The Stacks project

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 86.1.

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0DWF. Beware of the difference between the letter 'O' and the digit '0'.