Timeline for extending homomorphisms of Abelian schemes
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Apr 1, 2017 at 12:41 | history | edited | user19475 | CC BY-SA 3.0 |
deleted 18 characters in body
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| Mar 26, 2017 at 6:13 | vote | accept | CommunityBot | moved from User.Id=19475 by developer User.Id=69903 | |
| Mar 25, 2017 at 21:06 | answer | added | Jason Starr | timeline score: 5 | |
| Mar 25, 2017 at 11:01 | comment | added | Jason Starr | The Rigidity Lemma does not give etaleness (and how do you produce accent grave in comments?). Consider a fixed elliptic curve $E$ over a field $k$. Let $S$ be a smooth, connected $k$-curve. Let $\mathcal{B}\to S$ be a family of elliptic curves over $S$ that dominates moduli and such that $\mathcal{B}_s$ is isomorphic to $E$ for some $k$-point $s$ of $S$. Let $\mathcal{A}$ be $E\times_{\text{Spec}\ k} S$. Then the $S$-scheme $\text{Hom}_S(\mathcal{A},\mathcal{B})$ has a closed point over $s$ that does not extend to a section over any etale open of $S$. | |
| Mar 25, 2017 at 8:57 | comment | added | user19475 | @Jason Starr: Assuming $S$ normal is OK. By the Weil extension theorem, do you mean "If $f$ is a rational map between smooth separated group schemes, defined in codimension $\leq 1$, then it is defined everywhere."? And then apply the valuative criterion for properness? For $\mathrm{Hom}_S(\mathscr{A},\mathscr{B}) \to S$ being unramified: Do you mean I should use the infinitesimal lifting criterion? And why don't you make your comments into an answer? | |
| Mar 25, 2017 at 8:34 | comment | added | Jason Starr | I just realized, for the argument that I am sketching, you do need to know that $S$ is unibranch, e.g., $S$ is integrally closed in its fraction field. Otherwise there may exist finite and unramified morphisms that are not 'etale, e.g., the normalization of a nodal plane curve (that is irreducible) is finite and unramified, yet it is not 'etale. | |
| Mar 25, 2017 at 8:29 | comment | added | Jason Starr | There are two elements of the solution. The first is the Weil extension theorem, which implies that the connected components of the Hom scheme $\text{Hom}_S(\mathcal{A},\mathcal{B})$ are proper over $S$. The second is an infinitesimal analysis that implies that $\text{Hom}_S(\mathcal{A},\mathcal{B})\to S$ is an unramified morphism. One proof of this does follow from the Rigidity Lemma, as you suggest. Altogether, $\text{Hom}_S(\mathcal{A},\mathcal{B})$ is a disjoint union of finite unramified $S$-schemes. Every component $T$ that dominates $S$ is finite and 'etale over $S$. | |
| Mar 25, 2017 at 8:06 | history | asked | user19475 | CC BY-SA 3.0 |