Grothendieck Duality For Flat Morphisms

Abstract

Traditionally, the twisted inverse image pseudo-functor of Grothendieck duality (−)! is defined by means of compactification on a class of morphisms between noetherian schemes. Recently, Iyengar, Lipman, and Neeman introduced a formula for this pseudo- functor which works for flat, separated, essentially of finite-type morphisms between noetherian schemes. This formula raised some important questions. Not only it is not well understood why the formula is free of compactification but the process of how this formula satisfies all the standard properties of (−)! is also unclear. Another important question is whether this new formula can be expanded outside the class of flat, separated, essentially of finite-type morphisms between noetherian schemes. In this thesis, we talk about the motivations behind the two twisted inverse image pseudo-functors of Grothendieck duality (−)× and (−)!. We also recall the sufficient conditions and properties of these pseudo-functors. These properties are presented as the existence of some morphisms and compatibility diagrams satisfied by these mor- phisms. Then we discuss the surprising compactification-free formula of the pseudo- functor on the subclass of flat morphisms. A simplified proof that this formula is isomorphic to (−)! is also given. This recently discovered formula satisfies the properties of (−)! defined classically. As in the classical definition, the properties of this formula will also be presented via the existence of some morphisms and some compatibility diagrams. Extracting the essential information from the proofs, especially regarding the flat base change morphism, we discuss how understanding these proofs may enable us to generalize this Grothendieck Duality formula for flat morphisms to non-noetherian schemes Show more