Extensions of a Dualizing Complex by its Ring: Commutative Versions of a   Conjecture of Tachikawa
  Let $(R,\fm,k)$ be a commutative noetherian local ring with dualizing complex $\dua R$, normalized by $\Ext^{\depth(R)}_R(k,\dua R)\cong k$. Partly motivated by a long standing conjecture of Tachikawa on (not necessarily commutative) $k$-algebras of finite rank, we conjecture that if $\Ext^n_R(\dua R,R)=0$ for all $n>0$, then $R$ is Gorenstein, and prove this in several significant cases. 
L.L.Avramov, R.-O.Buchweitz, L.M.Sega