Suppose that $D(A)$ is the derived category of of a ring A. Let $b\in D(A)$ be a compact object and $B$ the localizing subcategory generated by b (having arbitrary coproduct).

Does the inclusion functor $ D(A)\leftarrow B: i$ have a left adjoint ? Let $W: D(A)\rightarrow B$ the right adjoint to the inclusion functor. Do we have that $W\circ i=id$ ? Does W preserve compact objects?

Remark: the existence of a right adjoint to the inclusion functor is well known.