since is a finitely generated -module, both and are finitely generated -modules. let be the unique maximal ideal of and put . then is a finite dimensional -vector space. let
you should know that, as a result of Nakayama's lemma, is generated, as an -module, by elements. (in fact, is the minimum number of generators of as an -module.) anyway, so there exists an onto -homomorphism . let . then we have a short exact sequence and thus, since is projective, this sequence splits, i.e. . so is a free -module of rank let
then and thus
now, since is a free -module of rank , we have . we also have by . thus by . so and hence , by Nakayama's lemma. it now follows from that and so is free.