Extending Grothendieck topologies to diagram categories and Serre functors on diagram schemes

We study Serre functors and related constructions. A Serre functor on a triangulated category was defined by Bondal and Kapranov to be an auto-equivalence inducing certain natural dualities on homorphism sets in the category. In the special case of the bounded derived category of complexes of coherent sheaves on a smooth scheme Y the Serre functor is given by twisting by the dualizing sheaf and shifting by the dimension of the scheme. In the work of Lunts it is shown that a Serre functor exists if the scheme Y is replaced by a diagram scheme, which is a collection of schemes connected by morphisms; or more precisely a functor X : D → Schemes where D is some category, often called the shape of the diagram. We will give a description of the Serre functor for certain diagram schemes.;Related to the description of the Serre functor for diagram schemes is the study of the category DiagSchemes of diagram schemes and how it inherits properties from the category of schemes. We will consider inheritance of Grothendieck topologies and construct a general method for diagrammatizing any topology in such a way that desirable properties are not lost in the process. We will then consider how to carry prestacks and stacks over along with a Grothendieck topology.;A part of this work is joint with Jonathan Wise, at Stanford University.