User - MathOverflow

What are the worst notations, in your opinion ?

2010-03-18T14:49:17Z

With which notation do you feel uncomfortable ?

Properties stable under base change in algebraic geometry

2010-04-08T15:56:23Z

I remember to have seen a big list in the EGA of properties $(P)$ such that: if $f : X \to Y$ has $(P)$ then, $f_{(S')} : X_{(S')}\to Y_{(S')}$ has $(P)$, where $f_{(S')}$ is the morphism $f$ after a base change $S'\to S$., etc. but I can't find it now...

Does anyone know where I can find such a list ?

(I am interested by $(P)$ = "to be a closed map")

Noether's normalization lemma over a ring A

2010-10-15T11:41:38Z

Given a field $k$ and a finitely generated $k$-algebra $R$ without zero divisors, one knows that there exist $x_1, \ldots, x_n$ algebraically independent such that $R$ is integral over $k[x_1, \ldots, x_n]$.

Does one have a similar statement, under good assumptions, if $k$ is not a field but a ring ? In this discussion, I am also interested by geometric explanations.

Answer by nicojo for What should I read before reading about Arakelov theory?

2011-02-07T07:26:06Z

You can try to read this wonderful paper by Burgos. It is in Catalan, though. If you are familiar with Spanish or French, you can manage to read it.

Non finitely-generated subalgebra of a finitely-generated algebra

2010-12-09T18:05:26Z

Ok, I feel a little bit ashamed by my question.

This afternoon in the train, I looked for a counter-example:
— k a field
— A a finitely generated k-algebra
— B a sub-k-algebra of A that is not finitely generated

Finally, I have found this:
— k any field
— A=k[x,y]
— B=k[xy, x.y^2, x.y^3, ...]

(proof : exercise)

My questions are :

1) What is your usual counter-example ?
2) Under which conditions can we conclude that B is f.g. ?
3) How would you interpret geometrically this counter-example ?

Choosing the algebraic independent elements in Noether's normalization lemma

2010-10-15T11:39:55Z

More precisely, if $a\in R$ is transcendent over $k$, can one find $x_2, \ldots x_n$ such that $R$ is integral over $k[a,x_2 \ldots, x_n]$ ? If it is false, can one get the conclusion under stronger assumptions ? In this discussion, I am also interested by geometric explanations.

Why the roots of unity are the analogs of constants ?

2010-06-14T13:55:23Z

Hello,

Joel Dogde, in a comment on his question "Roots of unity in different completions of a number field", says the following, about the analogy between number fields and function fields :

Number of roots of unity in number fields is something like the size of the constant field for function fields.

Could anyone explain that ?

Thanks.

Is there a reason for defining the differential forms before the vector fields ?

2010-06-03T18:34:22Z

Hi, my question is the following :

In EGA IV chapter 16, given $X$ a scheme over $S$, Grothendieck defines first $\Omega^1_{X/S}$, the $O_{X}$-module of the 1-differentials. He then defines the tangent sheaf : $T_{X/S}:= Hom_{O_X} (\Omega^1_{X/S}, O_X)$, which is equal to $Der(O_X, O_X)$. Why, one does not do the opposite and first define $T_{X/S}:=Der(O_X, O_X)$ and then $\Omega^1_{X/S}:=Hom_{O_X} (T_{X/S}, O_X)$ ?

I suspect there are several reasons for this :

1) This new object, whose definition seems to be easier, is maybe less handy to work with.

2) In some important cases, it gives the wrong object

3) Some other philosophical reason

I would like to have the opinion on this question of mathematicians who know more than me geometry and differential forms.

Thanks.

Answer by nicojo for Different ways of thinking about the derivative

2010-05-18T00:27:09Z

If $R$ is a ring, then a derivation $\partial : R \to R$ is a vector field over the scheme $\mathrm{Spec} R$.

Are two sheaves that are locally isomorphic globally isomorphic ?

