Say f: X → Y is a morphism of schemes. The sheaf direct image functor f★ always has a left adjoint, namely the sheaf inverse image functor f★ (with tensoring).
Under what (sufficient) conditions do we know that f★ has a right adjoint? What is it?
EditAnswer to a related question (edit): What I'm looking for are appropriately general sufficient conditions... Certainly exactness ofIf f★ ispreserves quasicoherence, then its restriction to quasicoherents f★: QCoh(X) → QCoh(Y) has a right adjoint when f is necessaryaffine condition(in particular, any closed immersion or finite morphism will do). As well 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.
In this question, perhaps I should be clear that I'm not assuming quasi-coherencenot in theserestricting to the quasi-coherent categories of O. One reason for working with non-quasicoherents is that jX!-Modules , the "extension by zero" right adjoint to j★ for an open immersion j, doesn't take qcoh to qcoh.
