Extensions of a Dualizing Complex by its Ring: Commutative Versions of a Conjecture of Tachikawa

. Let ( R, m ,k ) be a commutative noetherian local ring with dualizing complex D R , normalized by Ext depth( R ) R ( k,D R ) ∼ = k . Partly motivated by a long standing conjecture of Tachikawa on (not necessarily commutative) k -algebras of finite rank, we conjecture that if Ext nR ( D R ,R ) = 0 for all n > 0, then R is Gorenstein, and prove this in several significant cases.