Noncommutative Algebraic Geometry Representation Theory Interactions Related Topics The Structure of AS-regular Algebras Izuru Mori Department of Mathematics, Shizuoka University Noncommutative Algebraic Geometry Shanghai Workshop 2011, 9/12 Izuru Mori The Structure of AS-regular Algebras
Noncommutative Algebraic Geometry Representation Theory Interactions Related Topics Noncommutative algebraic geometry Classify noncommutative projective schemes ⇓ Classify finitely generated graded algebras Classify quantum projective spaces ⇓ Classify AS-regular algebras Izuru Mori The Structure of AS-regular Algebras
Noncommutative Algebraic Geometry Representation Theory Interactions Related Topics For simplicity, we assume that k = k , and A is a graded right coherent algebra over k . gr A = the abelian category of finitely presented graded right A -modules. tors A = the full subcategory of finite dimensional modules. Definition (Artin-Zhang) The noncommutative projective scheme associated to A is defined by tails A := gr A/ tors A . Izuru Mori The Structure of AS-regular Algebras
Noncommutative Algebraic Geometry Representation Theory Interactions Related Topics AS-regular algebras Definition (Artin-Schelter) An N -graded algebra A is AS-regular of dimension d and of Gorenstein parameter ℓ if A 0 = k (connected graded), gldim A = d , and 0 if i � = d A ( k, A ) ∼ Ext i = k ( ℓ ) if i = d. A quantum projective space is a noncommutative projective scheme associated to an AS-regular algebra. Izuru Mori The Structure of AS-regular Algebras
Noncommutative Algebraic Geometry Representation Theory Interactions Related Topics Theorem (Zhang) Every AS-regular algebra of dimension 2 and of Gorenstein parameter ℓ is isomorphic to n � k � x 1 , . . . , x n � / ( x i σ ( x n +1 − i )) i =1 where n ≥ 2 , deg x 1 ≤ · · · ≤ deg x n , deg x i + deg x n +1 − i = ℓ for all i , and σ ∈ Aut k k � x 1 , . . . , x n � . Izuru Mori The Structure of AS-regular Algebras
Noncommutative Algebraic Geometry Representation Theory Interactions Related Topics Theorem (Artin-Tate-Van den Bergh) Quadratic AS-regular algebras of dimension 3 and of finite GKdimension were classified by geometric triples ( E, σ, L ) where E ⊂ P 2 , σ ∈ Aut k E , and L ∈ Pic E . Izuru Mori The Structure of AS-regular Algebras
Noncommutative Algebraic Geometry Representation Theory Interactions Related Topics Representation theory Classify finite dimensional algebras � Classify finite dimensional algebras of finite global dimensions � Classify Fano algebras Izuru Mori The Structure of AS-regular Algebras
� Noncommutative Algebraic Geometry Representation Theory Interactions Related Topics Theorem (Gabriel) Every finite dimensional algebra of global dimension 1 is Morita equivalent to a path algebra of a finite acyclic quiver. Example � � ke 1 kα + kβ α kQ ∼ � 2 Q = 1 = 0 ke 2 β ke 1 kα k ( αβ ) β kQ ∼ α � 2 � 3 Q = 1 = 0 ke 2 kβ 0 0 ke 3 Izuru Mori The Structure of AS-regular Algebras
� � � � � Noncommutative Algebraic Geometry Representation Theory Interactions Related Topics The double Q of a quiver Q is defined by Q 1 = { α : i → j, α ∗ : j → i | α ∈ Q 1 } . Q 0 = Q 0 The preprojective algebra of Q is defined by α ∈ Q 1 αα ∗ − α ∗ α ) . Π Q := kQ/ ( � Example α β α Q = 1 � 2 Q = 1 2 α ∗ β β ∗ Π Q = kQ/ ( αα ∗ + ββ ∗ , α ∗ α + β ∗ β ) . Izuru Mori The Structure of AS-regular Algebras
Noncommutative Algebraic Geometry Representation Theory Interactions Related Topics Fano algebras Let R be a finite dimensional algebra. D := D b (mod R ) has a standard t -structure D ≥ 0 := { M ∈ D | h i ( M ) = 0 for all i < 0 } D ≤ 0 := { M ∈ D | h i ( M ) = 0 for all i > 0 } . For s ∈ Aut k D , we define D s, ≥ 0 := { M ∈ D | s i ( M ) ∈ D ≥ 0 for all i ≫ 0 } D s, ≤ 0 := { M ∈ D | s i ( M ) ∈ D ≤ 0 for all i ≫ 0 } . Izuru Mori The Structure of AS-regular Algebras
Noncommutative Algebraic Geometry Representation Theory Interactions Related Topics Definition (Minamoto) s ∈ Aut k D is ample if s i ( R ) ∈ D ≥ 0 ∩ D ≤ 0 ∼ = mod R for all i ≥ 0 , and ( D s, ≥ 0 , D s, ≤ 0 ) is a t-structure for D . Theorem (Minamoto) If s ∈ Aut k D is ample, then ( R, s ) is ample for H := D s, ≥ 0 ∩ D s, ≤ 0 in the sense of Artin-Zhang. Izuru Mori The Structure of AS-regular Algebras
