An Auslander-Reiten principle in derived categories
The Auslander-Reiten principle or (the AR duality) is a theorem which states that if \(R\) is a Cohen-Macaulay local ring of Krull dimension \(d\) such that it has only an isolated singularity, and \(M,\;N\) are maximal Cohen-Macaulay modules over \(R\) then there exists a natural isomorphism \(\underline{\mathrm{Hom}}_R(M,N)^\vee\cong \mathrm{Ext}^1_R(N,\tau M)\). In the main result of the present paper (Theorem 2.3) the author proves a general principle. This is a theorem for chain complexes of modules. This theorem is used to obtain: the standard version for the AR-duality (Section 3) or a generalized form proved (see Section 4) by \textit{O. Iyama} and \textit{M. Wemyss} [Invent. Math. 197, No. 3, 521--586 (2014; Zbl 1308.14007)]. The paper is closed with a discussion about the Auslander-Reiten conjecture for modules over Gorenstein rings.
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