Graded maximal Cohen-Macaulay modules over. Noncommutative graded Gorenstein isolated singularities. Kenta Ueyama. ICRA XV, Bielefeld, August 2012
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1 Graded maximal Cohen-Macaulay modules over noncommutative graded Gorenstein isolated singularities Shizuoka University, Japan ICRA XV, Bielefeld, August 2012
2 Notations Throughout this talk, k : an algebraically closed field of characteristic 0. A : a finitely generated noetherian connected graded k-algebra, ie, A = k x 1,..., x n /I where I : homogeneous, deg x i N +. grmod A : the category of f.g. graded right A-modules. tors A : the full subcategory of grmod A consisting of f.d. modules. tails A := grmod A/tors A. tails A is called the noncommutative projective scheme associated to A (cf. [Artin-Zhang]). Theorem 1 (Serre s theorem) If A = k[x 1,..., x n ]/I, deg x i = 1, then tails A = coh (Proj A).
3 Notations Definition 2 A is a graded isolated singularity def homological dimension of tails A is finite, ie, sup{i Ext i tails A(M, N ) 0 for some M, N tails A} <. If A = k[x 1,..., x n ]/I, deg x i = 1, then hdim(tails A) < A (p) is regular for any homogeneous prime ideal p m. Definition 3 A is AS-Gorenstein (AS-regular) of dim d and G-param l 1 id A A = id A op A = d < (gldim A = d < ), and { 2 Ext i A(k, A) = Ext i A op(k, A) k(l) if i = d, = 0 if i d. def
4 Notations The aim of this talk To study AS-Gorenstein isolated singularities!
5 ... Let A be an AS-Gorenstein algebra of dimension d. 1 H i m(m) := lim n Ext i A(A/A n, M). 2 CM gr (A) := {M grmod A H i m(m) = 0 i d}. 3 CM gr (A): the stable category of CM gr (A). (CM gr (A) is a triangulated category w.r.t. M[ 1] = ΩM.) Motivating Theorem (cf. [Iyama-Takahashi]) Let R be a noetherian commutative graded local Gorenstein ring. Assume that R is an isolated singularity. Then CM gr (R) has the Serre functor.
6 Let A be an AS-Gorenstein algebra of dimension d and G-param l. ω A := H d m(a) = Aν ( l) ν GrAut A where A ν is a graded A-A bimodule with action b x a = bxν(a). ω A is called the canonical module of A. Then the autoequivalence A ω A : grmod A grmod A induces an autoequivalence A ω A : CM gr (A) CM gr (A). Theorem 4 Let A be an AS-Gorenstein algebra of dimension d 2. TFAE. 1 A is a graded isolated singularity. 2 CM gr (A) has the Serre functor A ω A [d 1].
7 Proposition 5 Let A be an AS-regular algebra of dimension 2, and let G be a finite subgroup of GrAut A such that hdet σ = 1 for all σ G. Then the fixed subring A G is CM-representation-finite. Moreover, A G is an AS-Gorenstein isolated singularity. Example Let and let G = A = k x, y /(xy + yx), deg x = deg y = 1, ( ) 0 1 GrAut A. Then A 1 0 G is an AS-Gorenstein isolated singularity. Hence CM gr (A G ) has the Serre functor.
8 Definition 6 Let A be an AS-Gorenstein algebra. X CM gr (A) is called an n-cluster tilting module if add{x (s) s Z} = {M CM gr (A) Ext i A(M, X ) = 0 (0 < i < n)} = {M CM gr (A) Ext i A(X, M) = 0 (0 < i < n)}. A is CM-representation-finite A has a 1-cluster tilting module. The existence of n-cluster tilting module is a generalization of the notion of CM-representation finiteness.
9 Motivating Theorem (cf. [Iyama-Takahashi]) Let S = k[x 1,..., x d ], deg x i = 1, G a finite subgroup of SL d (k), and S G the fixed subring of S. Then 1 S G = End S G (S) as graded algebras. 2 Assume that S G is an isolated singularity. Then S CM gr (S G ) is a (d 1)-cluster tilting module...
10 Theorem 7 Let A = k x 1,..., x n /I be an AS-regular domain of dimension d 2 and G-param l, deg x i = 1. Take r N + such that r l. ξ ξ... G =..... GrAut A ξ where ξ is a primitive r-th root of unity. Then 1 A G = End A G (A) as graded algebras. 2 A G is an AS-Gorenstein isolated singularity, and A CM gr (A G ) is a (d 1)-cluster tilting module. 3 gldim End A G (A) = d. We can obtain examples of over non-orders.
11 Example Let A = k x, y /(αxy 2 + βyxy + αy 2 x + γx 3, αyx 2 + βxyx + αx 2 y + γy 3 ), deg x = deg y = 1 where α, β, γ k are generic scalars. Then A is an AS-regular algebra of dimension 3 and G-param 4. Let ( ) ξ 0 G = GrAut A 0 ξ where ξ is a primitive 4-th root of unity. Then A G is an AS-Gorenstein isolated singularity, and A CM gr (A G ) is a 2-cluster tilting module. Moreover, we have gldim End A G (A) = 3.
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