Cohen-Macaulay approximations from the viewpoint of triangulated categories. (English) Zbl 0955.13003
Summary: We introduce the notion of origin extension which is a formal analogue of Cohen-Macaulay approximation. However, as an approximation, origin extensions are not completely analogous to Cohen-Macaulay approximations. The most remarkably, there exist non-minimal origin extensions that do not include the minimal one as a direct summand. We also discuss when these non-trivially non-minimal origin extensions exist. Our aim is to find a standard method of classifying modules via Cohen-Macaulay approximations. When a module \(M\) has positive grade, we show no other module has the same Cohen-Macaulay approximation as \(M\).
See also the author’s same titled article in RIMS Kokyuroku 964, 1-28 (1996; Zbl 0930.13007).
See also the author’s same titled article in RIMS Kokyuroku 964, 1-28 (1996; Zbl 0930.13007).
MSC:
| 13C14 | Cohen-Macaulay modules |
| 13D25 | Complexes (MSC2000) |
| 18E30 | Derived categories, triangulated categories (MSC2010) |
| 13B02 | Extension theory of commutative rings |
Keywords:
Gorenstein ring; minimal origin extensions; Cohen-Macaulay approximation; classifying modulesCitations:
Zbl 0930.13007References:
| [1] | Auslander M., Soc. Math, de France, Mem 38 pp 5– (1989) |
| [2] | Auslander M., The stable module theory 94 (1969) |
| [3] | Miyata T., J. Math. Kyoto Univ 7 pp 65– (1967) |
| [4] | Yoshino Y., London Math.Soc 146 (1990) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
