Question on Ext for finite dimensional algebras Given a finite dimensional non-Gorenstein algebra $A$, do we have $Ext^i(D(A),A) \neq 0$ for infinitely many $i$? (We can assume A is local or commutative if that helps).
All I can show is that such algebras would be pretty exotic when this holds only for finitely many i.
 A: My modules are right modules.
Let $M$ be a non-projective module for a self-injective algebra $B$ such that $\text{Ext}^i_B(M,M)=0$ for $i>t$ (i.e., what you've called a "strange" module in some recent questions). Let
$$A=\pmatrix{k&M\\0&B}.$$
Then $A$ is not Gorenstein, as it has a one-dimensional injective module with infinite projective dimension, and according to my calculations, $\text{Ext}^i_A(DA,A)=0$ for $i>t+1$.
An $A$-module is a "row vector" $\pmatrix{V&X}$, where $V$ is a vector space and $X$ a $B$-module, together with a $B$-module map $V\otimes_kM\to X$.
$DA$ is the direct sum of two injectives $I_1=\pmatrix{k&0}$ and $I_2=\pmatrix{DM&DB}$ corresponding to the columns of $A$. 
If 
$$\dots P_1\to P_0\to M\to0$$ 
is a $B$-projective resolution of $M$, then
$$\dots\pmatrix{0&P_1}\to\pmatrix{0&P_0}\to\pmatrix{k&M}\to\pmatrix{k&0}\to0$$
is an $A$-projective resolution of $I_1$, and so 
$$\text{Ext}^{i+1}_A(I_1,A)=\text{Ext}^i_B(M,M\oplus B)$$
for $i>1$, which is zero for $i>t$ by the assumptions on $B$ and $M$.
There is a short exact sequence
$$0\to\pmatrix{0&DB}\to I_2\to\pmatrix{DM&0}\to0.$$
The last term is a direct sum of copies of $I_1$, and the first term is a projective $A$-module, so the $\text{Ext}^*_A(-,A)$ long exact sequence gives $\text{Ext}^i_A(I_2,A)=0$ for $i>t+1$.
