Happels functor and homologically well-graded Iwanaga-Gorenstein - PowerPoint PPT Presentation

Happel’s functor and homologically well-graded Iwanaga-Gorenstein algebras H. Minamoto and K. Yamaura Kota Yamaura University of Yamanashi Japan 1

For simplicity, • K : field • D = Hom K ( − , K ) • algebra = finite dimensional K -algebra • module = finitely generated right module • mod Λ : the category of finitely generated right Λ -modules 2

1. Motivation 3

Today. We study some functor from the derived category to the stable category. 4

Today. We study some functor from the derived category to the stable category. Def. An algebra A is called Iwanaga-Gorenstein if inj . dim A A < ∞ , inj . dim A A < ∞ 5

Today. We study some functor from the derived category to the stable category. Def. An algebra A is called Iwanaga-Gorenstein if inj . dim A A < ∞ , inj . dim A A < ∞ Def. A : IG-algebra 6

Today. We study some functor from the derived category to the stable category. Def. An algebra A is called Iwanaga-Gorenstein if inj . dim A A < ∞ , inj . dim A A < ∞ Def. A : IG-algebra def Ext > 0 • M ∈ mod A is Cohen-Macaulay A ( M, A ) = 0 ⇐ ⇒ { � } � Ext > 0 • CM ( A ) := M ∈ mod A A ( M, A ) = 0 � 7

Today. We study some functor from the derived category to the stable category. Def. An algebra A is called Iwanaga-Gorenstein if inj . dim A A < ∞ , inj . dim A A < ∞ Def. A : IG-algebra def Ext > 0 • M ∈ mod A is Cohen-Macaulay A ( M, A ) = 0 ⇐ ⇒ { � } � Ext > 0 • CM ( A ) := M ∈ mod A A ( M, A ) = 0 � Fact. Since A is IG, CM ( A ) is a Frobenius category. The stable category CM ( A ) has a structure of triangulated category. 8

Today. We study some functor from the derived category to the stable category. Def. An algebra A is called Iwanaga-Gorenstein if inj . dim A A < ∞ , inj . dim A A < ∞ Def. A : IG-algebra def Ext > 0 • M ∈ mod A is Cohen-Macaulay A ( M, A ) = 0 ⇐ ⇒ { � } � Ext > 0 • CM ( A ) := M ∈ mod A A ( M, A ) = 0 � Fact. Since A is IG, CM ( A ) is a Frobenius category. The stable category CM ( A ) has a structure of triangulated category. Rem. If A is self-injective, CM ( A ) = mod A . 9

Today. We study some functor from the derived category to the stable category. Def. An algebra A is called Iwanaga-Gorenstein if inj . dim A A < ∞ , inj . dim A A < ∞ Def. A = ⊕ ℓ i =0 A i : Z -graded IG-algebra def Ext > 0 • M ∈ mod Z A is Cohen-Macaulay A ( M, A ) = 0 ⇐ ⇒ { � } � Ext > 0 • CM Z ( A ) := M ∈ mod Z A A ( M, A ) = 0 � Fact. Since A is IG, CM Z ( A ) is a Frobenius category. The stable category CM Z ( A ) has a structure of triangulated category. Rem. If A is self-injective, CM Z ( A ) = mod Z A . 10

For a Z -graded IG-algebra A = ⊕ ℓ i =0 A i , ∃ H : D b ( mod ∇ A ) → CM Z ( A ) . 11

For a Z -graded IG-algebra A = ⊕ ℓ i =0 A i , ∃ H : D b ( mod ∇ A ) → CM Z ( A ) . Def. (X-W Chen, Mori) A = ⊕ ℓ i =0 A i : Z -graded algebra The algebra ∇ A is called the Beilinson algebra of A :   A 0 A 1 A 2 · · · A ℓ − 2 A ℓ − 1   A 0 A 1 · · · A ℓ − 3 A ℓ − 2       ∇ A :=  · · · · · · · · ·      A 0 A 1       O A 0 12

Def. A = ⊕ ℓ i =0 A i : Z -graded IG-algebra. H is defined as follows. ≃ D b ( mod Z A ) → D sg ( A ) → CM Z ( A ) H : D b ( mod ∇ A ) → − 13

Def. A = ⊕ ℓ i =0 A i : Z -graded IG-algebra. H is defined as follows. ≃ A ) → D b ( mod Z A ) → D sg ( A ) → CM Z ( A ) H : D b ( mod ∇ − . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An abelian subcategory � { } mod [0 ,ℓ − 1] A := M ∈ mod Z A � M i = 0 for i ̸∈ [0 , ℓ − 1] � of mod Z A 14

Def. A = ⊕ ℓ i =0 A i : Z -graded IG-algebra. H is defined as follows. ≃ A ) → D b ( mod Z A ) → D sg ( A ) → CM Z ( A ) H : D b ( mod ∇ − . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An abelian subcategory � { } mod [0 ,ℓ − 1] A := M ∈ mod Z A � M i = 0 for i ̸∈ [0 , ℓ − 1] � of mod Z A has a canonical projective generator T such that End Z A ( T ) ≃ ∇ A. 15

Def. A = ⊕ ℓ i =0 A i : Z -graded IG-algebra. H is defined as follows. ≃ A ) → D b ( mod Z A ) → D sg ( A ) → CM Z ( A ) H : D b ( mod ∇ − . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An abelian subcategory � { } mod [0 ,ℓ − 1] A := M ∈ mod Z A � M i = 0 for i ̸∈ [0 , ℓ − 1] � of mod Z A has a canonical projective generator T such that End Z A ( T ) ≃ ∇ A. So by Morita theory A ≃ mod [0 ,ℓ − 1] A ֒ → mod Z A. mod ∇ 16

