Grothendieck seemed to try to eliminate Noetherian conditions as much as possible in EGA. For example, he developed the cohomology theory of schemes without Noetherian conditions. On the other hand, the Hartshorne's book assumes Noetherian conditions to do it. If one's main interest is in the geometry of varieties or schemes of finite type over Dedekind domains, does one need non-Noetherian theory like Grothendieck's?

  • $\begingroup$ If your Dedekind domain happens to be the ring of integers of some number field, then one often encounters the associated adelic rings, which are non-Noetherian. $\endgroup$ May 10 '12 at 8:21
  • $\begingroup$ I would say "yes". For a related question with good answers, see mathoverflow.net/questions/61935/… $\endgroup$ May 10 '12 at 8:31

For the most part, the answer is "no"; most people who work with varieties will never need to worry about non-Noetherian rings. But there are reasons to be open to the non-Noetherian setting. First, they can actually come up (as pointed out in the comments). As just one example, the normalization of a Noetherian ring can be non-Noetherian. You can either work hard to show that the rings you care about don't have this pathology, or you can just relax and not worry about it. Second, there is often no harm in working with schemes in general. (One caution with Hartshorne: often the Noetherian conditions are unnecessary, and not used in the proofs; and in some cases, the arguments are more complicated because of the desire to work in the Noetherian setting.)

My suggestion is this: don't worry about reading how to remove Noetherian conditions until you have a need to remove them. But don't gratuitously add Noetherian conditions for no good reason. (Note also: sometimes Noetherian conditions can be removed with little pain, see for example my answer Flatness for family of hypersurfaces.)

  • $\begingroup$ Thank you for your nice answer. I agree with you on Hartshorne. Please let me wait for a few days before I accept yours. $\endgroup$ May 11 '12 at 3:13
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    $\begingroup$ Fiber products of noetherian schemes ... $\endgroup$ May 11 '12 at 11:22
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    $\begingroup$ @Martin Noetherian conditions are not necessarily preserved by a base change. I guess this is one of the reasons why Grothendieck endeavored to remove them. $\endgroup$ May 12 '12 at 12:13

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