Laurent Moret-Bailly

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Name	Laurent Moret-Bailly
Member for	2 years
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Location	Rennes, France
Age	61

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Jun 15	answered	Schemes over a noetherian local ring and its completion
May 21	comment	Degree of a finite locally free group scheme over a base scheme of characteristic p OK if "connected" means "with connected fibers". But I suspect the OP means "connected" literally.
May 12	awarded	● Enlightened
May 12	awarded	● Nice Answer
May 12	comment	First order decidability of rings vs Diophantine decidability @SJR: I agree, but my feeling is also (for the same reason) that examples should be easier to come by.
May 9	comment	First order decidability of rings vs Diophantine decidability @SJR: Thanks for this nice example. It is in a sense a bit disappointing since a key point here, as was noted in the comments, is the choice of $\mathbb{Z}$ as the subring of coefficients. In fact, if we choose $\mathbb{Q}(t)$ instead, then the existential theory of $\mathbb{R}(t)$ is not e.c: this goes back at least to Denef. What surprises me here is the first order undecidability without extra constants.
May 7	revised	First order decidability of rings vs Diophantine decidability edited tags
May 6	awarded	● Nice Question
May 6	comment	First order decidability of rings vs Diophantine decidability As SJR notes, I guess equations should have coefficients in some fixed (recursive) subring of $R$ in order for decidability to make sense. Likewise, first order formulas are in the language of rings augmented with constants from such a subring.
May 6	asked	First order decidability of rings vs Diophantine decidability
May 2	comment	morphism between two elliptic curves over a local field It is not trivial since there are always quadratic twists (besides, is not a group!). The point here is that any nontrivial twist of $X$ must have bad reduction.
May 1	accepted	Example of codim 1 regular embedding that is not an effective Cartier divisor?
May 1	answered	Example of codim 1 regular embedding that is not an effective Cartier divisor?
May 1	accepted	morphism between two elliptic curves over a local field
May 1	answered	morphism between two elliptic curves over a local field
Apr 24	accepted	Dimension of formal fiber
Apr 23	comment	The notion of multiplicity in algebraic geometry With the notation in the definition of $\overline{\mu}$, assume $P$ is a singular point. Then in general there is an $f\in\mathcal{O}_{X,P}$ such that $\mu_f(I)=1$, but none such that $\overline{\mu}_f(I)=1$.
Apr 23	comment	Bound for the degree of the field of definition for a closed point of a variety There is something wrong with your TeX. The math doesn't show properly: specifically, the math italic alphabetic characters don't show at all.
Apr 22	comment	Dimension of formal fiber Standard fact in commutative algebra: Let $A$ be a noetherian ring (I am not sure about the minimal assumptions), $J$ an ideal such that $A$ is $J$-adically complete and separated. Let $M$ be an $A$-module. If $M$ is $J$-adically complete and separated and $M/JM$ is finitely generated, then $M$ is finitely generated. Apply this to $A=\mathbb{C}[[f_1,\dots,f_d]]$, $J=(f_1,\dots,f_d)$ and $M=\mathbb{C}[[t]]$. Things are even simpler if, say, $f_1$ is a uniformizer in $\mathbb{C}[[t]]$: then it is easy to see directly that $\mathbb{C}[[f_1]]=\mathbb{C}[[t]]$.
Apr 21	comment	Kawamata-Viehweg Vanishing Theorem for Excellent Surfaces @Omprokash Das: it is not true in general that you can embed $k(x)$ into $\mathcal{O}_{X,x}$ or even into the function field of $X$. Example: assume $k$ is not algebraically closed, take $X=\mathbb{P}^n_k$ and take for $x$ a closed point with residue field $\neq k$. Variant: same $X$ ($n\geq2$, $k$ arbitrary), $x$= generic point of a non-unirational subvariety, e.g. a curve of positive genus.
Apr 20	comment	Kawamata-Viehweg Vanishing Theorem for Excellent Surfaces I don't understand "over $k(x)$". Clearly $S$ is a two-dimensional local scheme with closed point $\mathrm{Spec}\,k(x)$, but in general it is not a $k(x)$-scheme in any natural way.
