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Let $A$ be a commutative Frobenius algebra over a field $K$ (we can assume that $A$ is local). We can assume $A=K[x_1,...,x_r]/I$ for an ideal $I$ with $J^n \subseteq I \subseteq J^2$ where $J=<x_i>$, the ideal of $K[x_1,...,x_r]$.

Let $B=\{v_i \}$ be a monomial vector space basis (meaning $v_i$ is a homogeneous polynomial in $x_i$) of $A$ containing the unit of $A$. Let $M_i:=v_i A$ and $M:= \bigoplus_{}^{}{M_i}$ and $C:=\underline{End_A}(M)$ the stable endomorphism ring of $M$.

Question: Is $C$ independent of the choosen basis $B$ up to isomorphism?

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    $\begingroup$ You can choose a basis consisting of invertible elements and then get $C=0$. On the other hand you can choose another basis containing a non-invertible element and get $C\ne0$. $\endgroup$
    – Victor Ostrik
    Commented Apr 3, 2020 at 1:20
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    $\begingroup$ @VictorOstrik Thanks, I forgot to add that the basis should consist of monomials. $\endgroup$
    – Mare
    Commented Apr 3, 2020 at 9:28

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