Quasidualizing Modules and the Auslander and Bass Classes Bethany - PowerPoint PPT Presentation

Quasidualizing Modules and the Auslander and Bass Classes Bethany Kubik United States Military Academy 16 October 2011 Bethany Kubik Quasidualizing Modules and the Auslander and Bass Classes

Introduction Let ( R , m , k ) be a local noetherian ring with completion � R and let E = E R ( k ) be the injective hull of the residue field. Bethany Kubik Quasidualizing Modules and the Auslander and Bass Classes

Introduction Let ( R , m , k ) be a local noetherian ring with completion � R and let E = E R ( k ) be the injective hull of the residue field. Definition Given an R -module M , the Matlis dual is M ∨ = Hom R ( M , E ). Bethany Kubik Quasidualizing Modules and the Auslander and Bass Classes

Introduction Let ( R , m , k ) be a local noetherian ring with completion � R and let E = E R ( k ) be the injective hull of the residue field. Definition Given an R -module M , the Matlis dual is M ∨ = Hom R ( M , E ). We say that M is Matlis reflexive if the natural biduality map δ : M → M ∨∨ is an isomorphism. Bethany Kubik Quasidualizing Modules and the Auslander and Bass Classes

Introduction Let ( R , m , k ) be a local noetherian ring with completion � R and let E = E R ( k ) be the injective hull of the residue field. Definition Given an R -module M , the Matlis dual is M ∨ = Hom R ( M , E ). We say that M is Matlis reflexive if the natural biduality map δ : M → M ∨∨ is an isomorphism. Fact Assume that R is complete. If A is an artinian R-module, then A ∨ is noetherian. If N is a noetherian R-module, then N ∨ is artinian. The modules A and N are Matlis reflexive. Bethany Kubik Quasidualizing Modules and the Auslander and Bass Classes

Semidualizing Modules Definition An R -module C is semidualizing if it satisfies the following: Bethany Kubik Quasidualizing Modules and the Auslander and Bass Classes

Semidualizing Modules Definition An R -module C is semidualizing if it satisfies the following: 1. C is noetherian, i.e. finitely generated; Bethany Kubik Quasidualizing Modules and the Auslander and Bass Classes

Semidualizing Modules Definition An R -module C is semidualizing if it satisfies the following: 1. C is noetherian, i.e. finitely generated; χ R C 2. R → Hom R ( C , C ) is an isomorphism; and − − Bethany Kubik Quasidualizing Modules and the Auslander and Bass Classes

Semidualizing Modules Definition An R -module C is semidualizing if it satisfies the following: 1. C is noetherian, i.e. finitely generated; χ R C 2. R → Hom R ( C , C ) is an isomorphism; and − − 3. Ext i R ( C , C ) = 0 for all i > 0. Bethany Kubik Quasidualizing Modules and the Auslander and Bass Classes

Semidualizing Modules Definition An R -module C is semidualizing if it satisfies the following: 1. C is noetherian, i.e. finitely generated; χ R C 2. R → Hom R ( C , C ) is an isomorphism; and − − 3. Ext i R ( C , C ) = 0 for all i > 0. Example The R -module R is always semidualizing. Bethany Kubik Quasidualizing Modules and the Auslander and Bass Classes

Quasidualizing Modules Definition An R -module T is quasidualizing if it satisfies the following: Bethany Kubik Quasidualizing Modules and the Auslander and Bass Classes

Quasidualizing Modules Definition An R -module T is quasidualizing if it satisfies the following: 1. T is artinian; Bethany Kubik Quasidualizing Modules and the Auslander and Bass Classes

Quasidualizing Modules Definition An R -module T is quasidualizing if it satisfies the following: 1. T is artinian; χ � R 2. � T → Hom R ( T , T ) is an isomorphism; and R − − Bethany Kubik Quasidualizing Modules and the Auslander and Bass Classes

Quasidualizing Modules Definition An R -module T is quasidualizing if it satisfies the following: 1. T is artinian; χ � R 2. � T → Hom R ( T , T ) is an isomorphism; and R − − 3. Ext i R ( T , T ) = 0 for all i > 0. Bethany Kubik Quasidualizing Modules and the Auslander and Bass Classes

Quasidualizing Modules Definition An R -module T is quasidualizing if it satisfies the following: 1. T is artinian; χ � R 2. � T → Hom R ( T , T ) is an isomorphism; and R − − 3. Ext i R ( T , T ) = 0 for all i > 0. Example E is a quasidualizing R -module. Bethany Kubik Quasidualizing Modules and the Auslander and Bass Classes

Quasidualizing Modules Definition An R -module T is quasidualizing if it satisfies the following: 1. T is artinian; χ � R 2. � T → Hom R ( T , T ) is an isomorphism; and R − − 3. Ext i R ( T , T ) = 0 for all i > 0. Example E is a quasidualizing R -module. Example If R is complete, then T is a quasidualizing R -module if and only if T ∨ is a semidualizing R -module. Bethany Kubik Quasidualizing Modules and the Auslander and Bass Classes