Apr 19	comment	Equivalent definitions of ample bundles @Serge: the question is what is meant by "generate the stalk". With your phrasing, it should mean "generate the stalk $E_x$ as an $\mathcal{O}_{X,x}$ -module". And then of course it is true for $E=\mathcal{O}_{X}$ . In condition 1 of the question, "$E_x$" should be understood as the stalk at $x$ tensored with the residue field. An even then, that version of the definition works only if you work with rational points over an algebraically closed field.
Apr 19	comment	Equivalent definitions of ample bundles @Serge: $\mathcal{O}_X$ is always globally generated!
Apr 16	answered	Dimension of formal fiber
Apr 6	comment	Finite extensions of residue fields of Henselian DVRs Maybe I misunderstand the question, but what if you start with an extension $L/F$ which is not simple and then take $K=F((t))$?
Apr 2	comment	Is the image of discrete set under an open map discrete? @Nik: Oops, sorry.
Apr 2	comment	Is the image of discrete set under an open map discrete? Here is a simpler one: if $t\in\mathbb{R}$ is irrational, the image of $t\mathbb{Z}$ under the projection $\mathbb{R}\to\mathbb{R}/\mathbb{Z}$ is not discrete.
Mar 7	comment	Is the universal closed subscheme reduced? The $G$-invariant part of a reduced scheme is not reduced in general: take $\mathbb{Z}/2\mathbb{Z}$ acting on the curve $xy=0$ by exchanging coordinates.
Mar 6	accepted	Open idempotents in modules over a local ring
Mar 6	revised	Open idempotents in modules over a local ring added 360 characters in body
Mar 6	revised	Open idempotents in modules over a local ring added 767 characters in body
Mar 6	answered	Open idempotents in modules over a local ring
Mar 6	revised	Which schemes can be presented as limits of smooth varieties? deleted 12 characters in body
Mar 6	comment	Which schemes can be presented as limits of smooth varieties? I didn't say the morphisms were dominant. That would mean $B$ is the direct limit of its smooth subalgebras, a much stronger condition, it seems. And I know nothing about the non-affine case.
Mar 6	answered	Which schemes can be presented as limits of smooth varieties?
Mar 4	accepted	Artin approximation theorems over non-regular rings/non-Noetherian rings
Feb 28	answered	Artin approximation theorems over non-regular rings/non-Noetherian rings
Feb 19	comment	Zero-cohomology of birational varieties As you define it, $f_*$ depends on the choice of $U_1$.
Feb 18	comment	Formal power series over a henselian ring More generally: if $A$ is a local ring and $I$ an ideal such that $A$ is $I$-adically complete and separated, then $A$ is henselian if (and only if!) $A/I$ is henselian. This is easy by observing that each $A_n:=A/I^n$ is henselian (because $A_{n,\mathrm{red}}$ is) and then inductively lifting roots to $A$. See Raynaud, LNM 169, I, §2 where this result is given as an exercise.
Feb 16	comment	Are f.g. projective modules free over total quotient ring of a reduced non-noetherian commutative ring This ring is local, so projective=free.
Feb 11	comment	Open subset in the flat topology on Spec(R) The only Zariski neighborhood of the closed point in a local scheme $X$ is $X$. So, $U=\mathrm{Spec\,}(R_P)$ and your condition just means that $S$ is stable under generalization.
Feb 10	comment	Complementation in an extension field It is well known that if $\mathbb{Q}$ is complemented in $\mathbb{R}$ you get nonmeasurable sets in $\mathbb{R}$. I don't think you can get this without some form of AC.
Feb 8	comment	on trivialisation of T-torsors And what is $x$?
Feb 6	accepted	Are noetherian schemes generically Jacobson?
Feb 6	revised	Are noetherian schemes generically Jacobson? added 58 characters in body
Feb 5	answered	Are noetherian schemes generically Jacobson?
Feb 5	accepted	purity for finite flat group schemes
Feb 5	answered	purity for finite flat group schemes
Feb 4	revised	Are rational varieties simply connected? added 669 characters in body
Feb 4	awarded	● ag.algebraic-geometry

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