal{C}} and Tor i B ⁢ ( M , 𝒯 ′ ) = 0 {\mathrm{Tor}^{B}_{i}(M,\mathcal{T}^{\prime})=0} for all i ≥ 1 {i\geq 1} . Then ( 𝒞 , 𝒯 ) {(\mathcal{C},\mathcal{T})} and ( 𝒞 ′ , 𝒯 ′ ) {(\mathcal{C}^{\prime},\mathcal{T}^{\prime})} are n-tilting pairs respectively in A ⁢ - ⁢ mod {A\text{-}\mathrm{mod}} and B ⁢ - ⁢ mod {B\text{-}\mathrm{mod}} if and only if ( 𝔅 𝒞 ′ 𝒞 , 𝔅 𝒯 ′ 𝒯 ) {(\mathfrak{B}^{\mathcal{C}}_{\mathcal{C}^{\prime}},\mathfrak{B}^{\mathcal{T}}% _{\mathcal{T}^{\prime}})} is an n-tilting pair in Λ ⁢ - ⁢ mod {\Lambda\text{-}\mathrm{mod}} . (2) Assume that M ⊗ B 𝒱 ⊆ 𝒲 {M\otimes_{B}\mathcal{V}\subseteq\mathcal{W}} and Tor i B ⁢ ( M , 𝒱 ⊥ ) = 0 {\mathrm{Tor}^{B}_{i}(M,{{}^{\perp}\mathcal{V}})=0} for all i ≥ 1 {i\geq 1} . If 𝒲 {\mathcal{W}} and 𝒱 {\mathcal{V}} are Wakamatsu tilting subcategories respectively in A ⁢ - ⁢ mod {A\text{-}\mathrm{mod}} and B ⁢ - ⁢ mod {B\text{-}\mathrm{mod}} , then 𝔅 𝒱 𝒲 {\mathfrak{B}^{\mathcal{W}}_{\mathcal{V}}} is a Wakamatsu tilting subcategory in Λ ⁢ - ⁢ mod {\Lambda\text{-}\mathrm{mod}} .