## 48.1 Introduction

This chapter studies relative duality for morphisms of schemes and the dualizing complex on a scheme. A reference is [RD].

Dualizing complexes for Noetherian rings were defined and studied in Dualizing Complexes, Section 47.15 ff. In this chapter we continue this by studying dualizing complexes on schemes, see Section 48.2.

The bulk of this chapter is devoted to studying the right adjoint of pushforward in the setting of derived categories of sheaves of modules with quasi-coherent cohomology sheaves. See Sections 48.3, 48.4, 48.5, 48.6, 48.7, 48.8, 48.9, 48.11, 48.13, 48.14, and 48.15. Here we follow the papers [Neeman-Grothendieck], [LN], [Lipman-notes], and [Neeman-improvement].

We discuss the important and useful upper shriek functors $f^!$ for separated morphisms of finite type between Noetherian schemes in Sections 48.16, 48.17, and 48.18 culminating in the overview Section 48.19.

In Section 48.20 we explain alternative theory of duality and dualizing complexes when working over a fixed locally Noetherian base endowed with a dualizing complex (this section corresponds to a remark in Hartshorne's book).

In the remaining sections we give a few applications.

This chapter is continued by the chapter on duality on algebraic spaces, see Duality for Spaces, Section 84.1.

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