Let $A$ be the (symmetric Frobenius) algebra $A=K[x]/(x^3) \otimes_K K[x]/(x^3)$ over a field $K$, which is isomorphic to the group algebra of the cyclic group with 3 elements when $K$ has characteristic 3. It is an open problem (see problem 2 in http://www2.iazd.uni-hannover.de/~tholm/ARTIKEL/BI_repdimkG_new.pdf ) whether there exists a generator $M=A \oplus X$ for some non-projective $A$-module $X$ such that $End_A(M)$ has global dimension 3 or equivalently whether the representation dimension of $A$ is equal to 3 (it is known that this dimension is 3 or 4). Since $A$ is local, there is a unique simple $A$-modules $S$.

Now with the computer I noted a large class of such modules $M$ such that $End_A(M)$ has at least Cartan determinant equal to one, which is a strong indicator for finite global dimension.

Namely let $M_T:=A \oplus \bigoplus_{i \in T}^{}{\Omega^{i}(S)}$ for some subset $T$ of $\mathbb{Z}$.

For surprisingly many choices of such $T$ the algebra $B_T:=End_A(M_T)$ has Cartan determinant 1, for example for $T=[-i,i]$ for any $i \geq 1$.

Question 1: Is there a choice for $T$ such that $B_T$ has global dimension 3?

I kinda doubt it but for most choices of $T$ the calculation is too big already for the computer. For $T=[-1,1]$ for example the algebra should have global dimension larger 3.

Question 2: I noted that for a general symmetric local Frobenius algebra with simple module $S$ the choice $M_T:=A \oplus \bigoplus_{i \in T}^{}{\Omega^{i}(S)}$ for subsets $T$ such as $T=[-i,i]$ and $B_T:=End_A(M_T)$ leads often to Cartan determinant 1 algebras. Does in this case Cartan determinant 1 imply finite global dimension? Is there an explanation (for specific $A$) when for such a $T$ there is an $M_T$ with $B_T$ having finite global dimension?