Cartesian modules over representations of small categories

Abstract

We introduce the new concept of cartesian module over a pseudofunctor $ R$ from a small category to the category of small preadditive categories. Already the case when $ R$ is a (strict) functor taking values in the category of commutative rings is sufficient to cover the classical construction of quasi-coherent sheaves of modules over a scheme. On the other hand, our general setting allows for a good theory of contravariant additive locally flat functors, providing a geometrically meaningful extension of Crawley-Boevey'\''s Representation Theorem. As an application, we relate and extend some previous constructions of the pure derived category of a scheme.