The Structure of AS-regular Algebras

Transcription

1 Department of Mathematics, Shizuoka University Shanghai Workshop 2011, 9/12

2 Noncommutative algebraic geometry Classify noncommutative projective schemes Classify finitely generated graded algebras Classify quantum projective spaces Classify AS-regular algebras

3 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.

4 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 Ext i A (k, A) 0 if i d = k(l) if i = d. A quantum projective space is a noncommutative projective scheme associated to an AS-regular algebra.

5 Theorem (Zhang) Every AS-regular algebra of dimension 2 and of Gorenstein parameter l is isomorphic to where n 2, k x 1,..., x n /( deg x 1 deg x n, n x i σ(x n+1 i )) i=1 deg x i + deg x n+1 i = l for all i, and σ Aut k k x 1,..., x n.

6 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.

7 Representation theory Classify finite dimensional algebras Classify finite dimensional algebras of finite global dimensions Classify Fano algebras

8 Theorem (Gabriel) Every finite dimensional algebra of global dimension 1 is Morita equivalent to a path algebra of a finite acyclic quiver. Example ( ) Q = 1 α 2 kq ke 1 kα + kβ = β 0 ke 2 ke 1 kα k(αβ) Q = 1 α 2 β 3 kq = 0 ke 2 kβ 0 0 ke 3

9 The double Q of a quiver Q is defined by Q 0 = Q 0 Q 1 = {α : i j, α : j i α Q 1 }. The preprojective algebra of Q is defined by ΠQ := kq/( α Q 1 αα α α). Example Q = 1 α 2 Q = 1 β α β α ΠQ = kq/(αα + ββ, α α + β β). β 2

10 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}.

11 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.

12 Definition (Minamoto) An algebra R is Fano of dimension d if gldim R = d, and L R ω 1 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 ΠR := T R (ω 1 R ).

13 Example R is a Fano algebras of dimension 0 R is a semi-simple algebra In this case, ΠR = 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, ΠR = ΠQ.

14 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 [r] := A ri 1 A ri A ri+r 2. i Z A ri r+1 A ri r+2 A ri

15 Definition The Beilinson algebra of an AS-regular algebra A of Gorenstein parameter l is defined by Lemma A := (A [l] ) 0 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).

16 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] = gr T S ((ω 1 S ) σ) = gr ΠS. D b (tails A) = D b (tails ΠS) = 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).

17 Theorem (Minamoto-Mori) Let A, B be AS-regular algebras. 1 The following are equivalent:. gr A = gr B. A = B. Π( A) = Π( B). gr Π( A) = gr Π( B). 2 The following are equivalent: D b (tails A) = D b (tails B). D b (mod A) = D b (mod B).

18 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 0 ξ 3 root of unity.

19 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 ξ 2 0 GL(3, k) where ξ k is a primitive 0 0 ξ 3 4-th root of unity.

20 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.

21 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.

22 Generalizations Definition (Minamoto-Mori) A graded algebra A is AS-regular over R of dimension d and of Gorenstein parameter l if A 0 = R, gldim R <, gldim A = d, and Ext i A (R, A) 0 if i d = (DR)(l) if i = d. An AS-regular algebra A is symmetric if ω A := D H d m (A) = A( l) as graded A-A bimodules.

23 Theorem (Minamoto-Mori) If A is an AS-regular algebra over R 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 ΠS. D b (tails A) = D b (mod S). Theorem (Minamoto-Mori) A is a preprojective algebras of Fano algebras of dimension d A is a symmetric AS-regular algebras of dimension d + 1 and of Gorenstein parameter 1.

24 {AS-regular algebras over R of dimension d} Π {Fano algebras of dimension d 1} gr Π( A) = gr A (ΠS) = S Classifying AS-regular algebras over R of dimension d 1 up to graded Morita equivalence Classifying Fano algebras of dimension d 1 up to isomorphism.

25 Graded Frobenius Algebras Definition A finite dimensional graded algebra A is graded Frobenius of Gorenstein parameter l if DA = A(l) as graded A-modules. It is graded symmetric if DA = A(l) as graded A-A bimodules. Example The trivial extension of R is defined by R := R DR = T R (DR)/T R (DR) 2.

26 Theorem (Minamoto-Mori) A is a trivial extensions of finite dimensional algebras A is a graded symmetric algebras of Gorenstein parameter 1. Definition The Beilinson algebra of a graded Frobenius algebra A of Gorenstein parameter l is defined by A := (A [l] ) 0.

27 {graded Frobenius algebras} {finite dimensional algebras} gr ( A) = gr A ( S) = S Classifying graded Frobenius algebras up to graded Morita equivalence Classifying finite dimensional algebras up to isomorphism.