Reflexivity and rigidity for complexes, II: Schemes

TL;DR

This paper extends the theory of reflexivity and rigidity in derived categories from rings to schemes, establishing foundational results about semidualizing and Gorenstein-perfect maps, and building on prior work on commutative rings.

Contribution

It generalizes reflexivity and rigidity concepts to schemes, including theorems on rigid dualizing complexes, advancing the understanding of duality in algebraic geometry.

Findings

Proves basic facts about reflexivity over noetherian schemes.

Studies semidualizing and invertible complexes in the scheme context.

Includes theorems on rigid dualizing complexes on schemes.

Abstract

We prove basic facts about reflexivity in derived categories over noetherian schemes; and about related notions such as semidualizing complexes, invertible complexes, and Gorenstein-perfect maps. Also, we study a notion of rigidity with respect to semidualizing complexes, in particular, relative dualizing complexes for Gorenstein-perfect maps. Our results include theorems of Yekutieli and Zhang concerning rigid dualizing complexes on schemes. This work is a continuation of part I, which dealt with commutative rings.