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...esting to generalize our result to non-cyclic quotient singularities. This could then be regarded as an analog of a triangle equivalence CMZ(SG) ' Db(Λ) for some finite dimensional algebra Λ given in =-=[IT10]-=-. Results of a similar flavor have been shown in previous papers. In [Ami09, ART11, AIRT12], it was shown that the 2-Calabi-Yau categories Cw associated with elements w in Coxeter groups in [BIRS09] a...

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...G is small, then the natural map S#G → EndR(S) given by sg 7→ (t 7→ s · g(t)) (s, t ∈ S, g ∈ G) is an isomorphism. Proof. (1) See [Bou68] for example. (2) This is due to Auslander [Aus86, Yos90] (see =-=[IT]-=- for a detailed proof). This immediately gives us a rich source of CT modules: Theorem 5.8. Let k be a field of characteristic zero, and let S be the polynomial ring k[x1, . . . , xd] (resp. the for...

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...modules. The study of CohenMacaulay modules (or lattices) over orders is a classical subject in representation theory. We refer to [4, 13, 40, 41] for a general background on this subject and also to =-=[2, 3, 14, 25, 27, 28]-=- for recent results about connections with tilting theory. Recently a strong connection between Cohen-Macaulay representation theory and cluster categories has been found [2, 14, 25, 31]. We will enla...

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...g. [ARS, II.2.1], and Fact 1.7(2) can be found in most commutative algebra textbooks. Skew group rings are also a classical topic. There are now many proofs of 1.10, see for example [EG02], [M96] and =-=[IT]-=-. Auslander’s original proof is outlined in [Y90]. 1.5. Exercises. Exercise 1.12. Let R be a commutative ring and let M be an R-module. Define EndR(M) := {f :M →M | f is an R-module homomorphism}. (1)...

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...)i[M,Fi] = l∑ i=0 (−1)i[N,Fi] = [N, Y ]. In the paper we use the theory of Auslander-Reiten (abbr. AR) sequences of graded Cohen-Macaulay modules. For the detail, we recommend the reader to look at =-=[9, 3, 4]-=- and [13, Chapter 15.]. We denote by CMZ0 (R) the full subcategory of CMZ(R) consisting of M ∈ CMZ(R) such that M(p) is R(p)-free for any graded prime ideal p 6= m. Theorem 2.5. [3, 4, 13, 9] Let (R,m...

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...on Cohen-Macaulay modules (or lattices) over orders. Recently, strong connections between Cohen-Macaulay representation theory and tilting theory, especially cluster categories, have been established =-=[1, 2, 9, 20, 21, 22, 24]-=-. This paper enlarges some of these connections by dealing with frozen Jacobian algebras associated with tagged triangulations of once-punctured polygons from the viewpoint of Cohen-Macaulay represent...

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...nd G ≤ GL(d, k) a finite subgroup acting on S. The derived category Db(tails SG) and the graded stable category CMZ(SG) have been studied in two situations, namely, when S is generated in degree 1 in =-=[4]-=-, and when G = 〈diag(ξdeg x1 , . . . , ξdeg xd)〉 is a finite cyclic group where ξ ∈ k× is a primitive r-th root of unity so that SG = S(r) is the r-th Veronese algebra of S in [14] and [1, Section 5] ...