Approximation, Serre duality and recollements are three major tools, theories really, used to study triangulated categories in algebraic geometry and representation theory. However, the existence of Serre duality is a very strong condition; the bounded derived category of a finite dimensional algebra satisfies Serre duality if and only if it has finite global dimension. Replacing Serre duality by the weaker notion of a partial Serre functor, we get a new tool which applies to any approximable category, and in particular to any ring.
In this talk, I will explain how these tools interact. I will briefly (and badly) introduce Neeman's theory of approximation for triangulated categories, and show how approximability and Serre functors behave under recollements.