Maximal Cohen–Macaulay Modules Over Non–Isolated Surface Singularities and Matrix Problems

Abstract

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 ∨∨...

Cohen–Macaulay modules over the algebra of planar quasi–invariants and Calogero–Moser systems

“…Now we introduce a certain categorical construction [7, 8], playing the key role in our paper. Let be a reduced Cohen–Macaulay ‐algebra of Krull dimension two, either finitely generated or complete.…”

“…The proof of the following result can be found in [8, Lemma 3.1]. Lemma For any , the following statements are true: (1)the canonical morphism of ‐modules is an epimorphism; (2)the adjoint morphism of –modules is a monomorphism; …”

“…(2) The divisor class group of A. Let CM lf 1 (A) be the abelian group of Cohen-Macaulay A-modules of rank one, which are locally free in codimension one [7,8] (it is an analogue of the divisor class group of a normal domain [5,Section 7.3]). Then there exists an isomorphism of abelian groups More explicitly, let Γ Π, μ := h ∈ C z 1 , z 2 |Υ(h) ∈ K Π, μ .…”

“…First, we essentially use the theory of multivariate Baker-Akhieser functions of (generalized) Calogero-Moser systems [1,10,[12][13][14]. The second ingredient is the 'matrix-problem method' of [7,8] to study Cohen-Macaulay modules over singular surfaces with non-isolated singularities.…”

“…The main new feature about the algebra is the result asserting that it is Gorenstein in codimension one; see Theorem 2.22. (2) The divisor class group of . Let be the abelian group of Cohen–Macaulay ‐modules of rank one, which are locally free in codimension one [7, 8] (it is an analogue of the divisor class group of a normal domain [5, Section 7.3]). Then there exists an isomorphism of abelian groups where the group law on is the same as in the case of the one‐dimensional cuspidal curves (); see Theorem 3.2. (3) Cohen–Macaulay ‐modules of rank one .…”