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location Rennes, France
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Dec
22
awarded  Critic
Dec
19
comment Real algebraic solution
@Mostafa: If you know that your variety is absolutely irreducible and has a real point in the smooth locus, then it even has a totally real algebraic point. This follows from the main result of my paper avaliable here.
Dec
19
comment Real algebraic solution
@Alex and Jason: one "special property of the reals" is this: since the field $K$ of real algebraic numbers is real closed, the embedding $K\subset \mathbb{R}$ is "elementary" in the language of ordered fields. This immediately implies the result.
Dec
15
comment Torsors and twists of algebraic groups
@Mostafa: on second thought, we only have $\underline{\mathrm{Aut}}(\mathbb{G}_{a,\mathbb{Q}})=\mathbb{G}_{m,\mathbb{Q}}$. If $S$ has characteristic $p>0$ and $t\in\mathscr{O}_S$, then $x\mapsto x+tx^p$ is an automorphism of $\mathbb{G}_{a,S}$ iff $t$ is locally nilpotent. Thus we have a compatible family of automorphisms over $\mathbb{F}_p[t]/(t^{n+1})$ that does not extend to $\mathbb{F}_p[[t]]$.
Dec
14
comment Is there a scheme parametrizing the closed subgroups of an algebraic group?
And this raises a natural question: is $F$ a disjoint sum of quasicompact subspaces?
Dec
14
comment Is there a scheme parametrizing the closed subgroups of an algebraic group?
One can remove the characteristic zero restriction. Let $G_n$ be the $n$-th infinitesimal neighborhood of $G$ along the unit section. We have an obvious map $\ell_n: H\mapsto H_n$ from $F$ to the (projective) Hilbert scheme of flat closed $S$-subschemes of $G_n$. A simple Noetherian argument shows that if $U\subset F$ is quasicompact, the restriction of $\ell_n$ to $U$ is monic for large $n$, hence $U$ is quasiprojective.
Dec
13
comment Torsors and twists of algebraic groups
@Mostafa: sorry, $\mathbb{G}_a$ was too simple. But take $G:=\mathbb{G}_a\times\mathbb{G}_m$: over $\mathbb{Q}[t]/(t^{n+1})$ we have an automorphism of $G$ sending $(x,u)$ to $(x,u\exp(tx))$. These are compatible for varying $n$, but do not "glue" to give an automorphism over $\mathbb{Q}[[t]]$.
Dec
13
comment Torsors and twists of algebraic groups
To complement Jason's example, at least the automorphism sheaf of $\mathbb{G}_m\times\mathbb{G}_m$ is a scheme, namely the constant group scheme $\underline{SL(2,\mathbb{Z})}$. The automorphism sheaf of $\mathbb{G}_a$ is not even a scheme.
Dec
13
comment What is the etale fundamental group of Spec Z((x))?
@user74230: From Abhyankar's Lemma, I can only deduce that the induced cover of $\mathrm{Spec}(\mathbb{Q}[[x]])$ is dominated by $\mathrm{Spec}(K[[x^{1/e}]])$ for some number field $K$ and some $e$. Do you then manage to prove that $K$ must be unramified?
Dec
13
comment Torsors and twists of algebraic groups
I am not sure the terminology is completely fixed, but in my view, a "form" of $G$ (equivalently, a class in $H^1(S,\mathrm{Aut}\,(G))$) is called "inner" if it is in the image of $H^1(S,\mathrm{Int}\,(G))$, and is a "strong inner form" if it is in the image of $H^1(S,G)\to H^1(S,\mathrm{Int}\,(G))\to H^1(S,\mathrm{Aut}\,(G))$. As Daniel Litt observes, none of these maps has any reason to be injective or surjective in general. The (classes of) automorphism groups of $G$-torsors are the strong inner forms.
Dec
12
comment What is the etale fundamental group of Spec Z((x))?
@S.Carnahan: $\mathbb{Z}((x^{1/2}))$ is not étale over $\mathbb{Z}((x))$. To see this, tensor with $\mathbb{F}_2$.
Dec
12
comment What is the etale fundamental group of Spec Z((x))?
I'm not sure the notation $\mathbb{Z}((x))$ is standard. Just to be sure, do you mean $\mathbb{Z}[[x]][x^{-1}]$?
Dec
12
comment Degeneracy locus and flatness over local Artinian ring
@Jason: thanks, that was my mistake indeed!
Dec
11
comment Degeneracy locus and flatness over local Artinian ring
@user46578 : yes. It is correct if $t=1$, but clearly we have $\dim V(\sigma_1,\dots,\sigma_t)\leq\dim V(\sigma_1,\dots,\sigma_{t-1})$. My guess is $rt$ for the expected codimension. Also "globally generated" is probably not enough.
Dec
11
comment Degeneracy locus and flatness over local Artinian ring
The codimension should increase with $t$.
Dec
10
answered Relative identity component for group algebraic spaces
Dec
8
comment Covering a finite set of points of height 1 by an affine open
Chow's lemma works directly. There is a proper surjective birational $p:X′\to X$ with $X′$ quasiprojective over $R$. Since $X$ is normal, $p$ is an isomorphism over an open subscheme $U\subset X$ containing each $x_i$. Clearly $U$ is quasiprojective, as a subscheme of $X'$.
Dec
7
comment Smoothness and smoothness over formal neighborhood
@prochet: as ACL notes, there is no such example if $Y$ is locally noetherian.
Dec
6
comment Is there a scheme parametrizing the closed subgroups of an algebraic group?
Once we know $F$ is a separated algebraic space, then, in characteristic zero, it must be a scheme. Indeed, we have a morphism $\ell:F\to\mathrm{Gr}:=\mathrm{Grass}_{\mathrm{Lie}\,(G)/S}$ sending each subgroup to its Lie algebra. In char. 0 it is pointwise injective, i.e. $\ell(k)$ is injective for every $S$-field $k$ (presumably, $\ell$ is even a monomorphism). Since $\ell$ is locally of finite type (because $F$ is), it must be locally quasifinite, so $F$ is a scheme since $\mathrm{Gr}$ is. In fact, this shows that every quasicompact open subspace of $F$ is quasiprojective over $S$.
Dec
6
comment Is there a scheme parametrizing the closed subgroups of an algebraic group?
@JeskoHüttenhain: No problem!