In this article we develop a new method to deal with maximal Cohen-Macaulay modules over non-isolated surface singularities. In particular, we give a negative answer on an old question of Schreyer about surface singularities with only countably many indecomposable maximal Cohen-Macaulay modules. Next, we prove that the degenerate cusp singularities have tame Cohen-Macaulay representation type. Our approach is illustrated on the case of x, y, z /(xyz) as well as several other rings. This study of maximal Cohen-Macaulay modules over non-isolated singularities leads to a new class of problems of linear algebra, which we call representations of decorated bunches of chains. We prove that these matrix problems have tame representation type and describe the underlying canonical forms. like to thank Duco van Straten for helpful discussions and Wassilij Gnedin for pointing out numerous misprints in the previous version of this paper. Our special thanks go to Lesya Bodnarchuk for supplying us with TikZ pictures illustrating representation theory of decorated bunches of chains.
Generalities on maximal Cohen-Macaulay modulesLet (A, m) be a Noetherian local ring, = A/m its residue field and d = kr. dim(A) its Krull dimension. Throughout the paper A−mod denotes the category of Noetherian (i.e. finitely generated) A-modules, whereas A−Mod stands for the category of all A-modules, Q = Q(A) is the total ring of fractions of A and P is the set of prime ideals of height 1.Maximal Cohen-Macaulay modules over surface singularities. In this article we focus on the study of maximal Cohen-Macaulay modules over Noetherian rings of Krull dimension two, also called surface singularities. This case is actually rather special because of the following well-known lemma. Lemma 2.2. Let (A, m) be a surface singularity, N be a maximal Cohen-Macaulay Amodule and M a Noetherian A-module. Then the A-module Hom A (M, N ) is maximal Cohen-Macaulay.Proof. From a free presentation A n ϕ → A m → M → 0 of M we obtain an exact sequence:Proposition 2.6. The functor B ⊠ A − : CM(A) → CM(B) mapping a maximal Cohen-Macaulay module M to B ⊠ A M := (B ⊗ A M ) † is left adjoint to the forgetful functor CM(B) → CM(A). In other words, for any maximal Cohen-Macaulay A-module M and a maximal Cohen-Macaulay B-module N we have: Hom B (B ⊠ A M, N ) ∼ = Hom A (M, N ). Assume additionally A and B to be both reduced. Then for any Noetherian B-module M there exist a natural isomorphism M † A ∼ = M † B in the category of A-modules. For a proof, see for example [19, Proposition 3.18]. Lemma 2.7. Let (A, m) be a reduced Noetherian ring of Krull dimension one with a canonical module K. Then for any Noetherian A-module M we have a functorial isomorphism: M ∨∨ ∼ = M/ tor(M ), where ∨ = Hom A ( − , K). Proof. From the canonical short exact sequence 0 → tor(M ) → M → M/ tor(M ) → 0 we get the isomorphism M/ tor(M ) ∨ → M ∨ . Since M/ tor(M ) is a maximal Cohen-Macaulay A-module and ∨ is a dualizing functor, we get two natural isomorphisms M/ tor(M ) ∼ = −→ M/ tor(M ) ∨∨ ∼ = ←− M ∨∨...
