Tilting theory and Cohen-Macaulay representations Osamu Iyama - PowerPoint PPT Presentation

Tilting theory and Cohen-Macaulay representations Osamu Iyama Nagoya University Osamu Iyama (Nagoya) Tilting theory and CM representations 1 / 27

Introduction Results in dimension d ≤ 2 Examples from d -representation-infinite algebras Geigle-Lenzing complete intersections Osamu Iyama (Nagoya) Tilting theory and CM representations 2 / 27

T : triangulated category Definition U ∈ T : tilting object ⇐ ⇒ • ∀ i � = 0 Hom T ( U, U [ i ]) = 0 • T = thick U (:=the smallest triangulated subcategory of T closed under direct summands and containing U ) Example: Λ ∈ K b ( proj Λ) is a tilting object Theorem (Rickard, Keller) U ∈ T : tilting object If T satisfies mild conditions (algebraic and idempotent complete), then T ≃ K b ( proj End T ( U )) Osamu Iyama (Nagoya) Tilting theory and CM representations 3 / 27

Definition Λ : Iwanaga-Gorenstein ring ⇐ ⇒ Noetherian ring with inj . dim Λ Λ = inj . dim Λ Λ < ∞ Examples : • Finite dimensional selfinjective algebra over a field k • Commutative Gorenstein local ring • Gorenstein order Definition Λ : Iwanaga-Gorenstein ring • X ∈ mod Λ : (maximal) Cohen-Macaulay ⇐ ⇒ ∀ i > 0 Ext i Λ ( X, Λ) = 0 • CM Λ : category of Cohen-Macaulay Λ -modules • CM Λ : stable category Osamu Iyama (Nagoya) Tilting theory and CM representations 4 / 27

Properties (Happel, Buchweitz, Orlov) • CM Λ is a Frobenius category • CM Λ is an algebraic triangulated category • CM Λ ≃ D b ( mod Λ) / K b ( proj Λ) Question When does CM Λ has a tilting object? Observation Λ : semiperfect CM Λ has a tilting object ⇐ ⇒ CM Λ = 0 Osamu Iyama (Nagoya) Tilting theory and CM representations 5 / 27

G : abelian group Assume Λ is G -graded, i.e. Λ = � g ∈ G Λ g and Λ g Λ h ⊂ Λ g + h • mod G Λ : category of G -graded Λ -modules • CM G Λ := { X ∈ mod G Λ | ∀ i > 0 Ext i Λ ( X, Λ) = 0 } Properties • CM G Λ is a Frobenius category • CM G Λ is an algebraic triangulated category • CM G Λ ≃ D b ( mod G Λ) / K b ( proj G Λ) Question When does CM G Λ has a tilting object? Osamu Iyama (Nagoya) Tilting theory and CM representations 6 / 27

First examples k : field k [ x ] / ( x 2 ) : Z -graded by deg x = 1 Observation • mod Z ( k [ x ] / ( x 2 )) ≃ C b ( mod k ) • mod Z ( k [ x ] / ( x 2 )) ≃ D b ( mod k ) Osamu Iyama (Nagoya) Tilting theory and CM representations 7 / 27

• Λ : finite dimensional k -algebra • T (Λ) = Λ ⊕ D Λ : trivial extension algebra • T (Λ) is a Z -graded symmetric k -algebra with T (Λ) 0 = Λ and T (Λ) 1 = D Λ Theorem (Happel) Assume gl . dim Λ is finite • Λ ∈ mod Z T (Λ) is a tilting object • mod Z T (Λ) ≃ D b ( mod Λ) Application (Tachikawa) Q : Dynkin quiver = ⇒ T ( kQ ) is representation-finite Osamu Iyama (Nagoya) Tilting theory and CM representations 8 / 27

Dimension zero • Λ = � i ≥ 0 Λ i : Z -graded selfinjective k -algebra • For X ∈ mod Z Λ , let X ≥ 0 := � i ≥ 0 X i Theorem (Yamaura) ⇒ mod Z Λ has a tilting object • gl . dim Λ < ∞ ⇐ • In this case � i ≥ 0 Λ( i ) ≥ 0 gives a tilting object Π : preprojective algebra of an n -representation-finite algebra Λ (e.g. kQ for a Dynkin quiver Q ) Corollary (I-Oppermann, Yamaura) mod Z Π ≃ D b ( mod End Λ (Π)) Osamu Iyama (Nagoya) Tilting theory and CM representations 9 / 27