2010-05-12T10:21:48Z

Let $X$ be a topological space and let $\mathcal{F}$ and $\mathcal{G}$ be two sheaves over $X$.

Of course, if one has a morphism $f : \mathcal{F} \to \mathcal{G}$ such that for all $x\in X$, $f_x : \mathcal{F}_x \to \mathcal{G}_x$ is an isomorphism, then it is known that $f$ itself is an isomorphism.

My question is the following: if we don't have such a morphism $f$, but if we know that for all $x\in X$, $\mathcal{F}_x$ and $\mathcal{G}_x$ are isomorphic, is it true that $\mathcal{F}$ and $\mathcal{G}$ are isomorphic ?

Is a functor which has a left adjoint which is also its right adjoint an equivalence ?

2009-12-05T22:27:50Z

I am looking for a counter-example of two functors F : C -> D and G : D->C such that

1) F is left adjoint to G

2) F is right adjoint to G

3) F is not an equivalence (ie F is not a quasi-inverse of G)

Is there a relation between the tangent bundle of a scheme and the tangent bundle of the associated reduced scheme ?

2010-04-16T14:02:20Z

My question is vague and general, and, if you want, naive.

$\bullet$ Given $X/S$ a scheme. We denote by $X_{red}$ the reduced scheme associated to $X$.

Grothendieck defines in EGA 4 (16.5.12.1) the tangent bundle $T_{X/S}$ of $X$ relatively to $S$. Is there any relation between $T_{X_{red}/S}$ and $T_{X/S}$ ? More precisely, we know that there is a morphism $$ f : T_{X_{red}/S} \to T_{X/S}\times_{X} X_{red} ; $$ what can be said of $f$ ? I am interested, in particular, in the case when $S=Spec \ k$, with $k$ a field.

$\bullet$ A related question (but in which I am less interested) is the comparison of the Zariski tangent space at one point (as defined in Hartshorne p. 80) of $X$ and $X_{red}$. In general, if $\pi : X_{red} \to X$ denotes the canonical morphism, and if $x\in X_{red}$, the linear map $$ T_x \ \pi : T_x \ X_{red} \to T_{\pi(x)} \ X $$ is injective, since $\pi$ is a closed immersion. Is it an isomorphism ?

$\bullet$ Finally, a related question is the comparison of $\Omega^1_{X/S}$ and $\Omega^1_{X_{red}/S}$.

$\bullet$ Ideally, I would like to transport sections of $T_{X_{red}/S}$ to sections of $T_{X/S}$.

Topological limits as categorical limits

2010-04-01T12:28:57Z

Possible Duplicate:
limits in category theory and analysis

Let $X$ be topological space, $(u_n)$ a sequence converging to $l$.

Can one find a categorical setting such that $l$ appears to be the limit a of certain functor ?

Ie, can one find a category $C$, a functor $F : \mathbb{N}\to C$ that represents the sequence $(u_n)$ and such that $\lim F=l$ (or something like that) ?

Answer by nicojo for What are the worst notations, in your opinion ?

2010-03-18T15:07:40Z

I don't like (but maybe for a bad reason) the notation $F\vdash G$ for $F$ is left adjoint to $G$.

Any comment ?

Cohomology of sheaves for sites and Galois cohomology

2010-03-01T15:18:04Z

Hello,

I am looking for a reference (if it exists) that makes the link between cohomology of sheaves for sites and Galois cohomology :

quickly said, I would like to see Galois cohomology (at least in the commutative case) as the cohomology of a sheaf over the étale site of extensions of k.

By the way, what is a reference for cohomology of sites ?

Thanks

A good place where to learn about derived functors

2010-02-04T14:34:01Z

I would like to learn about derived functors. Which reference do you advise ?

Are coproducts and patchings filtered colimits ?

2009-12-14T15:25:01Z

1) Is the coproduct $\coprod_{i\in I} X_i$ a filtered colimit ?

