Ample group actions on AS-regular algebras. noncommutative graded isolated singularities. Kenta Ueyama

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1 and noncommutative graded isolated singularities Shizuoka Universit, Japan Perspectives of Representation Theor of Algebra Nagoa Universit November 15th 2013

2 AS-regular, AS-Gorenstein, noncomm graded isolated singularit This is a report on joint work with Izuru Mori k : a field A = k 1,, n /I, deg i > 0 for all i, I : homog ideal We alwas assume that A is two-sided noetherian grmod A : the categor of fin gen graded right A-modules tors A : the full subcategor consisting of fin dim modules [Artin-Zhang] The noncomm proj scheme is defined b tails A := grmod A/tors A If A is comm and deg i = 1, then tails A = coh (Proj A) [Serre] π : grmod A tails A : the quotient functor We denote objects in tails A b πm = M

3 Definition 1 (Artin-Schelter) A is AS-Gorenstein of dim d and Gor param l 1 injdim A A = injdim A op A = d <, and 2 Et i A (k, A) = Et i A op(k, A) = def { k(l) if i = d, 0 if i d def S is AS-regular S is AS-Gorenstein of gldim S = d If S is comm, then S : AS-regular of dim d S = k[ 1,, d ] Eample If S is an AS-regular algebra of dim 2, then S is isomorphic to k, /( α) (α 0) or k, /( + s ) (s N)

4 Jorgensen and Zhang introduced the notion of homological determinant for a graded algebra automorphism of S, which is the same as the usual determinant if S = k[ 1, n ], deg i = 1 Theorem 2 (Jorgensen-Zhang) S : an AS-regular algebra, G fin GrAut S, char k G If hdet σ = 1 for all σ G, then S G is AS-Gorenstein HSL(S) := {σ GrAut S hdet σ = 1} Definition 3 For a noetherian graded algebra A, A is a noncomm graded isolated singularit def gldim(tails A) < If A = k[ 1,, n ]/I, deg i = 1, then gldim(tails A) < gldim A (p) < for an homog prime ideal p m

5 The aim of this talk To stud the structure of D b (tails S G )! (S : an AS-regular algebra, G fin HSL(S)) For the purpose of this stud, we introduce a notion of ampleness of a group action on a graded algebra We will see that the ampleness of G on S is strongl related to the graded isolated singularit propert of S G

6 Definition 4 (Artin-Zhang) O tails A, s Aut(tails A) Then (O, s) is ample for tails A def s n O is projective generator for tails A for an n 0 For a graded algebra A, A (r) = i N A ri : the r-th Veronese algebra A ri A ri+1 A ri+r 1 A [r] = A ri 1 A ri i N A ri+1 A ri r+1 A ri 1 A ri : the r-th quasi-veronese algebra

7 Theorem 5 Under weak conditions, (A, (r)) : ample for tails A ( )e : tails A [r] tails A (r) is an equivalence functor where e = diag(1, 0,, 0) A [r] (idempt) and we identif ea [r] e = A (r) Proposition 6 A = k 1,, n /I, G = diag(ω deg 1,, ω deg n ) fin GrAut A where ω is a primitive r-th root of unit, then A (r) = A G, A [r] 1 = A G, e 1 σ G σ G Definition 7 G fin GrAut A is ample def ( )e : tails A G tails A G is an equivalence functor where e = 1 G σ G 1 σ

8 Definition 8 (Minamoto-Mori) S : an AS-regular algebra of Gor param l, then S 0 S 1 S l 1 S := (S [l] 0 S 0 S l 2 ) 0 = 0 0 S 0 : the quantum Beilinson algebra of S (fin dim algebra)

9 Theorem 9 (Mori-U) S : AS-regular of dim d 2, G fin HSL(S), char k G Then the following are equivalent 1 G is ample for S 2 S G is a graded isolated singularit, and S G End S G (S); s σ [t sσ(t)] is an isomorphism of graded algebras 3 S G/(e) is fin dim over k where e = 1 G σ G 1 σ S G Moreover, if one of these conditions is satisfied, then D b (tails S G ) = D b (mod ( S) G)

10 Proposition 10 (Auslander) S = k[ 1,, d ], deg i = 1, G fin SL(d, k), char k = 0 Then S G End S G (S); s σ [t sσ(t)] is an isomorphism of graded algebras Corollar 11 (MU) S = k[ 1,, d ], deg i = 1 and d 2, G fin SL(d, k), char k = 0 Then the following are equivalent 1 G is ample for S 2 S G is an isolated singularit 3 S G/(e) is fin dim over k where e = 1 G σ G 1 σ S G

11 For the rest, k : an alg closed field of char k = 0 S = k, /(f ) : an AS-regular algebra of dim 2 and of Gor param l (= deg + deg ) Assume gcd(deg, deg ) = 1 G fin HSL(S) Since S is AS-regular of dim 2, S = kq S for some quiver Q S ( S) G = kq S G = kq S,G for some quiver Q S,G Thus if G is ample, then D b (tails S G ) = D b (mod kq S,G )

12 Theorem 12 (MU) ( ) 1 ω 0 If G = 0 ω 1 where ω is a primitive r-th root of unit, then G is ample 2 If S = k, /( ± ), then ever G fin HSL(S) is of the above form ( ) ω 0 Assume that G = 0 ω 1 where ω = r 1 and r = G Ẉe now compute Q S,G such that D b (tails S G ) = D b (mod kq S,G )

13 Quiver Q S ( S = kq S ) l = deg + deg vertices: 0, 1,, l 1 arrows: i i + deg, Eample 1 deg = 1, deg = 3 : Q S = 0 2 deg = 3, deg = 5 : 0 i i + deg for an i = 0,, l B appropriate renumbering of vertices, it is equal to Q S =

14 Quiver Q S,G (( S) G = kq S,G, G = ( ) ω 0 0 ω ) r = G 1 vertices: (i, j) for all 0 i l 1 and 0 j r 1 arrows: For an i For an i i (= i + deg ) in Q S and an j Z/rZ, we draw (i, j 1) (i, j) i (= i + deg ) in Q S and an j Z/rZ, we draw (i, j + 1) (i, j) Eample 1 deg = 1, deg = 3, r = 2 : Q S = Q S,G = (0, 0) (1, 0) (2, 0) (3, 0) (0, 1) (1, 1) (2, 1) (3, 1)

15 Eample 2 deg = 3, deg = 5, r = 4 : Q S = Q S,G = (0, 0) (1, 0) (2, 0) (3, 0) (4, 0) (5, 0) (6, 0) (7, 0) (0, 1) (1, 1) (2, 1) (3, 1) (4, 1) (5, 1) (6, 1) (7, 1) (0, 2) (1, 2) (2, 2) (3, 2) (4, 2) (5, 2) (6, 2) (7, 2) (0, 3) (1, 3) (2, 3) (3, 3) (4, 3) (5, 3) (6, 3) (7, 3)