Existence of dualizing complexes

Duality on Gorenstein rings and canonical modules of Cohen Macaulay rings are generalized if we consider a complex instead of rings or m odules, and such a complex, called dualizing complex, is introduced by Grothendieck [7]. If a r in g A is a homomorphic im a g e o f a Gorenstein r in g , then A h as a dualizing complex a s is well known [5, Chapter V § 10], but other good sufficient condition of existence of dualizing complex is not known. O n the other hand Sharp showed in [21, (3.8) Theorem ] that i f a r i n g A h as a dualizing com plex, then A is an acceptable ring ; that is (1) universally catenary, (2) formal fibers a r e Gorenstein a n d (3 ) f o r any finitely generated A-algebra B , th e Gorenstein locus of Spec B is open. Again, it follows that if A has a dualizing complex, then A has a canonical module as the initial non-zero homology module of the complex. T h e purpose o f this note is to investigate how extent th e co nv erse holds. We show the following ; If (SO holds, then acceptable rings with canonical m odules have dualizing complexes (Theorem 5.2, Remark 5.3). Here both o f th e acceptability and the existence of canonical m odules are im portant. Really, there exists an acceptable ring with no canonical modules (§ 6, Example 1) and also exists a non-acceptable ring with canonical modules (§ 6, Example 2). If (S 2 ) does not hold bu t the ring is local, then slightly stronger condition on existence of canonical modules is necessary fo r u s (Theorem 5.5). A ll r in g s a re a ssu m e to b e com m utative ring w ith identity and, except section 3, noetherian. The terminologies and notations of [5 ], [13] and [16] are used freely.