In the category of finitely generated modules over a commutative Noetherian ring, the splitting of a short exact sequence can be checked locally at the maximal ideals of the ring. One reference for this is contained in the answer by Jeremy Rickard here Local property of split exact sequence .

My question is: Is there a reference of book or paper or some monograph where this fact (possibly along with a proof) appears ?

I would like to use this in a presentation and this result seems too natural to not have appeared anywhere before in literature.


  • 1
    $\begingroup$ Reiner's "Maximal Orders", Theorem 3.20, for instance. $\endgroup$ May 12, 2020 at 9:02
  • $\begingroup$ @UriyaFirst Shouldn't that be an answer? $\endgroup$
    – Pedro
    May 12, 2020 at 11:11
  • $\begingroup$ @PedroTamaroff I guess it should. $\endgroup$ May 12, 2020 at 12:53

1 Answer 1


Reiner's "Maximal Orders", Theorem 3.20, for instance.


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