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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

\[ A \longrightarrow R\mathop{\mathrm{Hom}}\nolimits _ A(\omega _ A^\bullet , \omega _ A^\bullet ) \]

is a quasi-isomorphism. After esthablishing 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.


Comments (6)

Comment #1235 by jojo on

It seems like the reference [R+D] doesn't work.

Comment #4193 by on

There is a theory of noncommutative dualizing complexes, developed by Amnon Yekutieli, James Zhang and Michel Van den Bergh.

It gave rise to the theory of rigid commutative dualizing complexes, developed by the same people.

See the book Derived Categories, by Amnon Yekutieli To be published by Cambridge University Press; prepublication version arXiv:1610.09640 https://arxiv.org/abs/1610.09640v4

Comment #4199 by on

There is a very recent theory of rigid dualizing complexes over commutative DG rings. See

Duality and Tilting for Commutative DG Rings, by Amnon Yekutieli https://arxiv.org/abs/1312.6411v4 and The twisted inverse image pseudofunctor over commutative DG rings and perfect base change, by Liran Shaul Advances in Mathematics 320 (2017) 279–328 https://doi.org/10.1016/j.aim.2017.08.041

There are intimate connections to J. Lurie's dualizing complexes

J. Lurie, Derived algebraic geometry XIV: representability theorems http://www.math.harvard.edu/~lurie/papers/DAG-XIV.pdf

and to D. Gaitsgory's

D. Gaitsgory, Ind-coherent sheaves, Mosc. Math. J. 13 (3) (2013) 399–528.

Comment #4385 by on

Yes, there are many, many references we could and should give here. It would be good to write a short section on the literature but I am not the right person to do this. On the other hand, this chapter in some sense is doing the absolute minimum to get started with the dualizing complex, in the spirit of Grothendieck and in spirit very close to Hartshorne's book (as it concerns the dualizing complex and how we work with it).


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