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?
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$\begingroup$ 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 $\endgroup$– Sam HopkinsCommented Oct 18, 2021 at 23:15
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8$\begingroup$ 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, $\endgroup$– WojowuCommented Oct 18, 2021 at 23:17
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2$\begingroup$ Couldn't find the user I was looking for, but did find this: mathoverflow.net/questions/224243/… $\endgroup$– Sam HopkinsCommented Oct 19, 2021 at 0:25
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$\begingroup$ @SamHopkins it seems like the OP was even part of the discussion in the linked question. $\endgroup$– user347489Commented Oct 19, 2021 at 0:50
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13$\begingroup$ 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. $\endgroup$– Donu ArapuraCommented Oct 19, 2021 at 1:46
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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. –