Grothendieck Duality on Diagrams of Schemes

Abstract

The Du Bois complex and Du Bois singularities, which extend results of Hodge theory to singular complex varieties, can be defined in terms of a cubical hyperresolution. In this dissertation I further develop the language of diagrams of schemes and then prove analogues of Grothendieck duality and other cohomological theorems for cubical diagrams. I then demonstrate the use of these by revisiting some known results about the Du Bois complex from a different perspective, and proving new results concerning the cohomology of a contraction.