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I want to prove that there exist projective covers in the category of graded modules over an algebra.

I am fairly new to "this" kind of mathematic and don't really know where to start: I found the answer to this question "graded modules have enough projectives" and thought this would be a first step, since it provides the existence of a projective graded module and a surjective map for every graded module. Now I have to get to the superfluousness of the kernel somehow – the constructed graded-free module $\oplus_i R[-i]^{\oplus B_{-i}}$ might be too large; and even if not, I have no idea how to relate the graded structure to the superfluos kernel.

And since there is no question mark in my question so far: Can anyone give me any hints on how to prove this or how to continue my "thoughts"?

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  • $\begingroup$ Do you want some added conditions on your algebra? For general algebras it's not true that every module has a projective cover. $\endgroup$ – Jeremy Rickard Sep 23 '14 at 12:55
  • $\begingroup$ (you mean it's not true for every graded(!) module over an general algebra, I guess?) Oh. I want it to be an algebra over a field k. It would be enough if it holds for the polynomials in n variables over k, I think. $\endgroup$ – Peter Peterson Sep 23 '14 at 13:09
  • $\begingroup$ Even for $k[x]$ (regarded as a graded $k$-algebra with $x$ in degree $1$), not every graded module has a projective cover: for example, $k[x,x^{-1}]$ doesn't. (Although in this case every finitely generated graded module does have a projective cover, I think.) $\endgroup$ – Jeremy Rickard Sep 24 '14 at 10:15
  • $\begingroup$ Thanks @jeremy. could you explain me how to see that k[x,x^{-1}] has no projective cover? $\endgroup$ – Peter Peterson Sep 24 '14 at 11:20

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