Derived equivalences for chain complexes with support

For a Serre subcategory $\mathscr L$ and a resolving subcategory $\mathscr A$ of an abelian category, we show that the derived equivalence $D^b(\overline{\mathscr A} \cap \mathscr L) \simeq D^b_{\mathscr L}(\mathscr A)$ holds under certain conditions. We apply this to obtain derived equivalences in the contexts of chain complexes of graded modules or coherent sheaves, with finite $\mathscr A$-dimension, supported on closed sets having eventually finite $\mathscr A$-dimension. Using this, we obtain descriptions of the homotopy fibers in (hermitian) K theory of the restriction maps to certain open sets.