User kiseki - MathOverflow

pro-$\ell$ etale fundamental group of a semi-abelian variety

2013-05-11T13:36:58Z

Let $A$ be a semi-abelian variety over $K$, $\ell$ a prime number which is not equal to char($K$).

Does the abelianization of geometrically pro-$\ell$ etale fundamental group $(\pi_{1}(A\otimes\overline K)^{\ell})^{ab}$ isomorphic to the $\ell$-adic Tate module of $A$?

morphism between two elliptic curves over a local field

2013-05-01T07:04:27Z

Let $X,Y$ be two elliptic curves over $K:=K(R)$ which have good models over $R$, Char($K$)=$0$, where $R$ complete DVR with algebraically residue field $k$.

If $L$ is a finite extension of $K$, such that $X \otimes L \cong Y \otimes L$.

My question is:

Is there $X \cong Y$ ?

morphism from adic spaces to schemes

2013-04-29T04:01:35Z

Let $X:=Spa A$ be an affinoid adic space, and $\underline X $ the ringed space of $X$. Let $Y:=Spec B$ be an affine scheme, $f: \underline X \longrightarrow Y$ a morphism of ringed spaces.

How to define a morphism of rings $g: B \longrightarrow A$ such that $f$ is induced by $g$?

trivial deformation of a smooth affine scheme over complete DVR

2013-04-06T07:13:18Z

Let $X$ be a affine smooth scheme finite type over $A/pA$, where $A$ a complete DVR and $chk=p>0$.

I know that since $H^{2}=0=H^{1}$, we have a unique lifting to $A/p^{2}$. In algebraic schemes case, we can obtain a trivial deformation by fiber product. But in this case, for example, $A$ is a p-adic number ring, What is the trivial deformation over $A/p^{2}$ ?

PS: I added some smooth conditions to $X$.

questions of localization of topos

2013-03-30T09:46:52Z

Let $T$ be a topos, and $F \in T$, $T/F$ a localization of $T$. So we have a natural morphism $i: T/F \longrightarrow T$.

My questions are:

1.What are the definitions of $i_{\ast}$ and $i^{\ast}$ without using site ?

2.Dose there exist a site $S$ such that $Sh(S) \cong T$ and there is an $U \in S$ which $U^{\sim} \cong F$ ?

morphism of injective objects

2013-03-28T12:57:50Z

Let $A,B$ be two bounded below complexes in module category, and $A \longrightarrow I$ (resp. $B \longrightarrow J$) a injective resolution. If $f: A \longrightarrow B$ is a morphism of complexes.

My question is: how to construct a morphism from $I$ to $J$ which induced by $f$?

question of topos and site

2013-03-28T07:12:42Z

Let $T, P$ be two topoi, and $f:T \longrightarrow P$.

Does there exist two site $S_{T}, S_{P}$ and a morphism $g: S_{T} \longrightarrow S_{P}$ such that $f$ is induced by $g$ ?

How to see the geometry and arithmetic of tannakian fundamental groups?

2013-03-07T11:05:54Z

The etale fundamental group is an inverse limit of automorphism groups of finite etale coverings. We can see the geometry of etale fundamental group very well from etale coverings just like topologically fundamental group. But the tannakian fundamental groups are defined as the automorphism of a fiber functor of a tensor category, this is definition abstract for me.

I want ask that how to see the geometry of tannakian fundamental groups?

What is the relation between first etale cohomology and pro-unipotent fundamental group?

I would like to extend my question. If we consider a hyperbolic curve $X$ over a local field $K$ with valuation ring $R$, there is an natural Galois action (i.e., outer Galois action) of $G_{K}$ on the etale fundamental group $\pi_{1}(X)$. From this outer Galois action, we can understand some geometry of $X$ and their reduction. For example, there is a good reduction criterion in terms of pro-$l$ fundamental groups (Oda, Tamagawa).

My question is: Dose there exist some similar Galois actions or criterions for Tannakian fundamental groups? or dose there exist some anabelian type theorems for Tannakian fundamental groups?