Dimension one • R = � i ≥ 0 R i : commutative Gorenstein ring in dimension one • Assume R 0 = k and has a Gorenstein parameter a ≤ 0 R ( k, R ) ≃ k ( a ) in mod Z R i.e. Ext 1 Theorem (Buchweitz-I-Yamaura) i =1 R ( i ) ≥ 0 is a tilting object in CM Z R ∃ p > 0 s.t. � p Osamu Iyama (Nagoya) Tilting theory and CM representations 10 / 27

Dimension two • Q : extended Dynkin quiver • Π = kQ/ � aa ∗ − a ∗ a | a ∈ Q 1 � : preprojective algebra Π is Z -graded by deg ( a ) = 0 , deg ( a ∗ ) = 1 • e ∈ Q 0 : extended vertex Theorem (Auslander-Reiten, Geigle-Lenzing, Kajiura-Saito-Takahashi) • R := e Π e is a simple singularity in dimension two • R is CM-finite with CM R = add e Π and has an Auslander algebra End R ( e Π) = Π • CM Z R ≃ D b ( kQ/ ( e )) Osamu Iyama (Nagoya) Tilting theory and CM representations 11 / 27

d -representation-infinite algebras (Joint work with Amiot and Reiten) • k : algebraically closed field • Λ : finite dimensional k -algebra of global dimension d L ⊗ Λ − : D b ( mod Λ) → D b ( mod Λ) • ν d := D Λ[ − d ] : derived d -AR translation Definition (Herschend-I-Oppermann) ⇒ ∀ i ≥ 0 , ν − i Λ d -representation-infinite ⇐ d (Λ) ∈ mod Λ Osamu Iyama (Nagoya) Tilting theory and CM representations 12 / 27

• Γ = � i ≥ 0 Γ i : positively graded k -algebra s.t. ∀ i ≥ 0 dim k Γ i < ∞ • Γ e := Γ ⊗ k Γ op Definition (cf. Ginzburg) Γ : n -Calabi-Yau algebra of Gorenstein parameter a ⇒ • Γ ∈ K b ( proj Z Γ e ) ⇐ • RHom Γ e (Γ , Γ e ) ≃ Γ[ − n ]( a ) in D ( Mod Z Γ e ) Theorem (Minamoto-Mori, Keller) ∃ bijection given by Λ �→ Π { d -reresentation-infinite algebras } ≃ { ( d + 1) -Calabi-Yau algebras of Gorenstein parameter 1 } Osamu Iyama (Nagoya) Tilting theory and CM representations 13 / 27

Setting • Λ : d -representation-infinite algebra • Assume Π := Π(Λ) is noetherian • e ∈ Λ : idempotent s.t. dim k (Π / ( e )) < ∞ and e Λ(1 − e ) = 0 Theorem (Amiot-I-Reiten) Let R := e Π e and Λ := Λ / ( e ) • R is Iwanaga-Gorenstein • gl . dim Λ ≤ d • CM Z R has a tilting object e Π • CM Z R ≃ D b ( mod Λ) Osamu Iyama (Nagoya) Tilting theory and CM representations 14 / 27

Definition • M ∈ CM R : d -cluster tilting ⇐ ⇒ add M = { X ∈ CM R | ∀ i ∈ [1 , d − 1] Ext i R ( M, X ) = 0 } = { X ∈ CM R | ∀ i ∈ [1 , d − 1] Ext i R ( X, M ) = 0 } • R : d -CM-finite ⇐ ⇒ ∃ M ∈ CM R : d -cluster tilting C d (Λ) : d -cluster category (Amiot-Guo-Keller) Theorem (Amiot-I-Reiten) • R is d -CM-finite with a d -cluster tilting module e Π • R has a d -Auslander algebra End R ( e Π) = Π • CM R ≃ C d (Λ) Osamu Iyama (Nagoya) Tilting theory and CM representations 15 / 27

