Timeline for Maximal ideals in polynomial rings over algebraically closed fields - when Weak Nullstellensatz does not apply
Current License: CC BY-SA 2.5
10 events
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| Feb 2, 2011 at 10:21 | comment | added | François Brunault | @David, thanks for the precisions. Although I'm unable to make your question more precise, I think I'm beginning to see your point. | |
| Feb 2, 2011 at 6:04 | comment | added | David Feldman | @François If I may make a comparison, an explicit description of even a single non-principal ultrafilter on $\Bbb N$ more than seems unlikely - such a description would surely embody a proof of an axiom, BPIT, provably independent of $ZF$ . But that has not prevented the development of a whole literature concerning the structure of $\beta{\Bbb N}$ (including more independence results). So my question which asks for a description of the maximal spectrum need not fall to the difficulty of describing the individual ideals. | |
| Feb 1, 2011 at 14:29 | comment | added | François Brunault | An explicit description of all maximal ideals seems unlikely to me in the general case. Lang's article gives an example where the residue field is isomorphic to $k(t)$, and this example can probably be adapted to find arbitrary residue fields $K/k$ with $\operatorname{trdeg}(K/k) \leq$ the number of variables. | |
| Feb 1, 2011 at 13:03 | comment | added | Martin Brandenburg | @David: You're right. | |
| Feb 1, 2011 at 9:20 | comment | added | David Feldman | Thanks Martin, but I think you accepted an answer there that ends where my question begins. | |
| Feb 1, 2011 at 8:49 | comment | added | Martin Brandenburg | mathoverflow.net/questions/41262/maximal-ideals-of-kx-1-x-2 | |
| Feb 1, 2011 at 6:51 | comment | added | David Feldman | Pete - I'm happy to have brought Lang's article to your attention! It always interests me how results/proofs do or don't get into the canon. For example, the American Mathematical Monthly has an endless supply of improvements on basic textbook proofs, but most seem ignored as new textbooks copy proofs out of old... | |
| Feb 1, 2011 at 6:35 | comment | added | Pete L. Clark | Never mind: it was easy enough to find: Lang, Serge Hilbert's Nullstellensatz in infinite-dimensional space. Proc. Amer. Math. Soc. 3, (1952). 407–410. I'll certainly take a look. | |
| Feb 1, 2011 at 6:29 | comment | added | Pete L. Clark | @David: I am pretty sure that "weak Nullstellensatz" does not include the extension by Lang of which you speak. In fact, I know the former pretty well (my notes on commutative algebra contain something like five proofs of it), but I think I have never heard of this result of Lang. It certainly sounds nice: could you give a reference? | |
| Feb 1, 2011 at 5:31 | history | asked | David Feldman | CC BY-SA 2.5 |