2) Is the colimit colim($X => Y$) (two arrows from X to Y) a filtered colimit ?

Definition of étale for rings

2009-12-10T13:05:03Z

Let $A \to B$ be a ring extension.

What is the definition of $B/A$ étale ?

When $A$ is a field, do we get a nice characterization ?

A technical question about derivations of sheaves on group schemes

2009-12-09T17:51:40Z

Let $G$ be a group scheme (for instance, over $k$ a field of characteristic 0). Let $e$ be its unit. I denote by $O_G$ the structural sheaf of $G$.

Let $D_e : O_{G,e} \to k$ a derivation.

I would like to get directly (ie, without any consideration about the cotangent bundle, or some canonical isomorphisms...) a derivation $D : O_G\to O_G$ that extends $D_e$, and which is compatible with the action of $G$. That is, I would like to get this derivation by the mean of the multiplication map : $m : G \times G \to G$, etc., etc.

I have guessed this question would not be difficult, and would only be a matter of technics, but I can't manage to do it.

Answer by nicojo for Is a functor which has a left adjoint which is also its right adjoint an equivalence ?

2009-12-05T23:36:25Z

The answer of Ben Webster, can be made easier. Consider the functor F : (A-mod) -> (A-mod) which maps any A-module on (0). Then, F is a left adjoint to F ; and so, is a also a right adjoint to F. This is clear because for all A-modules N, M, one has Hom_A(0,N)=Hom_A(M,0). But, F is not an equivalence.

Comment by

2011-03-03T08:38:02Z

Wow ! It is crazy how things turned about this question (see the above comments) !! which is, in my opinion, a very interesting question !!

Comment by

2011-01-24T06:01:45Z

@Mariano : i don't agree with you. There is a point in having two different lists, one with the good points, another one with the bad ones.

Comment by

2011-01-21T05:53:05Z

@Mariano ; what is the point of saving a question ?

Comment by

2010-12-09T19:53:13Z

I think that your description corresponds actually to B=k[x, xy, x.y^2, x.y^3, ...].

Comment by

2010-06-14T14:59:23Z

What is Malle conjecture ?

Comment by

2010-06-03T20:07:14Z

Yes, but, in the case $X/S$ not smooth, is $\Omega^1_{X/S}$ useful ?

Comment by

2010-05-19T09:06:42Z

This is true !! More precisely, if X=Spec A and if T is the tangent sheaf of X (over the base Z), as defined in EGA 4, then a global section of T is the same as a derivation of A.

Comment by

2010-05-05T10:53:50Z

The interview goes on like that : "Grothendieck provided a partial solution in terms of universes but a revolution of the foundations similar to what Cauchy and Weierstrass did for analysis is still to arrive."

Comment by

2010-04-16T14:47:34Z

Yes, I meant injective ! I've changed it in the question. Hence, with your example, we see that $T_x \ \pi$ is not surjective in general.

Comment by

2010-04-08T16:54:38Z

OK, sorry... Indeed, Poonen's notes contains references to EGA.

Comment by

2010-04-08T16:12:12Z

Thanks for your comment Emerton. Fortunately, in my case, $f$ is a closed immersion !

Comment by

2010-04-08T16:11:08Z

Thanks for this reference ! However, a reference to EGA should be better for me.

Comment by

2010-04-01T12:53:39Z

Yes, thank you for pointing it to me !

Comment by

2010-02-04T15:42:57Z

I don't really need them. But I feel like I need to know more about cohomology. I know a little bit of singular/simplicial/de Rham/Cech cohomologies. If often hear (in seminars) of the R^i f, and I would like to know who they are. Thanks for all for the answers. I was thinking of the Hartshorne, I know the Schapira-Kashiwara and I like it for it preciseness. I was more looking for some typed lectures than for a book. Thanks again.

Comment by

2009-12-23T15:01:53Z

You should add the reference to the previous question. Thanks.