For Nakayama algebras $A$ with a linear quiver and at most 7 points (which are 196 algebras) the following was true:
$max \{ dim(Ext_A^2(S,A)) | S $ simple $\}= max \{ dim(Ext_A^2(X,A)) | X $ indecomposable $\}.$
Question:
Is this true for general Nakayama algebras? And is it true for a more general class of algebras?
(counterexamples for some random finite dimensional algebras are welcome, I would guess this heavily fails for representation-infinite algebras so a representation-finite counterexample might be interesting)
I am able to translate the problem into elementary combinatorics, but it is very ugly. In case it is true, I would suspect a nice representation-theoretic argument. However, it is not true for $Ext_A^1$, so maybe it is rather non-trivial and hides a secret about $Ext_A^2$?
Note that $Ext_A^2(S,A)$ has a nice interpretation. Namely let $0 \rightarrow A \rightarrow I_0 \rightarrow I_1 \rightarrow I_2 \rightarrow \cdots $ be the beginning of a minimal injective coresolution of the regular module $A$, then $Ext_A^2(S,A)$ counts how often the injective envelope $I(S)$ of $S$ occurs as a direct summand of $I_2$, which is interesting information and determines $I_2$ when knowing it for all simples.
edit: I deleted the last part of the question, to avoid an overly complicated question.
