Timeline for Pushouts of noetherian rings
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16 events
| when toggle format | what | by | license | comment | |
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| Feb 23, 2020 at 15:07 | vote | accept | Martin Brandenburg | ||
| Aug 28, 2017 at 2:34 | answer | added | R. van Dobben de Bruyn | timeline score: 7 | |
| Jul 14, 2017 at 8:28 | history | edited | Martin Brandenburg | CC BY-SA 3.0 |
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Apr 8, 2011 at 17:22 | comment | added | Ravi Vakil | @Boyarsky, I like that Deligne quote. Do you have a reference? (Please email it too!) | |
| Jun 29, 2010 at 10:27 | comment | added | Boyarsky | @Tom: nope, it can't. The reason is that any noetherian local domain with dimension > 1 has infinitely many height-1 primes (since the union of finitely many non-maximal primes cannot equal the maximal ideal in any local ring, by one of those lemmas very early in Atiyah-MacDonald, 1.11 or so, and any nonzero non-unit in a noetherian domain lies in some height-1 prime by the Hauptidealsatz). So in fact the spectrum of such a ring must be infinite as a set. And $[0,n]$ with the order topology corresponds to being a local domain (after killing nilpotents) with Krull dimension $n$. | |
| Jun 29, 2010 at 2:04 | comment | added | Tom Goodwillie | No problem. I was being a little terse. By the way, I don't suppose a noetherian ring can have Spec [0,n] if n>1, can it? | |
| Jun 28, 2010 at 20:37 | comment | added | Boyarsky | @Tom: OK, sorry for my misunderstanding of what you were saying. | |
| Jun 28, 2010 at 19:42 | comment | added | Tom Goodwillie | @Boyarsky: If $R$ has an infinite ascending chain of prime ideals, then $R$ has an infinite ascending chain of ideals. If $R$ has an infinite ascending chain of ideals, then $R$ is non-noetherian. It is true that there are non-noetherian rings which cannot be detected by this method, but $L\otimes_KL$ is not one of them. | |
| Jun 28, 2010 at 17:00 | comment | added | Boyarsky | @Tom: Sorry, I don't understand. The example I gave was a non-noetherian ring whose spectrum reflects that it has no infinite ascending chain of prime ideals. So it seems to be a counterexample to your suggested method to prove a ring is not noetherian via a topological argument. What is the general mechanism by which you propose to prove a ring is non-noetherian by a topological method which would not apply to the non-noetherian example which I mentioned? | |
| Jun 28, 2010 at 16:08 | comment | added | Tom Goodwillie | @Boyarsky: Maybe one cannot infer that a ring is noetherian from such considerations, but one can sometimes infer that it is non-noetherian. | |
| Jun 28, 2010 at 15:04 | comment | added | Boyarsky | @Tom: there are non-noetherian valuation rings of finite Krull dimension (even spectrum homeomorphic to $[0,n]$ with the order topology), so one cannot infer that even a domain is not noetherian merely from topological considerations with Spec. So your second sentence is unclear. | |
| Jun 28, 2010 at 14:50 | comment | added | Tom Goodwillie | If R-->S is universal for maps from a given ring R to noetherian rings, then by using maps into fields you can show that every prime ideal of R is the contraction of a unique prime ideal of S. If you could also show that this bijection is an order isomorphism, then you could conclude that R has no infinite ascending chain of prime ideals, which would rule out the existence of such a universal map in the case you are looking at ($R=L\otimes_KL$). | |
| Jun 28, 2010 at 12:43 | comment | added | Boyarsky | By the way, the Picard functor is locally of finite presentation, and clearly the (locally) noetherian property is preserved under fiber products when at least one of the structure maps is locally of finite type. So the assertion that your Picard scheme is a group scheme makes perfectly good sense within the category of locally noetherian schemes if you insist on $S$ being locally noetherian and $X$ projective. (What lies deeper is that the Picard functor of an abelian scheme is an algebraic space when dropping projective hypotheses, and its relative identity component is always a scheme.) | |
| Jun 28, 2010 at 11:57 | comment | added | Boyarsky | (i) The "locally noetherian" restriction is easy to remove via the limit formalism of EGA IV$_3$, sections 8--12. (ii) It is not "ugly" that the noetherian property is generally lost of the formation of fiber products of noetherian schemes; what is ugly/unnatural is to insist on noetherian hypotheses when unnecessary. As Deligne once said, Grothendieck taught us that it is better to have a category with some "bad" objects and good operations. (iii) Faithfully flat descent provides natural/useful examples of non-noetherian schemes ($\widehat{A} \otimes_ A \widehat{A}$); you'll get over it. | |
| Jun 28, 2010 at 11:01 | history | asked | Martin Brandenburg | CC BY-SA 2.5 |