In this article the authors develop a new method to deal with maximal Cohen–Macaulay modules over non–isolated surface singularities. In particular, they give a negative answer on an old question of Schreyer about surface singularities with only countably many indecomposable maximal Cohen–Macaulay modules. Next, the authors prove that the degenerate cusp singularities have tame Cohen–Macaulay representation type. The authors'approach is illustrated on the case of $\mathbb{k}[[ x,y,z]]/(xyz)$ 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 the authors call representations of decorated bunches of chains. They prove that these matrix problems have tame representation type and describe the underlying canonical forms. Cover -- Title page -- Introduction, motivation and historical remarks -- Chapter 1. Generalities on maximal Cohen-Macaulay modules -- 1.1. Maximal Cohen-Macaulay modules over surface singularities -- 1.2. On the category \CM^{ }(\rA) -- Chapter 2. Category of triples in dimension one -- Chapter 3. Main construction -- Chapter 4. Serre quotients and proof of Main Theorem -- Chapter 5. Singularities obtained by gluing cyclic quotient singularities -- 5.1. Non-isolated surface singularities obtained by gluing normal rings -- 5.2. Generalities about cyclic quotient singularities -- 5.3. Degenerate cusps and their basic properties -- 5.4. Irreducible degenerate cusps -- 5.5. Other cases of degenerate cusps which are complete intersections -- Chapter 6. Maximal Cohen-Macaulay modules over \kk\llbracket, \rrbracket/(2+ 3- ) -- Chapter 7. Representations of decorated bunches of chains-I -- 7.1. Notation -- 7.2. Bimodule problems -- 7.3. Definition of a decorated bunch of chains -- 7.4. Matrix description of the category \Rep(\dX) -- 7.5. Strings and Bands -- 7.6. Idea of the proof -- 7.7. Decorated Kronecker problem -- Chapter 8. Maximal Cohen-Macaulay modules over degenerate cusps-I -- 8.1. Maximal Cohen-Macaulay modules on cyclic quotient surface singularities -- 8.2. Matrix problem for degenerate cusps -- 8.3. Reconstruction procedure -- 8.4. Cohen-Macaulay representation type and tameness of degenerate cusps -- Chapter 9. Maximal Cohen-Macaulay modules over degenerate cusps-II -- 9.1. Maximal Cohen-Macaulay modules over \kk\llbracket, \rrbracket/() -- 9.2. Maximal Cohen-Macaulay modules over \kk\llbracket, \rrbracket/(,) -- 9.3. Degenerate cusp \kk\llbracket, \rrbracket/(,) -- Chapter 10. Schreyer's question Develops a new method to deal with maximal Cohen-Macaulay modules over non-isolated surface singularities. In particular, the authors give a negative answer on an old question of Schreyer about surface singularities with only countably many indecomposable maximal Cohen-Macaulay modules.
