# Proper morphism sending coherent to coherent

Hello,

Is there a proof that the push forward by a proper morphism of Noetherian schemes sends coherent sheaves to coherent ones, without passing in the argument through projective morphisms?

Thank you, Sasha

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Well, you can use Chow's lemma, the result for projective morphisms, and the Leray spectral sequence to get it. Or you can find a different proof here EGA III 3.2.1 here: numdam.org/numdam-bin/feuilleter?id=PMIHES_1961__11_ – Parsa Dec 6 '11 at 15:38
@Parsa: If you use Chow's lemma, you are using the result for projective morphisms, which is what Sasha wants to avoid? – Timo Keller Dec 6 '11 at 17:22
If your schemes are locally of finite type over a field k, Antoine Ducros gave a proof that meets your requirements (see arXiv:math/0612521). Basically, the result is known for proper rigid analytic spaces (it is a theorem of Kiehl), so one also gets it for analytifications of proper morphisms (analytifications are to be taken in Berkovich sense, over k endowed with the trivial absolute value) and for proper morphisms themselves. – Jérôme Poineau Dec 6 '11 at 22:35
@Timo: I wasn't sure what he meant by "without passing in the argument through projective morphisms" so I gave an approach assuming proper but not projective (allowing the result for projective) and referenced another proof using only proper. – Parsa Dec 7 '11 at 0:21
Thank you very much for all the references. – Sasha Dec 18 '11 at 19:26