Examples of degenerations of Cohen-Macaulay modules (Q2839298)

Throughout this review \(K\)(\(F\)) stands for a field (an algebraically closed field), \(R\)(\(A\)) is an \(K\)-algebra (Artinian \(F\)-algebra) with identity and all modules are unital. In the affine variety consisting of finitely generated left \(A\)-modules of \(F\)-vector space dimension \(n\), a left \(A\)-module \(N\) is a degeneration of \(M\) precisely when \(N\) lies in the Zariski closure of the isomorphism class of \(M\) (see [\textit{M. Auslander} (ed.) and \textit{E. Lluis} (ed.), Representations of algebras. Workshop Notes of the Third International Conference on Representations of Algebras, Held in Puebla, Mexico, August 4--8, 1980. Lecture Notes in Mathematics. 944. Berlin-Heidelberg-New York: Springer-Verlag (1982; Zbl 0478.00010)]). Using the algebraic characterizations of degeneration, the second author of the paper under review, in a series of papers, have extended the degeneration theory of finitely generated modules beyond the realm of Artinian \(F\)-algebras.NEWLINENEWLINEIn the second part of the present paper it is proved that although the degeneration order and extension order are not identical in general but over either \(K[[x]]\) or \(K[[x]]/(x^m)\) the degeneration order coincides with the latter on the category of modules of finite length.NEWLINENEWLINEThe third section is devoted to proving that the degenerations of maximal Cohen-Macaulay modules over an even-dimensional simple hypersurface singularity of type \((A_n)\) are given by extensions thereby the degeneration order is the same as the extension order (on the category of maximal Cohen-Macaulay modules). This is accomplished with the aid of the theory of stable degeneration of maximal Cohen-Macaulay modules over a Gorenstein local ring (see [\textit{Y. Yoshino}, J. Algebra 332, No. 1, 500--521 (2011; Zbl 1267.13022)]).NEWLINENEWLINEIn the last section of the paper it is shown that on the category of maximal Cohen-Macaulay \(R\)-modules over a complete Cohen-Macaulay local ring \(R\) of finite Cohen-Macaulay representation type the extended degenerations is generated by extended degenerations of Auslander-Reiten sequences whence the extended extension order, extended degeneration order and extended \(AR\)-order are equivalent.