Noncommutative Algebraic Geometry Representation Theory Interactions Related Topics Definition (Minamoto) An algebra R is Fano of dimension d if gldim R = d , and R ω − 1 − ⊗ L R ∈ Aut k D is ample where DR := Hom k ( R, k ) and ω R := DR [ − d ] . The preprojective algebra of a Fano algebra R is defined by Π R := T R ( ω − 1 R ) . Izuru Mori The Structure of AS-regular Algebras
Noncommutative Algebraic Geometry Representation Theory Interactions Related Topics Example R is a Fano algebras of dimension 0 ⇔ R is a semi-simple algebra In this case, Π R ∼ = R [ x ] Example R is a basic Fano algebras of dimension 1 ⇔ R ∼ = kQ where Q is a finite acyclic non-Dynkin quiver. In this case, Π R ∼ = Π Q . Izuru Mori The Structure of AS-regular Algebras
Noncommutative Algebraic Geometry Representation Theory Interactions Related Topics Interactions Definition For a graded algebra A = ⊕ i ∈ Z A i and r ∈ N + , we define the r -th quasi-Veronese algebra of A by A ri A ri +1 · · · A ri + r − 1 A ri − 1 A ri · · · A ri + r − 2 A [ r ] := � . . . . ... . . . . . . i ∈ Z A ri − r +1 A ri − r +2 · · · A ri Izuru Mori The Structure of AS-regular Algebras
Noncommutative Algebraic Geometry Representation Theory Interactions Related Topics Definition The Beilinson algebra of an AS-regular algebra A of Gorenstein parameter ℓ is defined by ∇ A := ( A [ ℓ ] ) 0 Lemma For any graded algebra A and r ∈ N + , gr A [ r ] ∼ = gr A . Lemma For any algebra R , R - R bimodule M and σ ∈ Aut k R , gr T R ( M σ ) ∼ = gr T R ( M ) . Izuru Mori The Structure of AS-regular Algebras
Noncommutative Algebraic Geometry Representation Theory Interactions Related Topics Theorem (Minamoto-Mori) If A is an AS-regular algebra of dimension d ≥ 1 , then S := ∇ A is a Fano algebra of dimension d − 1 . A [ ℓ ] ∼ = T S (( ω − 1 S ) σ ) for some σ ∈ Aut k S . gr A ∼ = gr A [ ℓ ] ∼ S ) σ ) ∼ = gr T S (( ω − 1 = gr Π S . D b (tails A ) ∼ = D b (tails Π S ) ∼ = D b (mod S ) . Example (Beilinson) Applying to A = k [ x 1 , . . . , x n ] , deg x i = 1 , D b (coh P n − 1 ) ∼ = D b (tails A ) ∼ = D b (mod ∇ A ) . Izuru Mori The Structure of AS-regular Algebras
Noncommutative Algebraic Geometry Representation Theory Interactions Related Topics Theorem (Minamoto-Mori) Let A, B be AS-regular algebras. The following are equivalent: 1 gr A ∼ = gr B . ∇ A ∼ = ∇ B . Π( ∇ A ) ∼ = Π( ∇ B ) . gr Π( ∇ A ) ∼ = gr Π( ∇ B ) . The following are equivalent: 2 D b (tails A ) ∼ = D b (tails B ) . D b (mod ∇ A ) ∼ = D b (mod ∇ B ) . Izuru Mori The Structure of AS-regular Algebras
� � � � Noncommutative Algebraic Geometry Representation Theory Interactions Related Topics Example A = k [ x, y ] , deg x = 1 , deg y = 3 ⇒ A is an AS-regular algebra of dimension 2 ∇ A ∼ ⇒ = kQ is a Fano algebra of dimension 1 ⇒ Q = • • (extended Dynkin) • • Q is a reduced McKay quiver of �� �� ξ 0 ≤ SL(2 , k ) where ξ ∈ k is a primitive 4-th ξ 3 0 root of unity. Izuru Mori The Structure of AS-regular Algebras
� � � � � Noncommutative Algebraic Geometry Representation Theory Interactions Related Topics Example A = k � x, y, z � / ( xz + y 2 + zx ) deg x = 1 , deg y = 2 , deg z = 3 ⇒ A is an AS-regular algebra of dimension 2 ∇ A ∼ ⇒ = kQ is a Fano algebra of dimension 1 Q = • • ⇒ (not extended Dynkin) � � ������� � � � � � � • • Q is a reduced McKay quiver of � ξ 0 0 � ξ 2 0 0 ≤ GL(3 , k ) where ξ ∈ k is a primitive ξ 3 0 0 4-th root of unity. Izuru Mori The Structure of AS-regular Algebras
Noncommutative Algebraic Geometry Representation Theory Interactions Related Topics Example A = k � x, y � / ( x 2 y − yx 2 , xy 2 − y 2 x ) , deg x = deg y = 1 ⇒ A is an AS-regular algebra of dimension 3 ∇ A ∼ ⇒ = kQ/I is a Fano algebra of dimension 2 ⇒ Q = • �� • �� • �� • �� �� ξ 0 Q is a reduced McKay quiver of ≤ GL(2 , k ) 0 ξ where ξ ∈ k is a primitive 4-th root of unity. Izuru Mori The Structure of AS-regular Algebras
Noncommutative Algebraic Geometry Representation Theory Interactions Related Topics AS-regular algebras (of dimension 2) can be classified by (reduced) McKay quivers of a finite cyclic subgroups of GL( n, k ) up to graded Morita equivalence. Izuru Mori The Structure of AS-regular Algebras
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