Verdier-Type Category, Geometric Serre's conditions and perversity

Abstract

We generalize Serre's conditions for coherent sheaves to triangulated categories equipped with a $t$-structure and duality, which we call Verdier-type categories. We present examples of such categories and their associated Serre conditions, then focus on constructible sheaves, where we refer to these conditions as geometric Serre conditions. We investigate their relationship with perversity and illustrate the theory through a range of examples. Finally, we prove a geometric version of the Auslander-Reiten conjecture.