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location Rennes, France
age 63
visits member for 4 years, 10 months
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1d
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 $A\times Z$.
2d
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
Apr
29
comment Automorphisms of complete local rings
@YCor: Thanks, Yves! Corrected.
Apr
29
answered Automorphisms of complete local rings
Apr
13
answered Fibrations of projective varieties
Apr
11
comment Does this $\mathbb{Z}_p$-algebra morphism induce a closed immersion on the generic fiber?
Take $R=\mathbb{Z}_p$, $B=$ a nontrivial finite extension of $\mathbb{Q}_p$, $f$ mapping $x$ to zero. The image of $f[1/p]$ is $\mathbb{Q}_p$ but $f\bmod p$ is surjective since $B/(p)$ is zero.
Apr
6
comment Is a normal proper relative curve over a DVR projective?
If I remember correctly, Artin's contraction results give rise to algebraic spaces, not schemes (except, for instance, over a finite field). So, what kind of ``curve'' are we looking for?
Apr
2
comment Writing down gerbes explicitly over the projective line
Try the gerbe of square roots of $\mathscr{O}(1)$.
Mar
30
comment When does an algebraic space that is a torsor over a scheme have to be a scheme?
The paper cited by Ariyan (which I have not read) is presented as a simplified version of the example in Raynaud's thesis (LNM 119, XIII.3.2). See also loc. cit. VI.2.5 for Q3: if $S$ is normal and noetherian and $G$ is smooth with connected fibers on $S$, then every $G$-torsor is locally quasiprojective over $S$.
Mar
27
comment Is surjectivity for morphisms of schemes local on the domain?
It is wrong indeed.