MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

# Laurent Moret-Bailly

 7,356 Reputation 3411 views

## Registered User

 Name Laurent Moret-Bailly Member for 2 years Seen 1 hour ago Website Location Rennes, France Age 61
 Jun15 answered Schemes over a noetherian local ring and its completion May21 comment Degree of a finite locally free group scheme over a base scheme of characteristic pOK if "connected" means "with connected fibers". But I suspect the OP means "connected" literally. May12 awarded ● Enlightened May12 awarded ● Nice Answer May12 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. May9 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. May7 revised First order decidability of rings vs Diophantine decidabilityedited tags May6 awarded ● Nice Question May6 comment First order decidability of rings vs Diophantine decidabilityAs 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. May6 asked First order decidability of rings vs Diophantine decidability May2 comment morphism between two elliptic curves over a local fieldIt 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. May1 accepted Example of codim 1 regular embedding that is not an effective Cartier divisor? May1 answered Example of codim 1 regular embedding that is not an effective Cartier divisor? May1 accepted morphism between two elliptic curves over a local field May1 answered morphism between two elliptic curves over a local field Apr24 accepted Dimension of formal fiber Apr23 comment The notion of multiplicity in algebraic geometryWith 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 $\overline{\mu}_f(I)=1$. Apr23 comment Bound for the degree of the field of definition for a closed point of a varietyThere is something wrong with your TeX. The math doesn't show properly: specifically, the math italic alphabetic characters don't show at all. Apr22 comment Dimension of formal fiberStandard 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 $\mathbb{C}[[f_1]]=\mathbb{C}[[t]]$. Apr21 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. Apr20 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. Apr19 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. Apr19 comment Equivalent definitions of ample bundles@Serge: $\mathcal{O}_X$ is always globally generated! Apr16 answered Dimension of formal fiber Apr6 comment Finite extensions of residue fields of Henselian DVRsMaybe I misunderstand the question, but what if you start with an extension $L/F$ which is not simple and then take $K=F((t))$? Apr2 comment Is the image of discrete set under an open map discrete?@Nik: Oops, sorry. Apr2 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. Mar7 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. Mar6 accepted Open idempotents in modules over a local ring Mar6 revised Open idempotents in modules over a local ringadded 360 characters in body Mar6 revised Open idempotents in modules over a local ringadded 767 characters in body Mar6 answered Open idempotents in modules over a local ring Mar6 revised Which schemes can be presented as limits of smooth varieties?deleted 12 characters in body Mar6 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. Mar6 answered Which schemes can be presented as limits of smooth varieties? Mar4 accepted Artin approximation theorems over non-regular rings/non-Noetherian rings Feb28 answered Artin approximation theorems over non-regular rings/non-Noetherian rings Feb19 comment Zero-cohomology of birational varietiesAs you define it, $f_*$ depends on the choice of $U_1$. Feb18 comment Formal power series over a henselian ringMore 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. Feb16 comment Are f.g. projective modules free over total quotient ring of a reduced non-noetherian commutative ringThis ring is local, so projective=free. Feb11 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. Feb10 comment Complementation in an extension fieldIt 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. Feb8 comment on trivialisation of T-torsorsAnd what is $x$? Feb6 accepted Are noetherian schemes generically Jacobson? Feb6 revised Are noetherian schemes generically Jacobson?added 58 characters in body Feb5 answered Are noetherian schemes generically Jacobson? Feb5 accepted purity for finite flat group schemes Feb5 answered purity for finite flat group schemes Feb4 revised Are rational varieties simply connected?added 669 characters in body Feb4 awarded ● ag.algebraic-geometry