Descent of coherent sheaves and complexes to geometric invariant theory quotients

Fix a scheme X over a field of characteristic zero that is equipped with an action of a reductive algebraic group G. We give necessary and sufficient conditions for a G-equivariant coherent sheaf on X or a bounded-above complex of G-equivariant coherent sheaves on X to descend to a good quotient X // G. This gives a description of the coherent derived category of X // G as an admissible subcategory of the equivariant derived category of X.