Let $A$ be a right coherent ring and $\mathcal{X}$ be a contravariantly finite subcategory of ${\rm{mod}}\mbox{-}A$ containing projectives. In this paper, we construct a recollement of abelian categories $({\rm{mod}}_{0}\mbox{-}\mathcal{X}, {\rm{mod}}\mbox{-}\mathcal{X}, {\rm{mod}}\mbox{-}A)$, where ${\rm{mod}}_{0}\mbox{-}\mathcal{X}$ is a full subcategory of ${\rm{mod}}\mbox{-}\mathcal{X}$ consisting of all functors vanishing on projective modules. As a result, a relative version of Auslander's formula with respect to a contravariantly finite subcategory will be given. Some examples and applications will be provided.
J. Math. Soc. Japan
73(2):
329-349
(April, 2021).
DOI: 10.2969/jmsj/83308330
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