Existence of dualizing complexes
Tetsushi Ogoma · Kyoto journal of mathematics · 1984
Duality o n Gorenstein rin g s a n d canonical modules of Cohen Macaulay rings a re generalized if we consider a complex instead of rings or m odules, and such a com plex, 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 , th en A h a s a dualizing complex a s is well known [5, Chapter V § 10], b u t other good sufficient condition of existence of dualizing complex is not known.O n th e other hand Sharp showed in [21, (3.8) T heorem ] that i f a r i n g A h a s a dualizing com plex, th en A is an acceptable ring ; that is (1) universally catenary, ( 2) form al 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 m odule 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 c o n v e rse holds.We show the following ;If (SO holds, then acceptable rings w ith canonical m o d u le s h a v e 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 rin g with n o canonical modules ( § 6, Example 1) a n d also exists a non-acceptable rin g with canonical modules ( § 6, Example 2).If (S2 ) does not hold b u t th e rin g is local, th en sligh tly stro n ger condition on existence of canonical modules is necessary fo r u s (Theorem 5.5).A ll r in g s a r e a s s u m e to b e com m u tative ring w ith id en tity and, except section 3, noetherian.The term inologies and notations of [5 ], [1 3 ] and [16] are used freely.