Relative homology, higher cluster-tilting theory and categorified Auslander–Iyama correspondence

We study homological properties of contravariantly finite rigid subcategories of an arbitrary triangulated or abelian category, concentrating at the class of higher cluster tilting subcategories. In both the triangulated and abelian context we present several new characterizations of such subcategories and we give conditions ensuring that the associated cluster tilted category is Gorenstein and/or stably Calabi–Yau. Thus we generalize and improve on several recent results of the literature. In the context of abelian categories, we prove a categorified version of higher Auslander correspondence, as developed by Iyama in the case of module categories. In this connection we investigate in detail homological properties of (stable) Auslander categories associated to higher cluster tilting subcategories in an abelian category, and we present applications to (possibly infinitely generated) cluster tilting modules over an arbitrary ring. Finally we give bounds on the global and representation dimension of certain categories of coherent functors associated to Cohen–Macaulay objects.