Partial Serre duality and cocompact objects

Abstract

A successful theme in the development of triangulated categories has been the study of compact objects. A weak dual notion called 0-cocompact objects was introduced in arXiv:1801.07995, motivated by the fact that sets of such objects cogenerate co-t-structures, dual to the t-structures generated by sets of compact objects. In the present paper, we show that the notion of 0-cocompact objects also appears naturally in the presence of certain dualities. We introduce "partial Serre duality", which is shown to link compact to 0-cocompact objects. We show that partial Serre duality gives rise to an Auslander--Reiten theory, which in turn implies a weaker notion of duality which we call "non-degenerate composition", and throughout this entire hierarchy of dualities the objects involved are 0-(co)compact. Furthermore, we produce explicit partial Serre functors for multiple flavors of homotopy categories, thus illustrating that this type of duality, as well as the resulting 0-cocompact objects, are abundant in prevalent triangulated categories.