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Sep
30 |
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Scheme of irreducible components
You may also have a look at Matthieu Romagny, Manuscripta 136, 1–32 (2011) (in the context of algebraic stacks). |
Sep
17 |
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Is the Jacobian of curve self-dual?
@Felipe: The definition of a polarization involves a positivity condition (it should correspond to an ample divisor class). So, a principally polarized abelian variety is isomorphic to its dual, but I doubt that the converse is true. |
Sep
3 |
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Does bounded-degree base extension yield Zariski-dense Mordell-Weil group?
By a theorem of Zarhin, if $A'$ is the dual of $A$, then $(A\times A')^4$ admits a principal polarization. So the principally polarized case (in dimension $8n$) implies the general case. |
Aug
20 |
awarded | Popular Question |
Jul
28 |
awarded | Nice Answer |
Jul
16 |
awarded | Yearling |
Jul
1 |
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Vanishing of higher direct image of a morphism with generic fiber $\Bbb{P}^1$
If $Y$ is a DVR, take $X'=Y\times \mathbb{P}^1$, and $X=X'\smallsetminus\{z\}$ where $z$ is a point in the closed fiber. |
Jul
1 |
answered | Vanishing of higher direct image of a morphism with generic fiber $\Bbb{P}^1$ |
Jul
1 |
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The canonical bundle of an infinitesimal deformation
In fact $X$ need not be proper; or even of finite type (EGA II, 4.5.13). |
Jun
19 |
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If the direct sum of cyclic modules is cyclic, what happens to nontrivial extensions?
Sorry, I had indeed assumed your rings were commutative. |
Jun
18 |
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If the direct sum of cyclic modules is cyclic, what happens to nontrivial extensions?
The assumption that $M_1\oplus M_2$ is cyclic implies that $\mathrm{Ann}(M_1)+\mathrm{Ann}(M_2)=R$. Since $\mathrm{Ext}^1_R(M_2,M_1)$ is killed by $\mathrm{Ann}(M_1)$ and by $\mathrm{Ann}(M_2)$, it must be zero. |
Jun
11 |
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Can every commutative ring of characteristic $p\in\mathbb P$ be written as the form $R/(p)$ with $R$ being a ring of characteristic $0$?
For the existence, take $R=S\times\mathbb{Q}$. |
Jun
9 |
awarded | Civic Duty |
May
29 |
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Excellent rings
I suspect that in general, the universally catenary condition may be a source of trouble. |
May
26 |
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Varieties acted upon faithfully by an abelian variety
Standard construction: Let $G$ be a finite subgroup of $A$ acting freely on a variety $Y$. Then $G$ (resp. $A$) acts on $A\times Y$ by the diagonal action (resp. translation on $A$), and these actions commute so $A$ acts on $X:=A\times Y/G$, and it is easy to see that this action is free. Moreover, if you choose $Y$ so that every morphism $Y\to A$ is constant (e.g. $Y$ rational) then there is no $A$-equivariant morphism $X\to A$, and in particular $X$ is not a product $A\times Z$. |
May
25 |
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Automorphisms of a differential field and transcendence degree
Perhaps I am missing something, but from the definition it seems clear that $\mathcal{E}_\Phi$ is just $\Phi(\mathcal{E})$ and is isomorphic to $\mathcal{E}$ as a (differential) field. |
May
16 |
awarded | Citizen Patrol |
May
13 |
answered | Degree of sum of integral elements over a UFD |
May
7 |
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Infinitesimal deformations of the formal group of $\mathbb{G}_m$
Oh, right. I had read the question too fast. |
May
7 |
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Infinitesimal deformations of the formal group of $\mathbb{G}_m$
"The intervention of formal groups is a red herring": why is that? The question is about formal groups. |