Formal power series over a henselian ring

2013-02-18T12:48:25Z

Let $R$ be a henselian ring. Is $R[[x]]$ also a henselian ring?

relations between log schemes and toric varieties

2013-02-16T13:37:55Z

The theory of regular log scheme, roughly specking is that scheme theory with singularities like those of toric varieties.

Dose anyone could explain this idea in detail or give a reference about the relations between log schemes and toric varieties?

deformation of stable curve

2013-01-24T09:15:14Z

Let $R$ be a DVR, and $k$ residue field of $R$. We suppose that $X_{0}$ is a stable curve over Spec$k$.

Dose there exist a stable model $X$ over $R$ such that the special fiber isomorphic to $X_{0}$ ?

If we assume $R=C[[t]]$, where C is complex number field, how to find a deformation which make the generic fiber is smooth?

Galois action on special fiber of a stable model

2013-01-27T11:56:52Z

Let $X_{K}$ be a curve over a complete DVR $R$, $R/m:=k$ an algebraically closed field. We suppose the minimal field extension $L$ of $K$ such that $X_{L}$ has stable model $X_{R_{L}}$, and the special fiber is $X_{k}$. We obtain an action of $Gal(L/K)$ on $X_{k}$.

My question is:

Does $Gal(L/K) \longrightarrow Aut(X_{k})$ is an injection ?

Tannakian fundamental group for finitely linear representation of group

2012-12-25T02:49:00Z

Let $G$ be an arbitrary group and $k$ a field. Denote by $Rep_{k}(G)$ the category of finite dimensional representations of $G$ over $k$. The usual tensor product and dual operations for representations equip $Rep_{k}(G)$ with the structure of a $k$-linear rigid abelian tensor category. The forgetful functor $Rep_{k}(G) \longrightarrow Vec_{k}$ as fiber functor. So we get a neutral Tannakian category.

My question is :

If we consider the $G$ as constant affine group scheme over $k$, does the Tannakian fundamental group isomorphic to $G$?

An example of almost etale extension

2012-11-29T08:05:53Z

In the paper of Faltings' "p-adic Hodge theory", Faltings showed an example of almost etale extension before he proved the almost purity theorem. The example is following:

Let $k$ be a perfect field of characteristic $p$ and $W(k)$ the ring of Witt vectors, $W(k)(x_{1},...,x_{d})$ a localization of $W(k)[x_{1},...,x_{d}]$ at $p$. Let $V$ be the completion of $W(k)(x_{1},...,x_{2})$ with function field $K:=K(V)$, and $V_{n}$ be the extension of $V$ which generated by the $p^{n+1}$-th roots of unite together with $p^{n}$-th roots of $x_{i}$. If $W$ is a normalization of $V$ in another finite extension $L$ of $K$, and $W_{n}$ the composite of $W$ and $V_{n}$, and $p^{a_{n}}W_{n}$ the different of $W_{n}$ over $V_{n}$. If we use the notation $W_{\infty}$ (resp. $V_{\infty}$) to denote the union of $W_{n}$ (resp.$V_{n}$).

The Theorem 1.2 of Faltings' paper tell us $a_{n}$ converge to $0$ for $n \rightarrow \infty$, and then $V_{\infty} \longrightarrow W_{\infty}$ is almost etale extension.

My question is:

Why the almost etale extension of $W_{\infty}/V_{\infty}$ is deduced by theorem 1.2 ? Is this checked by the definition of almost etale ?

A question of line bundle for finite etale covering

2012-11-26T11:45:37Z

Hi, everyone, I want to ask a question about line bundle.

Let $X$ be a smooth curve over a algebraically closed field $k$, and $f:Y \longrightarrow X$ a Galois finite etale covering with Galois group $G$ and degree $n$. Suppose that $L$ is a line bundle on $X$.

Dose there exist a line bundle $M$ on $Y$ such that $M^{\otimes n}=f^{*}L$?

Thanks.

Does any connected formal $k$-group scheme is formally smooth?

2012-10-25T10:35:42Z

Let $G$ be a connected p-divisible group over a field $k$, then $G$ is a formally smooth group scheme over $k$, since the category of formal lie groups and the category of connected p-divisible groups over a complete noetherian local ring are equivalence.