Hom-tensor adjointness Fact Let A, B, and C be R-modules. Then the natural map ψ : Hom R ( A ⊗ R B , C ) → Hom R ( A , Hom R ( B , C )) is an isomorphism. This map is called Hom-tensor adjointness. Bethany Kubik Quasidualizing Modules and the Auslander and Bass Classes

Hom-tensor adjointness Fact Let A, B, and C be R-modules. Then the natural map ψ : Hom R ( A ⊗ R B , C ) → Hom R ( A , Hom R ( B , C )) is an isomorphism. This map is called Hom-tensor adjointness. Hom-tensor adjointness explains the first and second steps in the following sequence: = Hom R ( T ∨ ⊗ R T , E ) Hom R ( T ∨ , T ∨ ) ∼ ∼ = Hom R ( T , Hom R ( T ∨ , E )) ∼ = Hom R ( T , T ) . Bethany Kubik Quasidualizing Modules and the Auslander and Bass Classes

M -reflexive modules Definition Let M be an R -module. Then an R -module L is derived M -reflexive if it satisfies the following: Bethany Kubik Quasidualizing Modules and the Auslander and Bass Classes

M -reflexive modules Definition Let M be an R -module. Then an R -module L is derived M -reflexive if it satisfies the following: 1. the natural biduality map δ M L : L → Hom R (Hom R ( L , M ) , M ) defined by l �→ [ φ �→ φ ( l )] is an isomorphism; and Bethany Kubik Quasidualizing Modules and the Auslander and Bass Classes

M -reflexive modules Definition Let M be an R -module. Then an R -module L is derived M -reflexive if it satisfies the following: 1. the natural biduality map δ M L : L → Hom R (Hom R ( L , M ) , M ) defined by l �→ [ φ �→ φ ( l )] is an isomorphism; and 2. one has Ext i R ( L , M ) = 0 = Ext i R (Hom R ( L , M ) , M ) for all i > 0. Bethany Kubik Quasidualizing Modules and the Auslander and Bass Classes

M -reflexive modules Definition Let M be an R -module. Then an R -module L is derived M -reflexive if it satisfies the following: 1. the natural biduality map δ M L : L → Hom R (Hom R ( L , M ) , M ) defined by l �→ [ φ �→ φ ( l )] is an isomorphism; and 2. one has Ext i R ( L , M ) = 0 = Ext i R (Hom R ( L , M ) , M ) for all i > 0. Remark We write G artin ( R ) to denote the class of all artinian derived M M -reflexive R -modules, G noeth ( R ) to denote the class of all M noetherian derived M -reflexive R -modules, and G mr M ( R ) to denote the class of all Matlis reflexive M -reflexive R -modules. Bethany Kubik Quasidualizing Modules and the Auslander and Bass Classes

Auslander Class Remark When M = C is a semidualizing R -module, the class G noeth ( R ) is M the class of totally C-reflexive R-modules , sometimes denoted G C ( R ). Bethany Kubik Quasidualizing Modules and the Auslander and Bass Classes

Auslander Class Remark When M = C is a semidualizing R -module, the class G noeth ( R ) is M the class of totally C-reflexive R-modules , sometimes denoted G C ( R ). Definition Let L and M be R -modules. We say that L is in the Auslander class A M ( R ) with respect to M if it satisfies the following: Bethany Kubik Quasidualizing Modules and the Auslander and Bass Classes

Auslander Class Remark When M = C is a semidualizing R -module, the class G noeth ( R ) is M the class of totally C-reflexive R-modules , sometimes denoted G C ( R ). Definition Let L and M be R -modules. We say that L is in the Auslander class A M ( R ) with respect to M if it satisfies the following: 1. the natural homomorphism γ M L : L → Hom R ( M , M ⊗ R L ), defined by l �→ ψ l where ψ l ( m ) = m ⊗ l , is an isomorphism; and Bethany Kubik Quasidualizing Modules and the Auslander and Bass Classes

Auslander Class Remark When M = C is a semidualizing R -module, the class G noeth ( R ) is M the class of totally C-reflexive R-modules , sometimes denoted G C ( R ). Definition Let L and M be R -modules. We say that L is in the Auslander class A M ( R ) with respect to M if it satisfies the following: 1. the natural homomorphism γ M L : L → Hom R ( M , M ⊗ R L ), defined by l �→ ψ l where ψ l ( m ) = m ⊗ l , is an isomorphism; and 2. one has Tor R i ( M , L ) = 0 = Ext i R ( M , M ⊗ R L ) for all i > 0. Bethany Kubik Quasidualizing Modules and the Auslander and Bass Classes

Theorem Lemma (–,Leamer, Sather-Wagstaff) Let A and M be R-modules such that A is artinian and M is Matlis reflexive. Then A ⊗ R M is Matlis reflexive. Bethany Kubik Quasidualizing Modules and the Auslander and Bass Classes