The Stacks project

47.1 Introduction

In this chapter we discuss dualizing complexes in commutative algebra. A reference is [RD].

We begin with a discussion of essential surjections and essential injections, projective covers, injective hulls, duality for Artinian rings, and study injective hulls of residue fields, leading quickly to a proof of Matlis duality. See Sections 47.2, 47.3, 47.4, 47.5, 47.6, and 47.7 and Proposition 47.7.8.

This is followed by three sections discussing local cohomology in great generality, see Sections 47.8, 47.9, and 47.10. We apply some of this to a discussion of depth in Section 47.11. In another application we show how, given a finitely generated ideal $I$ of a ring $A$, the “$I$-complete” and “$I$-torsion” objects of the derived category of $A$ are equivalent, see Section 47.12. To learn more about local cohomology, for example the finiteness theorem (which relies on local duality – see below) please visit Local Cohomology, Section 51.1.

The bulk of this chapter is devoted to duality for a ring map and dualizing complexes. See Sections 47.13, 47.14, 47.15, 47.16, 47.17, 47.18, 47.19, 47.20, 47.21, 47.22, and 47.23. The key definition is that of a dualizing complex $\omega _ A^\bullet $ over a Noetherian ring $A$ as an object $\omega _ A^\bullet \in D^{+}(A)$ whose cohomology modules $H^ i(\omega _ A^\bullet )$ are finite $A$-modules, which has finite injective dimension, and is such that the map

is a quasi-isomorphism. After establishing some elementary properties of dualizing complexes, we show a dualizing complex gives rise to a dimension function. Next, we prove Grothendieck's local duality theorem. After briefly discussing dualizing modules and Cohen-Macaulay rings, we introduce Gorenstein rings and we show many familiar Noetherian rings have dualizing complexes. In a last section we apply the material to show there is a good theory of Noetherian local rings whose formal fibres are Gorenstein or local complete intersections.

In the last few sections, we describe an algebraic construction of the “upper shriek functors” used in algebraic geometry, for example in the book [RD]. This topic is continued in the chapter on duality for schemes. See Duality for Schemes, Section 48.1.