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Cohen-Macaulay modules on hypersurface singularities. I. (English) Zbl 0617.14033
Let $$R=P/(f)$$ be an analytic hypersurface ring. The author investigates first the relation between maximal Cohen-Macaulay modules (MCM) over R and over $$R_ 1=P_ 1/(f+y^ 2)$$, where $$P_ 1=P<y>$$ (and $$P=k<x_ 0,...,x_ n>$$, k algebraically closed and char(k)$$\neq 2)$$. He proves in $$corollary^ 2.8$$ that there are only finitely many isomorphism classes of indecomposable MCM’s over R if and only if this is true for $$R_ 1$$. In theorem 3.1 it is shown - in a more general frame - that there is a canonical bijection between the sets of isomorphism classes of MCM’s over R and over $$R_ 2=P_ 2/(f+y^ 2+z^ 2)$$ respectively, where $$P_ 2=P<y,z>.$$
Since the two-dimensional simple singularities, i.e. the rational double points, have only finitely many isomorphism classes of MCM’s over their local rings [by M. Artin and J.-L. Verdier, Math. Ann. 270, 79-82 (1985; Zbl 0553.14001)], one gets by iterated application of corollary 2.8 the main result of this paper: There are only finitely many isomorphism classes of indecomposable MCM’s over the local ring of an isolated simple hypersurface singularity $$(A_ k, D_ k, E_ 6, E_ 7, E_ 8$$ in Arnold’s classification). The author also gives a conceptional description of the Auslander-Reiten quivers of the simple plane curve singularities in $$char(k)=0$$, showing that these quivers coincide with certain graphs associated to representations of finite reflection groups in $$Gl(2,k).$$