1,423 reputation
918
bio website perso.univ-rennes1.fr/…
location Rennes
age 39
visits member for 5 years
seen 1 hour ago

May
7
comment When is the Hom-scheme connected?
No, your functor is not representable by a scheme, already e.g. for $A=B=k[X]$.
May
6
awarded  Yearling
Apr
17
comment Reference for a lemma on étale maps
With all due respect to the Stacks Project, it is not peer-reviewed...
Mar
31
comment When does an algebraic space that is a torsor over a scheme have to be a scheme?
For Q1, a precise reference is SGA1, Exp. VIII, Cor. 7.9. In fact SGA1, Exp. VIII, sec. 7 gives a list of criteria as in your Q3.
Mar
31
comment Flatness and intersection of fibers
Certainly not. Geometrically $Y_1=Spec(B_1)$ and $Y_2=Spec(B_2)$ are closed subvarieties in some affine space $\mathbb{A}^{n+1}_X$ with $X=Spec(A)$ and you are asking: is it true that $Y_1,Y_2$ flat over $X$ imply $Y_1\cap Y_2$ flat over $X$? A counterexample is e.g. $X=\mathbb{A}^1_{\mathbb{C}}$ and $Y_1,Y_2=$ the coordinate axes in $\mathbb{A}^2_{\mathbb{C}}$.
Mar
31
comment Structure of $\text{Aut}_R(R[X])$
All $R$-automorphisms of $R[X]$ are substitutions $X\mapsto a_0+a_1X+a_2X^2+\dots+a_nX^n$ with $a_i\in R$ and $a_0$ arbitrary, $a_1$ invertible, $a_i$ nilpotent for $i\ge 2$. This is either an exercise, or (I believe) stated somewhere in Demazure-Gabriel.
Mar
27
comment How far is it to extend the results of SGA III Exp. VIB from group schemes to group spaces?
You're perfectly right that the assumptions of condition (ii) of sga3 exp.viB Cor.4.4 are met. The point I was worried about is whether (ii) should imply (i) in loc. cit., and it is quite striking that for reduced $S$ (as in your case) this implication holds. Apologies!
Mar
26
comment How far is it to extend the results of SGA III Exp. VIB from group schemes to group spaces?
Maybe could you have a look at Lemma 4.2.8 in the paper S. Brochard, Foncteur de Picard d’un champ algébrique, Math. Ann. 2009. Also note that this has been extended to alg. stacks in an Erratum to the paper, available on the author's webpage.
Mar
26
comment How far is it to extend the results of SGA III Exp. VIB from group schemes to group spaces?
Dear Heer, do you in fact assume that $G$ is smooth over $S$? If I'm not mistaken, with your assumptions it could be that $G=G_s\amalg G_t$ mapping to $S$ in the natural way, which I doubt you want to consider.
Feb
25
comment Implicit Function Theorem on Singular Varieties
Dear Giulio, in Algebraic Geometry the property of finiteness has a specific meaning which implies properness. The example of Laurent is bijective but not proper since the image of the closed subset $X_2$ is not closed in $Y$.
Feb
18
comment $R^{\dim X-\dim Y}f_{\ast}\omega_X \simeq \omega_Y$ in positive characteristic?
Isn't such vanishing the result of Chatzistamatiou and Rülling in "Higher direct images of the structure sheaf in positive characteristic", ANT 2011?
Feb
18
comment Is the quotient of a scheme by the free action of an elliptic curve an algebraic space?
Why Keel-Mori? The original Artin theorem proves that the quotient fppf sheaf, which since there are no stabilizers is also the quotient stack [X/E], is an algebraic space. See e.g. Artin, The implicit function theorem in algebraic geometry, Thm 7.1.
Dec
22
awarded  Popular Question
Dec
19
comment How do you calculate the group scheme of E[p] for a an elliptic curve E in characteristic p?
In your first sentence, the p-torsion kernel $E[p]$ of a supersingular elliptic curve is not $\alpha_p$. Indeed, the former has order $p^2$.
Dec
9
comment Quotient of a (non-linear) algebraic group by a closed subgroup
Also Perrin, Approximation des schémas en groupes quasi-compacts sur un corps, Bull. SMF 1976, shows the following: if G is quasi-compact and H is either a normal subgroup, or defined by a finitely generated ideal, then the fpqc sheaf G/H is representable by a scheme.
Dec
3
answered Regular rings and formally smooth algebras
Nov
16
revised Is this formally étale morphism of schemes an isomorphism?
deleted 193 characters in body
Nov
15
revised Is this formally étale morphism of schemes an isomorphism?
added 5 characters in body
Nov
15
answered Is this formally étale morphism of schemes an isomorphism?
Nov
8
comment Is this formally étale morphism of schemes an isomorphism?
Two questions: 1) I'd be happy to have details on the proof that $\mathfrak{m}'=\mathfrak{m}B$. 2) For formal glueing in the very last step, the quoted lemma from the Stacks Project requires to know that $B$ is a finite $A$-module, how do you deal with this issue?