Stable equivalence and the stable triangle category (Q1802209)

Let \(A\) be a finite dimensional algebra. Then one may consider the following categories associated with the given algebra \(A\): The module category mod-\(A\) and its stable category \(\overline{\text{mod}}\text{-}A\) which is the quotient of mod-\(A\) by the projective modules, the bounded derived category \(D^ b(A)\) and its stable derived category \(\overline{D}^ b(A)\) which is the quotient of \(D^ b(A)\) by the homology-projective complexes, and the category \(T^ b(A)\) of triangles in the derived category of \(A\) and its stable triangle category \(\overline{T}^ b(A)\) which is the quotient of \(T^ b(A)\) by the split homology-projective triangles. Suppose that \(A\) and \(A'\) are finite dimensional selfinjective algebras over a field and that \(R:\text{ mod-}A \to \text{mod-}A'\) and \(I:\text{ mod-}A' \to \text{mod-}A\) are exact functors such that \(R\) is a right adjoint of \(I\). Then they induce functors \(\overline{R}: \overline{\text{mod}}\text{-}A \to \overline{\text{mod}}\text{-}A'\) and \(\overline{I}: \overline{\text{mod}}\text{-}A' \to \overline{\text{mod}}\text{-}A\). In addition, \(R\) and \(I\) induce functors \(\overline{R}\cdot\) and \(\overline{I}\cdot\) between the stable categories, and functors \(\overline{R}^ T\) and \(\overline{I}^ T\) between stable triangle categories. The main result in the paper is the following: If \(\overline{R}\) and \(\overline{I}\) are fully faithful, then (1) the functors \(\overline{R}\cdot\) and \(\overline{I}\cdot\) define equivalences between \(\overline{D}^ b(A)\) and \(\overline{D}^ b(A')\) that are quasi-inverses. (2) The functors \(\overline{R}^ T\) and \(\overline{I}^ T\) are equivalences between \(\overline{T}^ b(A)\) and \(\overline{T}^ b(A')\) that are quasi-inverses.