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Maximal Cohen-Macaulay modules over Gorenstein rings and Bourbaki- sequences. (English) Zbl 0641.13014
Commutative algebra and combinatorics, US-Jap. Joint Semin., Kyoto/Jap. 1985, Adv. Stud. Pure Math. 11, 65-92 (1987).
[For the entire collection see Zbl 0632.00003.]
Consider a simple surface singularity X (or Du Val singularity). There is a very nice correspondence between the dual graph of the resolution of singularities of X and indecomposable Cohen-Macaulay modules over A [See M. Artin and J.-L. Verdier, Math. Ann. 270, 79-82 (1985; Zbl 0553.14001)]. Following this direction the authors study MCM (= maximal Cohen-Macaulay) modules over a Gorenstein ring A and sometimes over a hypersurface in order to give more precise results. The question of existence of MCM with given data \((n,m)\) (\(n\)= minimal number of generators and \(m=\text{rank}(M))\) is considered. The main technique is to study relations of MCM and Bourbaki sequences \((0\to F\to M\to I\), where F is A-free and I a codimension 2 CM-ideal or \(I=A)\). Then the problem of MCM is translated into an ideal problem.
Another important fact is the Rao correspondence: If R is a normal Gorenstein domain then there exists a bijection between stable isomorphism classes of orientable MCM-modules and even linkage classes of codimension 2 CM-ideals.
Reviewer: M.Morales

MSC:
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
14J17 Singularities of surfaces or higher-dimensional varieties
14B05 Singularities in algebraic geometry