Timeline for Can the projective line be provided with a ring structure?
Current License: CC BY-SA 3.0
12 events
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| Jan 7, 2020 at 13:48 | comment | added | Tom De Medts | @WolfgangTintemann: Notice that the formulas on the Wikipedia page are only partially defined: $0 \cdot \infty$ and $\infty + \infty$ are undefined. So this does not make $\mathbf{P}^1_K$ into a ring. | |
| Aug 2, 2015 at 19:51 | comment | added | Todd Trimble | If $\phi: X \to Y$ is a bijection and $X$ carries a group structure (denoted by $+_X$), then $Y$ carries a group structure by "conjugation": $y +_Y y' = \phi(\phi^{-1}(y) +_X \phi^{-1}(y'))$. This I'm sure is what Eric meant. Another term for this (which generalizes to any sort of structure you like) is "transport of structure". It's a general thing all mathematicians pick up on at some point. | |
| Aug 2, 2015 at 18:56 | history | edited | Ben Webster♦ | CC BY-SA 3.0 |
added 277 characters in body
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| Aug 2, 2015 at 18:26 | comment | added | Wolfgang Tintemann | In en.wikipedia.org/wiki/Projective_line I see under section "Line extended by a point at infinity" some formulas for addition and multiplication - alas no citation of the source of these rules. | |
| Aug 2, 2015 at 18:17 | comment | added | Wolfgang Tintemann | Sorry - I dont understand what "conjugating addition" means. | |
| Aug 2, 2015 at 17:52 | comment | added | Eric Wofsey | @WolfgangTintemann: Yes, it is obtained by conjugating addition on $\mathbb{R}/\pi\mathbb{Z}$ by the homeomorphism $\tan:\mathbb{R}/\pi\mathbb{Z}\to\mathbb{P}^1_{\mathbb{R}}$. | |
| Aug 2, 2015 at 17:46 | comment | added | Wolfgang Tintemann | This looks like the addition formula of the $\tan$-function. | |
| Aug 2, 2015 at 17:34 | comment | added | Noam D. Elkies | Actually there is an algebraic group structure on ${\bf P}^1_{\bf R}$, namely $x \oplus y = (x+y) \, / \, (1-xy)$. But it can't extend to ${\bf P}^1_{\bf C}$, nor to a ring structure on ${\bf P}^1_{\bf R}$. | |
| Aug 2, 2015 at 17:29 | vote | accept | Wolfgang Tintemann | ||
| Aug 2, 2015 at 17:07 | comment | added | Will Chen | @WolfgangTintemann You can find the case for $K = \mathbb{C}$ here on page 10: jmilne.org/math/CourseNotes/AV.pdf You should probably read the earlier pages to see how he defines an abelian variety. The general case for other fields $K$ should follow from either the Lefschetz principle when $char(K) = 0$ or by (trivial) deformation theory when $char(K) > 0$ by lifting to characteristic 0. | |
| Aug 2, 2015 at 16:05 | comment | added | Wolfgang Tintemann | It is far beyond my knowledge that "a projective curve with an algebraic group structure (as addition would be) must be an elliptic curve". In what kind of books do I find this remarkable result ? Algebraic Geometry, Elliptic Curves, or Projective Geometry ? | |
| Aug 2, 2015 at 15:54 | history | answered | Ben Webster♦ | CC BY-SA 3.0 |