Say f: X &rarr; Y is a morphism of schemes.  The sheaf direct image functor f<sub>&#9733;</sub> always has a left adjoint, namely the sheaf inverse image functor f<sup>&#9733;</sup> (with tensoring).

>Under what (sufficient) conditions do we know that f<sub>&#9733;</sub> has a *right* adjoint?  What is it?

**Answer to a related question (edit):** If f<sub>&#9733;</sub> preserves quasicoherence, then its restriction to quasicoherents f<sub>&#9733;</sub>: QCoh(X) &rarr; QCoh(Y) has a right adjoint when f is *affine* (in particular, any closed immersion or finite morphism will do).  The basic idea is to globalize the affine case (see Eric Wofsey's answer below); thanks to Pablo Solis for pointing me to [page 6 of Ravi Vakil's notes explaining this][1].

In this question, I'm *not* restricting to the quasi-coherent categories.  One reason for working with non-quasicoherents is that j<sub>!</sub> , the "extension by zero" right adjoint to j<sup>&#9733;</sup> for an open immersion j, doesn't take qcoh to qcoh.


  [1]: http://math.stanford.edu/~vakil/0506-216/216class5354.pdf