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In an example I have worked out for my work, I have constructed a category whose objects are graded $R$-modules (where $R$ is a graded ring), and with morphisms the usual morphisms quotient the following class of morphisms:

$\Sigma=\left\lbrace f\in \hom_{\text{gr}R\text{-mod}}\left(A,B\right) \ | \ \ker\left(f\right)_0\neq 0, \ \mathrm{coker}\left(f\right)_0\neq 0\right\rbrace$

(by quotient I mean simply that this class of morphisms are isomorphisms, thus creating an equivalence relation) I am wondering if this category has a better (more canonical) description, or if I can show it is equivalent to some other interesting category.

Thanks!

BBischof
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    Since $\Sigma$ isn't an ideal in the category of graded $R$-modules, I don't think that quotienting by it makes much sense. – Rasmus Aug 13 '10 at 10:38
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    Do you mean to make the maps in $\Sigma$ all the zero maps (which is quotienting) or do you mean to make the maps in $\Sigma$ isomorphisms (which is localizing)? As $\Sigma$ is not an ideal the first isn't well defined, and as $\Sigma$ can contain the zero map between modules the second doesn't seem to make much sense either. – Jim Feb 27 '13 at 07:29
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    This is very weird, really: you are making all modules with non-zero zero component isomorphic... Does this really make sense in some context? – Mariano Suárez-Álvarez Jun 30 '13 at 02:54
  • @MarianoSuárez-Alvarez I was trying to get the altruist badge, so I found the oldest unanswered question and put a bounty on it. It makes sense that this has been unanswered for three years... – Brian Rushton Jul 04 '13 at 03:21
  • Well, I don't remember. I do remember that I was localizing by these morphisms and that was what I meant by quotienting. Judging by the date, I was probably trying to work out a ncag example. I don't have a clue what it was! – BBischof Jul 06 '13 at 13:20
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    Wow this is the first non-deleted question! Good job. – Тyma Gaidash Apr 24 '21 at 22:19

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