The homotopy category of pure injective flats and Grothendieck duality

TL;DR

This paper establishes an equivalence between the homotopy categories of pure injective flats and injective modules on a noetherian scheme with a dualizing complex, extending Grothendieck duality to a pure derived setting.

Contribution

It proves a new equivalence of triangulated categories involving pure injective flats and injectives, generalizing Grothendieck duality to the pure derived category context.

Findings

Equivalence of homotopy categories of pure injective flats and injective modules.

Extension of Grothendieck duality to the pure derived category.

Induction of equivalence between pure derived categories of flats and absolutely pure modules.

Abstract

Let (X;OX) be a locally noetherian scheme with a dualizing complex D. We prove that DOX - : K(PinfX)----> K(InjX) is an equivalence of triangulated categories where K(InjX) is the homotopy category of injective quasi-coherent OX- modules and K(PinfX) is the homotopy category of pure injective flat quasi-coherent OX-modules. Where X is affine, we show that this equivalence is the infinite completion of the Grothendieck duality theorem. Furthermore, we prove that D OX - induces an equivalence between the pure derived category of flats and the pure derived category of absolutely pure quasi-coherent OX-modules.