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It seems that noetherian assumptions are not necessary in many results by Hartshorne, in his book "Algebraic Geometry". How much is this true? Could you please give examples?

Wojowu
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Al-Amrani
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  • I remember one user a while back who asked many questions on MO about removing the noetherian assumption from various basic results in algebraic geometry. Possibly can find these questions via the search feature – Sam Hopkins Oct 18 '21 at 23:15
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    It would make for an unbelievably long list if you were to list all results which do and ones which don't generalize to non-Noetherian case, and if they do, how they have to be modified. If I recall correctly, SGA texts develop the theory without the Noetherian condition as much as possible, so you may want to consult that, – Wojowu Oct 18 '21 at 23:17
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    Couldn't find the user I was looking for, but did find this: https://mathoverflow.net/questions/224243/examples-of-noetherian-overkill – Sam Hopkins Oct 19 '21 at 00:25
  • @SamHopkins it seems like the OP was even part of the discussion in the linked question. – user347489 Oct 19 '21 at 00:50
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    Please keep in mind that Hartshorne's book is a textbook which attempts to give an overview under reasonable assumptions in a reasonable number of pages. So he assumes schemes are noetherian, or that fields are algebraically closed when not strictly necessary. If you want things in more generality, please consult EGA, SGA, or the Stacks Project. – Donu Arapura Oct 19 '21 at 01:46
  • In fact the SGA text develops methods to remove the noetherian assumptions. – Wilberd van der Kallen Oct 19 '21 at 07:17
  • To PhD Students , first of all, and to Donu Arapura @ : let me explain the Genesis of my question. I am Jouanolou's student (1970's, 1980's, 1990's). He is one of the most appreciated , by Grothenthendieck himself ! See "Semailles et Récoltes". So, and I am not going to put this as a self-answer to my own question. there's a PhD course by Jouanolou (Strasbourg), à la Grothendieck, without any noeterian hypothesis : " Elimination theory, the case of one variable (i.e. two variables in the homogeneous case)" . Published in French : "Elimination , Le cas d'une variable" (Hermann , 2006,Paris.). – Al-Amrani Oct 19 '21 at 18:59
  • To PhD students : the book mentioned above (by F.Apéry and J.-P. Jouanolou) is 477 pages, with exercises and solutions (required by the editor Hermann). F. Apéry, son of the well-known APERY, helped in writing the book, no more. It is a good practice in Algebraic Geometry , "à la GROTHENDIECK". The content geralizes to any number of variables ! See Jouanolou's work as published in different issues of Advances in Math. In French ! Who will/would , in English , translate ?! – Al-Amrani Oct 19 '21 at 19:30
  • @ Donu Arapura .It would be interesting to give an example showing how Noetherianity assumption allows a simple proof, when non Noetherianity does not. Thas is in fact the meaning of my question. – Al-Amrani Nov 05 '21 at 23:00
  • Here is a typical example which was given to me.The proof of Serre-affinity theorem, in the Noetherian case , uses that sheaves associated to injective modules are flasque. In the general case, that is not true. – Al-Amrani Nov 29 '21 at 22:10

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Here is a typical example Jouanolou quoted to me. The proof of Serre-affinity theorem, in the Noetherian case , uses that sheaves associated to injective modules are flasque. In the general case, that is not true. –

Al-Amrani
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