Auslanders formula in dualizing variaties Shijie Zhu (Joint with - PowerPoint PPT Presentation

Bibliography Auslander’s formula in dualizing variaties Shijie Zhu (Joint with Ron Gentle, Job Rachowicz and Gordana Todorov) GMRT, University of Iowa November 19, 2017 Shijie Zhu (Joint with Ron Gentle, Job Rachowicz and Gordana Todorov) GMRT, University of Iowa Auslander’s formula in dualizing variaties

Bibliography The Auslander’s formula Theorem (Auslander) Let Λ be an artin algebra. (Λ – mod) – mod : the category of finitely presented (contravariant) functors, (Λ – mod) – mod 0 : the category of finitely presented functors vanishing on projective modules. Then (Λ – mod) – mod ∼ = Λ – mod (Λ – mod) – mod 0 Shijie Zhu (Joint with Ron Gentle, Job Rachowicz and Gordana Todorov) GMRT, University of Iowa Auslander’s formula in dualizing variaties

Bibliography Remark ( − , f ) (Λ – mod) – mod 0 = { F | ( − , X ) → ( − , Y ) → F → 0 for some epimorphism f : X → Y } ∼ = (Λ – mod) – mod Shijie Zhu (Joint with Ron Gentle, Job Rachowicz and Gordana Todorov) GMRT, University of Iowa Auslander’s formula in dualizing variaties

Bibliography Let A be an additive category with pseudo-kernel. i.e. for any morphism f : A → B , there is a morphism g : K → A such that Hom( − , g ) Hom( − , f ) Hom( − , K ) → Hom( − , A ) → Hom( − , B ) is exact. For example, triangulated categories have pseudo-kernels. Shijie Zhu (Joint with Ron Gentle, Job Rachowicz and Gordana Todorov) GMRT, University of Iowa Auslander’s formula in dualizing variaties

Bibliography Let A be an additive category with pseudo-kernel. i.e. for any morphism f : A → B , there is a morphism g : K → A such that Hom( − , g ) Hom( − , f ) Hom( − , K ) → Hom( − , A ) → Hom( − , B ) is exact. For example, triangulated categories have pseudo-kernels. Proposition An additive category A has pseudo-kernel if and only if A – mod is abelian. Shijie Zhu (Joint with Ron Gentle, Job Rachowicz and Gordana Todorov) GMRT, University of Iowa Auslander’s formula in dualizing variaties

Bibliography Let A be an additive category with pseudo-kernel. i.e. for any morphism f : A → B , there is a morphism g : K → A such that Hom( − , g ) Hom( − , f ) Hom( − , K ) → Hom( − , A ) → Hom( − , B ) is exact. For example, triangulated categories have pseudo-kernels. Proposition An additive category A has pseudo-kernel if and only if A – mod is abelian. If A has pseudo-kernel, then any contravariantly finite subcategory X has pseudo-kernel. Shijie Zhu (Joint with Ron Gentle, Job Rachowicz and Gordana Todorov) GMRT, University of Iowa Auslander’s formula in dualizing variaties

Bibliography Let X be a contravariantly finite subccategory of A ; A – mod be the category of finitely presented functors on A ; T X = { F ∈ A – mod | ( − , X ) → F → 0 for some X ∈ X} ; F X = { F ∈ A – mod | F ( X ) = 0 for all X ∈ X} . Shijie Zhu (Joint with Ron Gentle, Job Rachowicz and Gordana Todorov) GMRT, University of Iowa Auslander’s formula in dualizing variaties

Bibliography Definition ( F , T ) is a torsion theory in abelian category C , if (1) T ⊥ = F and ⊥ F = T . (2) For any M ∈ C , there is an exact sequence 0 → tM → M → rM → 0 , where tM ∈ T and rM ∈ F . Theorem (Gentle, Todorov) ( F X , T X ) is a torsion theory. Shijie Zhu (Joint with Ron Gentle, Job Rachowicz and Gordana Todorov) GMRT, University of Iowa Auslander’s formula in dualizing variaties

Bibliography Let F ∈ A – mod. � ( − , B ) � ( − , C ) � F � 0 ( − , A ) Shijie Zhu (Joint with Ron Gentle, Job Rachowicz and Gordana Todorov) GMRT, University of Iowa Auslander’s formula in dualizing variaties

� � � � � � � Bibliography Let F ∈ A – mod. 0 � ( − , E ) ( − , A ) ( − , X C ) tF 0 ( p . b . ) ( − , f C ) � ( − , B ) � ( − , C ) � F � 0 ( − , A ) Shijie Zhu (Joint with Ron Gentle, Job Rachowicz and Gordana Todorov) GMRT, University of Iowa Auslander’s formula in dualizing variaties

� � � � � � � � � � � � � � � � � Bibliography Let F ∈ A – mod. 0 � ( − , E ) ( − , A ) ( − , X C ) tF 0 ( p . b . ) ( − , f C ) ( − , A ) ( − , B ) ( − , C ) F 0 � ( − , B ⊕ X C ) � ( − , C ) � rF ( − , E ) 0 0 Shijie Zhu (Joint with Ron Gentle, Job Rachowicz and Gordana Todorov) GMRT, University of Iowa Auslander’s formula in dualizing variaties

