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.



localizing subcategories of a nice triangulated category
As noted in the comments, there is no reason for the functor to have a left adjoint in general, as the inclusion will not preserve limits.


For the other two questions, the inclusion functor is fully-faithful, which occurs if and only if the unit map $\text{id} \to W \circ i$ is an equivalence. Finally, $A \in D(A)$ is compact, but $W(A) \in \text{Loc}(b)$ will typically not be.


Remark: In the literature, $W$ would be called a colocalization functor. A lot of facts and theorems can be found in the literature - for example, Section 3 of [Hovey, Palmieri, Strickland](https://hopf.math.purdue.edu/Hovey-Palmieri-Strickland/axiomatic.pdf).


