  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.
