Partial Serre duality and cocompact objects

Let \(\mathsf{T}\) be a triangulated category with coproducts and let \(\mathsf{X}\) be a set of compact objects in \(\mathsf{T}\). It is known that \(\mathsf{X}\) generates a certain t-structure, and in particular describes explicitly a left adjoint to the inclusion of the coaisle. Unfortunately, it does not make much sense to consider the naive dual of this setup; cocompact objects rarely appear in categories which occur naturally. Motivated by this we introduced a weaker version of cocompactness called 0-cocompactness, and showed that such objects cogenerate t-structures.

In this talk we will present some new features of this dual theory. In particular we introduce Serre duality with respect to a subcategory and show that the Serre dual of a compact object is always 0-cocompact. We thus obtain new examples of 0-cocompact objects by producing explicit relative Serre functors.

This is joint work with Steffen Oppermann and Torkil Stai.