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As i understand, finitely generated graded modules over Noetherian graded rings admit a finite free resolution (FFR). What are other classes of modules that admit a FFR? How about finitely generated modules over local Noetherian rings?

Manos
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    Finite free resolutions admit all modules that have finite projective dimension. (If this means to have also finite rank free modules, then just add the noetherian property to the module>) Maybe you want to ask for "minimal" free resolutions. –  Dec 18 '13 at 18:51
  • @YACP: Minimal free resolutions need not be finite, though, correct? – Manos Dec 18 '13 at 18:52
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    The minimal free resolution gives exactly the projective dimension of the module. –  Dec 18 '13 at 18:53
  • @YACP: So far i know only about minimal free resolutions of modules over local rings (paragraph 1.3 from Bruns and Herzog), in which case i can see that the minimal free resolution gives the projective dimension. Does the book treat MFR for modules over non-local rings as well? – Manos Dec 18 '13 at 19:00
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    No, because these don't exist. –  Dec 18 '13 at 19:01
  • @YACP: That's useful to know. Is the following correct: i) MFRs exist for graded modules over graded Noetherian rings ii) for a graded module over a graded Noetherian ring the MFR need not be finite – Manos Dec 18 '13 at 19:03
  • i) No. The graded ring has to be *local, that is, to have only one homogeneous maximal ideal. ii) Yes, if such a resolution there exists. –  Dec 18 '13 at 19:17
  • @YACP: Great, things are clearer now. I could accept all your comments as an answer. – Manos Dec 18 '13 at 19:19

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