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I am looking at graded $k[x]$-modules. I am also interested in the subcategory of modules $M$ all whose graded components are finite dimensional, but that's an optional restriction. In either case, I would like to know if there are any graded modules that are both projective and injective.

My thoughts about this so far: graded $k[x]/(x^{N+1})$-modules should be the same as $k$-representations of the quiver $\mathrm{A}_N$. This has the projective-injective representation that assigns $k$ to every vertex. Now, graded $k[x]$-modules should be the same as $\mathrm{A}_\infty$-representations. Here I'm not sure if the representation that assigns $k$ to every vertex still is projective-injective.

It corresponds to the $k[x]$-module $k[x, x^{-1}]$. Probably not, if I understand this answer correctly.

  1. Are there any other projective-injective modules?
  2. Is $k[x, x^{-1}]$ projective in the category of modules with finite dimensinoal components? At least, using $\bigoplus_{n \in \mathbf{Z}} k[x]\langle n\rangle$ to prove that $k[x, x^{-1}$ is not projective does not work in this case.
Bubaya
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  • I don't get why graded $k[x]/(x^{N+1})$-modules would be like representations of $A_N$, you only quotient your ring, it shouldn't truncate your graded modules? – julio_es_sui_glace Sep 29 '23 at 13:23
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    https://math.stackexchange.com/questions/585495/a-problem-about-an-r-module-that-is-both-injective-and-projective no nonzero module over a domain which isn't a field can be both projective and injective – Matthew Towers Sep 29 '23 at 15:00

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