TITLE: State of the art on a question on the existence of dualizing complex
QUESTION [8 upvotes]: 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 ?

REPLY [15 votes]: 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.