Bibliography Denote by res X the restriction functor. Shijie Zhu (Joint with Ron Gentle, Job Rachowicz and Gordana Todorov) GMRT, University of Iowa Auslander’s formula in dualizing variaties

Bibliography Denote by res X the restriction functor. Define a functor e : X – mod → A – mod: If F ∈ X has a presentation ( X , f ) ( X , X 1 ) → ( X , X 0 ) → F → 0 , define eF by ( A , f ) ( A , X 1 ) → ( A , X 0 ) → eF → 0 . Shijie Zhu (Joint with Ron Gentle, Job Rachowicz and Gordana Todorov) GMRT, University of Iowa Auslander’s formula in dualizing variaties

Bibliography Denote by res X the restriction functor. Define a functor e : X – mod → A – mod: If F ∈ X has a presentation ( X , f ) ( X , X 1 ) → ( X , X 0 ) → F → 0 , define eF by ( A , f ) ( A , X 1 ) → ( A , X 0 ) → eF → 0 . Theorem For any F ∈ A – mod , res X F ∈ X – mod . Shijie Zhu (Joint with Ron Gentle, Job Rachowicz and Gordana Todorov) GMRT, University of Iowa Auslander’s formula in dualizing variaties

� � � � � � � � � � � � Bibliography Let F ∈ A – mod. �� e res X F ( − , K ) ( − , X E ) ( − , X C ) 0 ( p . b . ) ( − , f E ) � ( − , E ) ( − , A ) ( − , X C ) tF 0 ( p . b . ) ( − , f C ) � ( − , B ) � ( − , C ) � F � 0 ( − , A ) Shijie Zhu (Joint with Ron Gentle, Job Rachowicz and Gordana Todorov) GMRT, University of Iowa Auslander’s formula in dualizing variaties

Bibliography One can show Shijie Zhu (Joint with Ron Gentle, Job Rachowicz and Gordana Todorov) GMRT, University of Iowa Auslander’s formula in dualizing variaties

Bibliography One can show • r : A – mod → F X is a left adjoint of the inclusion i : F X → A – mod. Shijie Zhu (Joint with Ron Gentle, Job Rachowicz and Gordana Todorov) GMRT, University of Iowa Auslander’s formula in dualizing variaties

Bibliography One can show • r : A – mod → F X is a left adjoint of the inclusion i : F X → A – mod. • res X : A – mod → X – mod is a right adjoint of e : X – mod → A – mod. Shijie Zhu (Joint with Ron Gentle, Job Rachowicz and Gordana Todorov) GMRT, University of Iowa Auslander’s formula in dualizing variaties

Bibliography One can show • r : A – mod → F X is a left adjoint of the inclusion i : F X → A – mod. • res X : A – mod → X – mod is a right adjoint of e : X – mod → A – mod. Proposition There is an exact sequence of categories res X � X – mod � F X i � A – mod � O O where i is the inclusion functor with r ⊣ i and e ⊣ res X . Shijie Zhu (Joint with Ron Gentle, Job Rachowicz and Gordana Todorov) GMRT, University of Iowa Auslander’s formula in dualizing variaties

Bibliography Question: when does the functor res X has a right adjoint? Shijie Zhu (Joint with Ron Gentle, Job Rachowicz and Gordana Todorov) GMRT, University of Iowa Auslander’s formula in dualizing variaties

� � � � Bibliography Question: when does the functor res X has a right adjoint? Theorem (Asadollahi,J., Hafezi, R., Keshavarz, M.H, 2017) When A is a contravariantly finite subcategory of Λ – mod for some artin algebra Λ containing all the projective Λ modules and X is the category of projective Λ modules, there is a recollement res X � i � A – mod F X X – mod , Shijie Zhu (Joint with Ron Gentle, Job Rachowicz and Gordana Todorov) GMRT, University of Iowa Auslander’s formula in dualizing variaties

� � � � Bibliography Question: when does the functor res X has a right adjoint? Theorem (Asadollahi,J., Hafezi, R., Keshavarz, M.H, 2017) When A is a contravariantly finite subcategory of Λ – mod for some artin algebra Λ containing all the projective Λ modules and X is the category of projective Λ modules, there is a recollement res X � i � A – mod F X X – mod , Notice in this situation, F X ∼ = A – mod 0 and X – mod ∼ = Λ – mod. Shijie Zhu (Joint with Ron Gentle, Job Rachowicz and Gordana Todorov) GMRT, University of Iowa Auslander’s formula in dualizing variaties

Bibliography Definition Let k be a commutative artin ring with radical r and E ( k / r ) be the injective envelope of the k module k / r. Denote by D = Hom k ( − , E ( k / r )) the duality. Then a Hom -finite additive k category C is called a dualizing k -variety if there is an equivalence C op – mod C – mod → F �→ DF . For example, Λ – mod is a dualizing variety. Any functorially finite subcategory of a dualizing variety is again a dualizing variety. Shijie Zhu (Joint with Ron Gentle, Job Rachowicz and Gordana Todorov) GMRT, University of Iowa Auslander’s formula in dualizing variaties