Rigid Cohen-Macaulay modules over aquotient singularities of dimension threeYuji Yoshino
Math. Department, Faculty of ScienceOkayama University, Okayama 700-8530, Japanyoshino@math.okayama-u.ac.jp
This is partly a joint work with O. Iyama.
Let G be a finite subgroup of GL(3, C) which naturally acts on the formal power
series ring S = k[[x, y, z]]. We denote by R the invariant subring of S by this action ofG. We employ the following assumptions on the group G:(i) Any element σ 6= 1 of G does not have eigen value 1.
(ii) G is a subgroup of SL(3, C).
It is easy to see that the conditions (i) and (ii) are equivalent to that R is a Gorensteinring with isolated singularity. We are interested in the category of maximal CohenMacaulay modules CM(R) over R.
Under the assumptions, we know that the Auslander-Reiten-Serre duality holds for
the Tate cohomologies, i.e.
i2−iExt3R (ĚxtR (M, N ), R) ∼= ĚxtR (N, M ),for any i ∈ Z and M, N ∈ CM(R). In other words, the stable category CM(R) is a2-Calabi-Yau category.
To stateP the first main theorem, we denote by Λ the skew-group ring of G over S,i.e. Λ = σ∈G Sσ with multiplication (sσ)(tτ ) = sσ(t)στ for s, t ∈ S and σ, τ ∈ G.
P1Now let ρ be an idempotent |G|σ∈G σ of Λ and consider the residue ring Λ = Λ/ΛρΛ.
Note that Λ is a finite R-algebra which is an artinian ring. We denote by Λ-mod thecategory of finite left Λ-modules.
Theorem 0.1 There is a cateory equivalence between CM(R)/[S] and Λ-mod.
This theorem can be applied to several cases to classify particular modules. To givesuch an example, letζσ =  ζ ,
ζ1where ζ is a primitive cubic root of unity, and let G be a cyclic group of order threewhich is generated by σ. In this case, the invariant subring R is (the completion of)
the Veronese subring of degree three :R = k[[{monomials of degree three in x, y, z, }]]The action of G gives the G-graded structure on S in such a way that S = S0 ⊕ S1 ⊕ S2 ,where each Sj is the R-module of semi-invariants that is defined as
Sj = {f ∈ S | f σ = ζ j f }.
Note that S0 = R. It is known that Sj (j ∈ G) are maximal Cohen-Macaulay modules
over R, and in particular they are reflexive R-modules of rank one, whose classes form
the divisor class group of R. Particularly, {Sj } are all of the maximal Cohen-Macaulay
modules of rank one over R.
On the other hand, it follows from Theorem 0.1 that the category CM(R) of maximal Cohen-Macaulay modules over R is of wild representation type. Actually, Λ-modis nothing but the category of representations of the wild quiver :µ¶−→
Q = • −→ •−→Henceforth, we are interested in rigid maximal Cohen-Macaulay modules that isdefined as follows:
Definition 0.2 An R-module M is called rigid if Ext1R (M, M ) = 0.
By computation, the modules Sj (j ∈ G) and their any syzygies and any cosyzygies
are rigid (and indecomposable). The second main theorem is the following:
Theorem 0.3 Under such circumstances as above, let S be a sequence of indecomposable rigid maximal Cohen-Macaulay modules defined as follows:
S = (· · · , Ω−2 S1 , Ω−2 S2 , Ω−1 S1 , Ω−1 S2 , S1 , S2 , Ω1 S1 , Ω1 S2 , Ω2 S1 , Ω2 S2 , · · · ).
Then any rigid maximal Cohen-Macaulay module over R is isomorphic to a module ofthe following form:
P a ⊕ Qb ⊕ R c ,where a, b, c are nonnegative integers and {P, Q} is a pair of two adjacent modules inthe sequence S.

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