Motivation Primary Definitions and Remarks Relationship between DG Ext and Yoneda Ext DG Ext and Yoneda Ext for DG modules Saeed Nasseh Sean Sather-Wagstaff Department of Mathematics North Dakota State University The 2011 Fall Central Sectional Meeting of the AMS, Special Session on Local Commutative Algebra S. Nasseh and S. Sather-Wagstaff DG Ext and Yoneda Ext for DG modules
Motivation Primary Definitions and Remarks Relationship between DG Ext and Yoneda Ext Assumption In this talk ( R , m , k ) is assumed to be a local commutative noetherian ring with unity. S. Nasseh and S. Sather-Wagstaff DG Ext and Yoneda Ext for DG modules
Motivation Primary Definitions and Remarks Relationship between DG Ext and Yoneda Ext Assumption In this talk ( R , m , k ) is assumed to be a local commutative noetherian ring with unity. Definition The finitely generated R -module C is semidualizing if The homothety map χ R C : R → Hom R ( C , C ) given by 1 χ R C ( r )( c ) = rc is an isomorphism, and Ext i R ( C , C ) = 0 for all i > 0. 2 S. Nasseh and S. Sather-Wagstaff DG Ext and Yoneda Ext for DG modules
Motivation Primary Definitions and Remarks Relationship between DG Ext and Yoneda Ext Assumption In this talk ( R , m , k ) is assumed to be a local commutative noetherian ring with unity. Definition The finitely generated R -module C is semidualizing if The homothety map χ R C : R → Hom R ( C , C ) given by 1 χ R C ( r )( c ) = rc is an isomorphism, and Ext i R ( C , C ) = 0 for all i > 0. 2 in Vasconcelos’ analysis of divisors associated to modules 1 S. Nasseh and S. Sather-Wagstaff DG Ext and Yoneda Ext for DG modules
Motivation Primary Definitions and Remarks Relationship between DG Ext and Yoneda Ext Assumption In this talk ( R , m , k ) is assumed to be a local commutative noetherian ring with unity. Definition The finitely generated R -module C is semidualizing if The homothety map χ R C : R → Hom R ( C , C ) given by 1 χ R C ( r )( c ) = rc is an isomorphism, and Ext i R ( C , C ) = 0 for all i > 0. 2 in Vasconcelos’ analysis of divisors associated to modules 1 in studies of local ring homomorphisms by Avramov and 2 Foxby S. Nasseh and S. Sather-Wagstaff DG Ext and Yoneda Ext for DG modules
Motivation Primary Definitions and Remarks Relationship between DG Ext and Yoneda Ext Assumption In this talk ( R , m , k ) is assumed to be a local commutative noetherian ring with unity. Definition The finitely generated R -module C is semidualizing if The homothety map χ R C : R → Hom R ( C , C ) given by 1 χ R C ( r )( c ) = rc is an isomorphism, and Ext i R ( C , C ) = 0 for all i > 0. 2 in Vasconcelos’ analysis of divisors associated to modules 1 in studies of local ring homomorphisms by Avramov and 2 Foxby in Wakamatsu’s work on tilting theory 3 S. Nasseh and S. Sather-Wagstaff DG Ext and Yoneda Ext for DG modules
Motivation Primary Definitions and Remarks Relationship between DG Ext and Yoneda Ext Conjecture (W. V. Vasconcelos, 1974) The set of isomorphism classes of semidualizing modules over a Cohen-Macaulay local ring is finite. S. Nasseh and S. Sather-Wagstaff DG Ext and Yoneda Ext for DG modules
Motivation Primary Definitions and Remarks Relationship between DG Ext and Yoneda Ext Theorem (L. W. Christensen and S. Sather-Wagstaff, 2008) If R is Cohen-Macaulay and equicharacteristic, then the set of isomorphism classes of semidualizing modules is finite. S. Nasseh and S. Sather-Wagstaff DG Ext and Yoneda Ext for DG modules
Motivation Primary Definitions and Remarks Relationship between DG Ext and Yoneda Ext Theorem (L. W. Christensen and S. Sather-Wagstaff, 2008) If R is Cohen-Macaulay and equicharacteristic, then the set of isomorphism classes of semidualizing modules is finite. Outline of the proof: Reduce to the case where k is algebraically closed and R 1 is a finite dimensional k -algebra S. Nasseh and S. Sather-Wagstaff DG Ext and Yoneda Ext for DG modules
Motivation Primary Definitions and Remarks Relationship between DG Ext and Yoneda Ext Theorem (L. W. Christensen and S. Sather-Wagstaff, 2008) If R is Cohen-Macaulay and equicharacteristic, then the set of isomorphism classes of semidualizing modules is finite. Outline of the proof: Reduce to the case where k is algebraically closed and R 1 is a finite dimensional k -algebra Parametrize the set of R -modules M such that 2 dim k ( M ) = r by an algebraic scheme S. Nasseh and S. Sather-Wagstaff DG Ext and Yoneda Ext for DG modules
