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$\newcommand\End{\operatorname{End}}$Let $A=k[x_1, \dotsc, x_n]$ and $M$ be a graded $A$-module. Let $E=\End_*(M)$ be the graded endomorphism ring of $M$ (i.e., $E_i$ consists of all degree-$i$-endomorphisms of $M$). What can be said about the Jacobson radical $J(E)$ of $E$ in terms of the module $M$? For instance, if $M$ is finite dimensional, all indeterminates $x_i$ are centrally nilpotent in $E$, and hence are contained in $J(E)$. But how to treat this in general?

Specific questions are:

  • If $M$ is finite dimensional and indecomposable, is it true that $J(E)=(x_1, \dotsc, x_n)$?
  • Is there a correspondence between maximal ideals of $E$ and certain properties of $M$?
  • How to think of elements of $J(E)$ which are not nilpotent?
  • what are nilpotent elements of $E$ not contained in $J(E)$?
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