Def. A = ⊕ ℓ i =0 A i : Z -graded IG-algebra. H is defined as follows. ≃ A ) → D b ( mod Z A ) → D sg ( A ) → CM Z ( A ) H : D b ( mod ∇ − . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Def. (Buchweitz) A = ⊕ i ≥ 0 A i : Z -graded algebra. The following Verdier quotient is called the singular derived category. D sg ( A ) := D b ( mod Z A ) / K b ( proj Z A ) 17

Def. A = ⊕ ℓ i =0 A i : Z -graded IG-algebra. H is defined as follows. ≃ A ) → D b ( mod Z A ) → D sg ( A ) → CM Z ( A ) H : D b ( mod ∇ − . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Def. (Buchweitz) A = ⊕ i ≥ 0 A i : Z -graded algebra. The following Verdier quotient is called the singular derived category. D sg ( A ) := D b ( mod Z A ) / K b ( proj Z A ) Thm. (Buchweitz) If A is IG, then ≃ ∃ CM Z ( A ) − → D sg ( A ) 18

Def. A = ⊕ ℓ i =0 A i : Z -graded IG-algebra. H is defined as follows. ≃ A ) → D b ( mod Z A ) → D sg ( A ) CM Z ( A ) H : D b ( mod ∇ − → . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Why we study H ? This functor H often becomes fully faithful or an equivalence. In the case A is self-injective, it is known a necessary and sufficient condition for H to be fully faithful or an equivalence. 19

Def. A = ⊕ ℓ i =0 A i : Z -graded algebra A is right ℓ -strictly well-graded (right swg) 20

Def. A = ⊕ ℓ i =0 A i : Z -graded algebra def Hom Z A is right ℓ -strictly well-graded A ( A 0 , A ( j )) ⇐ ⇒ (right swg) 21

Def. A = ⊕ ℓ i =0 A i : Z -graded algebra def Hom Z A is right ℓ -strictly well-graded A ( A 0 , A ( j )) = 0 for all j ̸ = ℓ ⇐ ⇒ (right swg) 22

Def. A = ⊕ ℓ i =0 A i : Z -graded algebra def Hom Z A is right ℓ -strictly well-graded A ( A 0 , A ( j )) = 0 for all j ̸ = ℓ ⇐ ⇒ (right swg) Ex. A = A 0 ⊕ A 1 ⊕ A 2 A is right 2 -swg ⇔ deg deg deg 0 0 A 0 → A 0 − 1 A 0 − 2 A 0 − 0 1 A 1 0 A 0 → A 1 − 1 A 1 − 2 A 2 1 A 2 0 A 0 → A 2 Hom Z Hom Z Hom Z A ( A 0 , A ) = 0 A ( A 0 , A (1)) = 0 A ( A 0 , A (2)) = Hom A ( A 0 , A ) 23

Def. A = ⊕ ℓ i =0 A i : Z -graded algebra def Hom Z A is right ℓ -strictly well-graded A ( A 0 , A ( j )) = 0 for all j ̸ = ℓ ⇐ ⇒ (right swg) Rem. • A = ⊕ ℓ i =0 A i : Z -graded self-injective algebra A is right ℓ -swg ⇔ A is left ℓ -swg • A = ⊕ ℓ i =0 A i : basic Z -graded algebra A is swg self-injective ⇔ DA ≃ A ( ℓ ) in mod Z A . 24

Thm. (X-W Chen, Happel, Minamoto-Mori Orlov, Y) A = ⊕ ℓ i =0 A i : Z -graded self-injective algebra A ) → mod Z ( A ) H : D b ( mod ∇ (1) H is fully faithful ⇔ A is swg.  A is swg   (2) H is an equivalence ⇔ gl . dim A 0 < ∞   25

Thm. (X-W Chen, Happel, Minamoto-Mori Orlov, Y) A = ⊕ ℓ i =0 A i : Z -graded self-injective algebra A ) → mod Z ( A ) H : D b ( mod ∇ (1) H is fully faithful ⇔ A is swg.  A is swg   (2) H is an equivalence ⇔ gl . dim A 0 < ∞   Rem. Original result due to Happel. He had studied the case that A = Λ ⊕ D Λ is the trivial extension of an algebra Λ by D Λ . So we call H Happel’s functor. 26

Thm. (X-W Chen, Happel, Minamoto-Mori Orlov, Y) A = ⊕ ℓ i =0 A i : Z -graded self-injective algebra A ) → mod Z ( A ) H : D b ( mod ∇ (1) H is fully faithful ⇔ A is swg.  A is swg   (2) H is an equivalence ⇔ gl . dim A 0 < ∞   Rem. Original result due to Happel. He had studied the case that A = Λ ⊕ D Λ is the trivial extension of an algebra Λ by D Λ . So we call H Happel’s functor. Aim. Give an IG-analogue of this result. 27

2. Our results 28

Rem. A = ⊕ ℓ i =0 A i : Z -graded IG-algebra A is swg ⇒ H is fully faithful ?? 29

Rem. A = ⊕ ℓ i =0 A i : Z -graded IG-algebra A is swg ⇒ H is fully faithful ?? No !! ⇝ 30

Rem. A = ⊕ ℓ i =0 A i : Z -graded IG-algebra A is swg ⇒ H is fully faithful ?? No !! ⇝ Recall. A = ⊕ ℓ i =0 A i : Z -graded algebra Hom Z A is right strictly well-graded A ( A 0 , A ( j )) = 0 for all j ̸ = ℓ ⇐ ⇒ 31