Timeline for What elementary problems can you solve with schemes?
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Jul 16, 2013 at 12:49 | comment | added | Zsbán Ambrus | @NoahS: True, thanks for the explanation. | |
| Jul 16, 2013 at 11:52 | comment | added | Noah Schweber | By the compactness theorem, since $\phi$ is a consequence of $T$ we know that $\phi$ is a consequence of some finite $T'\subset T$; this $T'$ can only contain finitely many $\chi_j$, so in fact $\phi$ is a consequence of the characteristic of the algebraically closed field being at least $n$ for $n=\max\lbrace j: \chi_j\in T'\rbrace$. (Incidentally, this also proves that we only need roots of polynomials of sufficiently large degree!) | |
| Jul 16, 2013 at 11:51 | comment | added | Noah Schweber | @Zsban: Yes. Suppose $\phi$ is a first-order sentence true in all algebraically closed fields of characteristic zero. The set of algebraically closed fields of characteristic zero is the set of models of $T=\lbrace \psi\rbrace\cup\lbrace \theta_i: i\in\omega\rbrace\cup\lbrace \chi_j: j\in\omega\rbrace$, where $\psi$ is the field axiom(s), $\theta_i$ asserts that all degree-$i$ polynomials have roots, and $\chi_j$ asserts that the characteristic is at least $j$. (cont'd) | |
| Jul 16, 2013 at 8:01 | comment | added | Zsbán Ambrus | @MartinBrandenburg: logic proves the if part, but does it also prove the only if part? | |
| Apr 4, 2013 at 0:18 | history | edited | Daniel Miller | CC BY-SA 3.0 |
added 48 characters in body
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| Mar 24, 2011 at 3:22 | history | made wiki | Post Made Community Wiki by Kim Morrison | ||
| Mar 22, 2011 at 17:11 | comment | added | Donu Arapura | While I also agree that the logic proofs are simpler in this case (I'd probably use ultraproducts myself), the technique of specializing into characteristic $p$ can pushed much further, and scheme theory provides a very powerful language for this. | |
| Mar 22, 2011 at 9:31 | comment | added | Martin Brandenburg | But this is just an application of the compactness theorem in mathematical logic! | |
| Mar 21, 2011 at 23:12 | comment | added | Thierry Zell | I find this application especially nice. | |
| Mar 21, 2011 at 20:18 | history | answered | Dustin Clausen | CC BY-SA 2.5 |