My computer suggests that the following is true for Nakayama algebra (it also found not counterexample for arbitrary algebras):
Let $A$ be an algebra of finite Gorenstein dimension $g \geq 1$, then there exists an indecomposable module $M$ with $Ext^{i}(D(A),M)=0$ for $i=1,2,...,g-2$ but $Ext^l(D(A),M) \neq 0$ for $l=g-1$ or $l=g$.
(If you want you can replace Gorenstein dimension by global dimension first, remembering that if the global dimension is finite it is equal to the Gorenstein dimension)
Questions:
-Is it true for Nakayama algebras?
-Is it true for general algebras?
