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The homological theory of maximal Cohen-Macaulay approximations. (English) Zbl 0697.13005

The main purpose of the paper is to show theorem A:
Let R be a commutative noetherian local ring with dualizing module \(\omega\). For any finitely generated R-module N there exist finitely generated R-modules \(M_ N\) and \(I^ N\) together with an R-linear map \(d_ N: M_ N\to I^ N\) such that
(a) \(Im(d_ N)\cong N.\)
(b) depth\((M_ N)=\dim (R)\) and \(id_ RI_ N<\infty\) where \(I_ N=Ker(d_ N).\)
(c) depth(Coker\((d_ N))=\dim (R).\)
(d) There exists a nonnegative integer n such that \(d_ N=p\circ j\) with an injection \(j: M_ N\to \omega^{\oplus n}\) and a surjection \(p: \omega^{\oplus n}\to I^ N.\)
The essential uniqueness theorem is also proven.
The authors discuss their theory in abelian categories with a suitable subcategory of “maximal Cohen-Macaulay objects”.
Reviewer: Y.Aoyama

MSC:

13D99 Homological methods in commutative ring theory
18G99 Homological algebra in category theory, derived categories and functors
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
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References:

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