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) |
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