Timeline for Can the simplicity of abelian varieities be implied by the reduction
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 29, 2010 at 17:05 | vote | accept | TJCM | ||
| Nov 27, 2010 at 21:01 | answer | added | Qing Liu | timeline score: 5 | |
| Nov 27, 2010 at 0:35 | comment | added | BCnrd | Formal GAGA is not needed; it is simpler. Each $\mathcal{A}[\ell^n]$ is a finite etale $R$-scheme ($R$ is base dvr), and any map between finite etale $R$-schemes is determined by its effect on a single geometric fiber. (This is true way more generally over any connected scheme, but let's not get into that.) Indeed, can assume by faithfully flat scalar extension to completion, then maximal unramified extension, then completion again, that $R$ is complete with sep. closed residue field. Then by Hensel every finite etale algebra is a product of copies of $R$. So it is "physically obvious". | |
| Nov 26, 2010 at 21:23 | comment | added | TJCM | Just want to make sure I understand the way to prove the injectivity correctly: Do I need to first go to endomorphism of $A[l^n]$ over formal scheme $Spf(R)$, then use formal GAGA and lifting property of etale scheme $A[l^n]$ over $R$? | |
| Nov 26, 2010 at 17:46 | history | edited | Andrey Rekalo | CC BY-SA 2.5 |
Typo in the title
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| Nov 26, 2010 at 17:08 | comment | added | BCnrd | Yes. Let $\mathcal{A}$ denote the Neron model over the dvr local ring at the chosen good place. The natural map ${\rm{End}}(\mathcal{A}) \rightarrow {\rm{End}}(A)$ is an equality and the natural specialization map ${\rm{End}}(\mathcal{A}) \rightarrow {\rm{End}}(\mathcal{A}_{\kappa})$ is injective (using $\ell$-adic Tate modules for $\ell \ne {\rm{char}}(\kappa)$), so we get an injective map ${\rm{End}}^0(A) \hookrightarrow {\rm{End}}^0(\mathcal{A}_{\kappa})$. The target here is a division algebra, so the source is too (as everything finite-dim'l over $\mathbf{Q}$). | |
| Nov 26, 2010 at 16:50 | history | asked | TJCM | CC BY-SA 2.5 |