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This paper studies vanishing of Ext modules over Cohen--Macaulay local rings. The main result of this paper implies that the Auslander--Reiten conjecture holds for maximal Cohen--Macaulay modules of rank one over Cohen--Macaulay normal local rings. It also recovers a theorem of Avramov--Buchweitz--Şega and Hanes--Huneke, which shows that the Tachikawa conjecture holds for Cohen--Macaulay generically Gorenstein local rings. [ABSTRACT FROM AUTHOR]