Timeline for Affine neighborhood of an $S$-valued point
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Jun 25, 2013 at 3:02 | review | First posts | |||
| Jun 26, 2013 at 9:08 | |||||
| Jun 2, 2013 at 5:10 | answer | added | user29283 | timeline score: 2 | |
| May 30, 2013 at 7:20 | comment | added | NotFromBrazil | @Emerton @ACL Yes, the setup here is that $E$ is an elliptic curve. I'm sorry that I forgot to mention this. | |
| May 30, 2013 at 7:19 | comment | added | ACL | If you can find a relatively ample line bundle $L$ on $E$ (if, as I suspect, $E$ is an elliptic curve or an abelian scheme, this exists, or is a quite harmless hypothesis), and if $S$ is affine and has trivial (torsion is enough) Picard, then you can assume that the restriction of $L$ to the $0$-section is trivial, and taking higher powers of $L$, obtain a global section $s$ of $L$ which is invertible on the $0$-section. The invertibility locus of $s$ is an affine neighborhood of this section. | |
| May 30, 2013 at 2:39 | comment | added | user28172 | If the base $S$ is local then any open affine $U$ around the closed point of the identity section $e$ will contain $e$ (since the only open subscheme of a local scheme that contains the closed point is the entire space), so by passing to a local base you get such an affine open. It may be unnecessary to have such a $U$, depending on the goal. If the aim is to discussion formal completion along the identity via a power series ring then one wants the base to be local, or at least the relative tangent space along $e$ to be globally free (of rank 1). Passing to a local base is usually harmless. | |
| May 30, 2013 at 2:30 | comment | added | Emerton | ... context explaining what he means.) Regards, | |
| May 30, 2013 at 2:30 | comment | added | Emerton | Dear NFB, There is no such general notion, and certainly in the general set-up of $E\to S$, there need not be an affine open subset of $E$ that contains the zero section. (Imagine that $S$ was a positive-dimensional projective variety, which could occur if $E$ was just a product $E_0 \times S$, so that the zero section was again a positive-dimensional projective variety.) But probably Hida wrote one thing and meant another, e.g. perhaps he is implicitly restricting to some small open subset of the base. (I haven't read the text apart from what you quoted, so there may also be addional ... | |
| May 29, 2013 at 23:20 | history | asked | NotFromBrazil | CC BY-SA 3.0 |