� � � � � � � � � � � � � � � � • 2 ≤ d ≤ n • ( a 1 , . . . , a d ) s.t. 0 < a i < n , ( n, a i ) = 1 , � d i =1 a i = n • G := � diag ( ζ a 1 , . . . , ζ a d ) � ⊂ SL d ( k ) • Π := k [ x 1 , . . . , x d ] ∗ G : skew group algebra vertices : { 1 , 2 , . . . , n } ≃ Z /n Z arrows : Q 1 := { x j : i → i + a j | i ∈ Q 0 , 1 ≤ j ≤ d } relations : x j x j ′ = x j ′ x j � 0 ( i ≤ i + a j ) grading : deg ( x j : i → i + a j ) = 1 ( i > i + a j ) x 1 d = n = 4 1 x 2 2 x 3 a 1 = a 2 = a 3 = a 4 = 1 Π x 4 x 3 x 2 x 1 x 4 x 4 x 1 x 2 x 3 x 4 4 3 x 3 x 2 x 1 Osamu Iyama (Nagoya) Tilting theory and CM representations 16 / 27

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Corollary Let R := k [ x 1 , . . . , x d ] G and Λ := Π 0 / ( e d ) • CM Z R ≃ D b ( mod Λ) • CM R ≃ C d (Λ) x 1 x 1 x 1 1 x 2 2 1 x 2 2 1 x 2 2 x 3 x 3 x 3 Π 0 Λ Π x 4 x 4 x 4 x 3 x 2 x 2 x 2 x 1 x 4 x 4 x 1 x 4 x 1 x 4 x 1 x 2 x 3 x 3 x 3 x 4 x 4 4 3 4 3 3 x 3 x 3 x 2 x 2 x 1 x 1 Osamu Iyama (Nagoya) Tilting theory and CM representations 17 / 27

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Corollary Let R := k [ x 1 , . . . , x d ] G and Λ := Π 0 / ( e d ) • CM Z R ≃ D b ( mod Λ) • CM R ≃ C d (Λ) x 1 x 1 x 1 1 x 2 2 1 x 2 2 1 x 2 2 x 3 x 3 x 3 Π 0 Λ Π x 4 x 4 x 4 x 3 x 2 x 2 x 2 x 1 x 4 x 4 x 1 x 4 x 1 x 4 x 1 x 2 x 3 x 3 x 3 x 4 x 4 4 3 4 3 3 x 3 x 3 x 2 x 2 x 1 x 1 Osamu Iyama (Nagoya) Tilting theory and CM representations 17 / 27

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Corollary Let R := k [ x 1 , . . . , x d ] G and Λ := Π 0 / ( e d ) • CM Z R ≃ D b ( mod Λ) • CM R ≃ C d (Λ) x 1 x 1 x 1 1 x 2 2 1 x 2 2 1 x 2 2 x 3 x 3 x 3 Π 0 Λ Π x 4 x 4 x 4 x 3 x 2 x 2 x 2 x 1 x 4 x 4 x 1 x 4 x 1 x 4 x 1 x 2 x 3 x 3 x 3 x 4 x 4 4 3 4 3 3 x 3 x 3 x 2 x 2 x 1 x 1 Osamu Iyama (Nagoya) Tilting theory and CM representations 17 / 27

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Corollary Let R := k [ x 1 , . . . , x d ] G and Λ := Π 0 / ( e d ) • CM Z R ≃ D b ( mod Λ) • CM R ≃ C d (Λ) x 1 x 1 x 1 1 x 2 2 1 x 2 2 1 x 2 2 x 3 x 3 x 3 Π 0 Λ Π x 4 x 4 x 4 x 3 x 2 x 2 x 2 x 1 x 4 x 4 x 1 x 4 x 1 x 4 x 1 x 2 x 3 x 3 x 3 x 4 x 4 4 3 4 3 3 x 3 x 3 x 2 x 2 x 1 x 1 Osamu Iyama (Nagoya) Tilting theory and CM representations 17 / 27