Motivation Primary Definitions and Remarks Relationship between DG Ext and Yoneda Ext Theorem (L. W. Christensen and S. Sather-Wagstaff, 2008) If R is Cohen-Macaulay and equicharacteristic, then the set of isomorphism classes of semidualizing modules is finite. Outline of the proof: Reduce to the case where k is algebraically closed and R 1 is a finite dimensional k -algebra Parametrize the set of R -modules M such that 2 dim k ( M ) = r by an algebraic scheme GL n ( R ) acts on this scheme so that orbits are exactly the 3 isomorphism classes S. Nasseh and S. Sather-Wagstaff DG Ext and Yoneda Ext for DG modules
Motivation Primary Definitions and Remarks Relationship between DG Ext and Yoneda Ext Theorem (L. W. Christensen and S. Sather-Wagstaff, 2008) If R is Cohen-Macaulay and equicharacteristic, then the set of isomorphism classes of semidualizing modules is finite. Outline of the proof: Reduce to the case where k is algebraically closed and R 1 is a finite dimensional k -algebra Parametrize the set of R -modules M such that 2 dim k ( M ) = r by an algebraic scheme GL n ( R ) acts on this scheme so that orbits are exactly the 3 isomorphism classes Ext vanishing implies that every semidualizing R -module 4 has an open orbit S. Nasseh and S. Sather-Wagstaff DG Ext and Yoneda Ext for DG modules
Motivation Primary Definitions and Remarks Relationship between DG Ext and Yoneda Ext Theorem (L. W. Christensen and S. Sather-Wagstaff, 2008) If R is Cohen-Macaulay and equicharacteristic, then the set of isomorphism classes of semidualizing modules is finite. Outline of the proof: Reduce to the case where k is algebraically closed and R 1 is a finite dimensional k -algebra Parametrize the set of R -modules M such that 2 dim k ( M ) = r by an algebraic scheme GL n ( R ) acts on this scheme so that orbits are exactly the 3 isomorphism classes Ext vanishing implies that every semidualizing R -module 4 has an open orbit There can only be finitely many open orbits, so there are 5 only finitely many isomorphism classes of semidualizing R -modules S. Nasseh and S. Sather-Wagstaff DG Ext and Yoneda Ext for DG modules
Motivation DG algebras and DG modules Primary Definitions and Remarks Semiprojective DG A -Modules and DG Ext Relationship between DG Ext and Yoneda Ext Definition A commutative differential graded algebra over R ( DG R-algebra for short) is an R -complex A with A i = 0 for i < 0 equipped with a chain map µ A : A ⊗ R A → A denoted µ A ( a ⊗ b ) = ab (which is called the product) that is S. Nasseh and S. Sather-Wagstaff DG Ext and Yoneda Ext for DG modules
Motivation DG algebras and DG modules Primary Definitions and Remarks Semiprojective DG A -Modules and DG Ext Relationship between DG Ext and Yoneda Ext Definition A commutative differential graded algebra over R ( DG R-algebra for short) is an R -complex A with A i = 0 for i < 0 equipped with a chain map µ A : A ⊗ R A → A denoted µ A ( a ⊗ b ) = ab (which is called the product) that is associative: for all a , b , c ∈ A we have ( ab ) c = a ( bc ) ; 1 unital: there is an element 1 ∈ A 0 such that for all a ∈ A we 2 have 1 a = a ; graded commutative: for all a , b ∈ A we have 3 ab = ( − 1 ) | a || b | ba and a 2 = 0 when | a | is odd. S. Nasseh and S. Sather-Wagstaff DG Ext and Yoneda Ext for DG modules
Motivation DG algebras and DG modules Primary Definitions and Remarks Semiprojective DG A -Modules and DG Ext Relationship between DG Ext and Yoneda Ext Definition A commutative differential graded algebra over R ( DG R-algebra for short) is an R -complex A with A i = 0 for i < 0 equipped with a chain map µ A : A ⊗ R A → A denoted µ A ( a ⊗ b ) = ab (which is called the product) that is associative: for all a , b , c ∈ A we have ( ab ) c = a ( bc ) ; 1 unital: there is an element 1 ∈ A 0 such that for all a ∈ A we 2 have 1 a = a ; graded commutative: for all a , b ∈ A we have 3 ab = ( − 1 ) | a || b | ba and a 2 = 0 when | a | is odd. Examples R is a DG R -algebra. 1 S. Nasseh and S. Sather-Wagstaff DG Ext and Yoneda Ext for DG modules
Motivation DG algebras and DG modules Primary Definitions and Remarks Semiprojective DG A -Modules and DG Ext Relationship between DG Ext and Yoneda Ext Definition A commutative differential graded algebra over R ( DG R-algebra for short) is an R -complex A with A i = 0 for i < 0 equipped with a chain map µ A : A ⊗ R A → A denoted µ A ( a ⊗ b ) = ab (which is called the product) that is associative: for all a , b , c ∈ A we have ( ab ) c = a ( bc ) ; 1 unital: there is an element 1 ∈ A 0 such that for all a ∈ A we 2 have 1 a = a ; graded commutative: for all a , b ∈ A we have 3 ab = ( − 1 ) | a || b | ba and a 2 = 0 when | a | is odd. Examples R is a DG R -algebra. 1 The Koszul complex K R ( x 1 , · · · , x n ) with x 1 , · · · , x n ∈ R is 2 a DG R -algebra. S. Nasseh and S. Sather-Wagstaff DG Ext and Yoneda Ext for DG modules
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