Grothendieck groups for hypersurface rings

Solberg, Øyvind

doi:10.1090/S0002-9939-1990-1027101-5

Grothendieck groups for hypersurface rings
HTML articles powered by AMS MathViewer

by Øyvind Solberg PDF

Proc. Amer. Math. Soc. 109 (1990), 943-950 Request permission

Abstract:

Let $R$ and $S$ be commutative complete noetherian local Gorenstein domains. If the category of finitely generated maximal Cohen-Macaulay modules over $R$ and $S$ are stably equivalent and the equivalence commutes with the first syzygy functors, then we show that the Grothendieck groups for $R$ and $S$ are isomorphic. In particular, we apply this result to hypersurface rings.

References

Maurice Auslander, Functors and morphisms determined by objects, Representation theory of algebras (Proc. Conf., Temple Univ., Philadelphia, Pa., 1976) Lecture Notes in Pure Appl. Math., Vol. 37, Dekker, New York, 1978, pp. 1–244. MR 0480688
Maurice Auslander, Isolated singularities and existence of almost split sequences, Representation theory, II (Ottawa, Ont., 1984) Lecture Notes in Math., vol. 1178, Springer, Berlin, 1986, pp. 194–242. MR 842486, DOI 10.1007/BFb0075297
Maurice Auslander and Idun Reiten, Grothendieck groups of algebras and orders, J. Pure Appl. Algebra 39 (1986), no. 1-2, 1–51. MR 816889, DOI 10.1016/0022-4049(86)90135-0
Maurice Auslander and Idun Reiten, Representation theory of Artin algebras. VI. A functorial approach to almost split sequences, Comm. Algebra 6 (1978), no. 3, 257–300. MR 472919, DOI 10.1080/00927877808822246
Maurice Auslander and Idun Reiten, Stable equivalence of Artin algebras, Proceedings of the Conference on Orders, Group Rings and Related Topics (Ohio State Univ., Columbus, Ohio, 1972) Lecture Notes in Math., Vol. 353, Springer, Berlin, 1973, pp. 8–71. MR 0335575
Jürgen Herzog and Herbert Sanders, The Grothendieck group of invariant rings and of simple hypersurface singularities, Singularities, representation of algebras, and vector bundles (Lambrecht, 1985) Lecture Notes in Math., vol. 1273, Springer, Berlin, 1987, pp. 134–149. MR 915171, DOI 10.1007/BFb0078840
R.-O. Buchweitz, G.-M. Greuel, and F.-O. Schreyer, Cohen-Macaulay modules on hypersurface singularities. II, Invent. Math. 88 (1987), no. 1, 165–182. MR 877011, DOI 10.1007/BF01405096
Horst Knörrer, Cohen-Macaulay modules on hypersurface singularities. I, Invent. Math. 88 (1987), no. 1, 153–164. MR 877010, DOI 10.1007/BF01405095