## State of the art on a question on the existence of dualizing complex

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Let A be a noetherian ring and D(A) be the derived category of modules on A. Recall that a dualizing complex for A is an object R in D(A) of finite injective dimension, with cohomology of finite type and such that the natural morphism of functors $Id \longrightarrow R\mathcal{H}om(R\mathcal{H}om(., R), R )$ is an isomorphism of functors.

In the book "Residues and Duality" (R. Hartshorne) (V.10), it is presented as an open problem to know if a noetherian local domain of dimension 1 admits a dualizing complex.

What is the state of the art on this question ?

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## 2 Answers

R.Y. Sharp proved in the 80s that a Noetherian ring with a dualizing complex must be "acceptable". This is a weakening of "excellent" obtained by replacing every occurence of the word "regular" with "Gorenstein", so means (1) universally catenary, (2) Gorenstein formal fibers, and (3) Gorenstein locus open in any finitely generated algebra.

There is an example due to Ferrand and Raynaud of a one-dimensional noetherian domain for which the generic formal fiber is not Gorenstein. Such a ring does not, by Sharp's result, have a dualizing complex.

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As reference, I mention that the title of the article of Ferrand and Raynaud is "Fibres formelles d'un anneau local noetherien" (available on numdam). The fact that a local domain with a dualizing complex has Gorenstein formal fiber is in Hartshorne's book so I don't need the stronger results of Sharp. Thanks a lot for the answer. – unknown (google) Oct 18 at 9:37

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Actually, more has been determined since Sharp's work in the 1980s. It is easy to see that a homomorphic image of a Gorenstein ring of finite Krull dimension has a dualizing complex. Sharp conjectured that this is the only way for a Noetherian ring to have a dualizing complex. In a pair of papers (2000 and 2002) in Transactions of the AMS, Takesi Kawasaki showed that Sharp's conjecture is true. That is, a Noetherian ring has a dualizing complex if and only if (1) it has finite Krull dimension and (2) it is the homomorphic image of a Gorenstein ring.

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