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Let $M$ be a finitely generated 2-graded $k[x,y]$-module, concentrated in nonnegative degrees, and $f$ be a degree $(n,m)$-endomorphism of $M$. We can consider $f$ as a morphism $M\langle n,m\rangle\to M_{\geq (n,m)}$, where $M_{\geq (n,m)}$ is the submodule of $M$ in degrees ${}\geq (n,m)$.

Definition: An element $r\in R$ of some ring is called strongly nilpotent if $\forall s\in R$ the element $rs$ is nilpotent.

Are the following two conditions equivalent?

  1. $f$ factors through a strongly nilpotent degree $(0,0)$-endomorphism of $M_{\geq (n,m)}$,
  2. $f$ factors through a strongly nilpotent degree $(n,m)$-endomorphism of $_{\geq (n',m')}$ for some $(0,0)\leq (n',m')\leq (n,m)$?

Of course, 1. implies 2., but how about the converse? If $f$ factors through a strongly nilpotent element for some $(n', m')$, then $f=g'hg$ for $g$ of degree $(n,m)$, $h$ of degree $(0,0)$ strongly nilpotent, and $g'$ of degree $(n-n', m-m')$.

Can anyone provide me with a counter example where 2. does not imply 1.?

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