Derived equivalences for chain complexes with support

For a Serre subcategory $\mathcal{L}$ and a resolving subcategory $\mathcal{A}$ of an abelian category, we show that the derived equivalence $D^b(\bar{\mathcal{A}}\cap \mathcal{L})\simeq D_{\mathcal{L}}^b(\mathcal{A})$ 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 $\mathcal{A}$-dimension, supported on closed sets having eventually finite $\mathcal{A}$-dimension. Using this, we obtain descriptions of the homotopy fibers in (hermitian) K theory of the restriction maps to certain open sets.