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location Rennes, France
age 63
visits member for 4 years, 11 months
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Jul
1
comment 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
comment 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
comment 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
comment 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
comment 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
comment Excellent rings
I suspect that in general, the universally catenary condition may be a source of trouble.
May
26
comment 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
comment 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
comment Infinitesimal deformations of the formal group of $\mathbb{G}_m$
Oh, right. I had read the question too fast.
May
7
comment 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.
May
7
comment Is a normal proper relative curve over a DVR projective?
@Lisa: I suppose you may do that. In a more down-to-earth way, you can pick a closed point $x$ of the closed fiber $X_0$ which is in the regular locus $U$, and then a (Weil) divisor $D$ through $x$, finite over $R$. If $R$ is henselian, $D$ splits as $D'\coprod D"$ where $x$ is the only point of $D'$ in the closed fiber, so $D'\subset U$. Repeat this with one $x$ in each component of $X_0$ and take the sum of the $D'$'s: this is an ample Cartier divisor.
May
6
comment Is a normal proper relative curve over a DVR projective?
I think the answer is also yes if $R$ is henselian, if this helps.
May
1
comment Why are unramified maps not required to be locally of finite presentation?
For one thing, it seems reasonable that all immersions should be unramified.
Apr
30
comment Examples of quotients by infinitesimal group schemes
The construction itself was known before: for instance it is mentioned in Mumford's Abelian Varieties. What I proved in my paper is that the resulting quotient is the Jacobian of a genus 2 fibration over $\mathbb{P}^1$.
Apr
30
awarded  ac.commutative-algebra
Apr
29
revised Automorphisms of complete local rings
Error corrected, pointed out by Ycor