My question is following:

Does any connected formal $k$-group scheme is formally smooth?

reduction of elliptic curves

2012-09-10T07:31:22Z

Let $X$ be an elliptic curve over a complete local field.

The definition of semi-abelian reduction is: "the Neron model of $X$ is a semi-abelian scheme". On the other hand, the definition of semi-stable reduction is: "the minimal regular model of $X$ is semi-stable."

For elliptic curves, are the two definitions equivalent? How to prove it?

relationship between etale cohomology group of vanishing cycle and irreducible components

2012-09-05T12:54:23Z

Hi everyone, I want to ask a question about vanishing cycle.

Let $X$ be a stable curve over a complete DVR $A$ with residue field $k=\overline k$, and $X_{k}$ the special fiber. Suppose that $X_{i}$ are irreducible components of $X_{k}$, $S_{i}:=X_{i} \cap (\cup_{j, j \neq i} X_{j})$

If we denote $X_{i}-${nodes, $S_{i}$} by $U_{i}$. My question is following.

What's the relationship between $H^{1}(X_{k},R\Phi(Q_{\ell})$) and $H^{1}(U_{i},Q_{\ell})$

Curves whose stable reductions do not contain rational curves

2012-07-11T13:19:33Z

Let $X$ be a smooth projective curve over $K:=K(A)$. $A$ is a strict henselian ring, $A/m=k=\bar k$. Suppose $\cal X$ is a stable model of $X$, ${\cal X}_{s}$ is the special fiber.

My question is:

what are the conditions on $X$ so that $\cal X_{s}$ does not contain any rational curves?

A problem of graph theory

2012-07-11T09:47:03Z

Hi, everyone, I want to ask a question about graph theory.

Let $G$ be a finite graph, and $E$ the set of edges of $G$. For each vertex $a$, we denote that $E_{a}$ by the edges of $G$ which pass to $a$. Suppose for any $a \in G$, $\sharp E_{a} >3$. Let $f: E \longrightarrow$ {$1,2$}.

Does there exists a map $f$ such that for each $a$, $\sharp (f^{-1}(1) \cap E_{a})$ is a nonzero even number.

Thank you very much!

References about the Grothendieck's way of algebraizing the notions of calculus and differential geometry

2012-06-27T13:02:50Z

Hi everyone,

I'm looking for some references about the differential operators on schemes(connection, curvature, etc...). I am reading the EGA IV 16, but EGA does not treats connection, curvature, etc....

Are there any articles/books that deal with the the Grothendieck's way of algebraizing the notions of calculus and differential geometry?

Thank you very much!

roadmap for studying arithmetic geometry

2010-04-16T11:01:00Z

hi everybody, I have already finished the Hartshorne's algebraic geometry from chapter 1 to chapter 4, so I'd like to find some suggestions about the next step to study arithmetic geometry. I want to know that how to use the scheme theory and their cohomology to solove the arithmeic problem.Would you like to recommend me some of these kind of books and papers?

Thank you very much!

PS: I also want to learn some materials about moduli theory, if you like, could you recommend me some books or papers ?

English version of Manin's "Correspondences, motifs and monoidal transformations "

2012-06-25T04:10:06Z

Hi everyone, I want to find a classical paper "Correspondences, motifs and monoidal transformations " by Manin. But I only found the Russian version. Dose this paper exist a english version? and where I could get it ?

Thank you very much!

good references in moduli stack and stable reduction

2011-08-02T12:58:57Z

Hello,everyone. I'm looking for some good references in moduli stack and stable reduction, so I ask here for some advice.

I knew the famous paper of Deligne-Mumford, but this paper is hard for me now, I will very happy if someone could give me some suggestions for reading Deligne-Mumford, or tell me some good references about moduli-stack and stable reduction of curves or abelian varieties.

THANK YOU VERY MUCH!

strict henselian and excellent henselian

2012-06-04T12:54:51Z

Hello, everyone. I want to ask a problem about strict henselian ring.

Let $A$ be a strict henselian DVR.

Dose there exist subrings $A_{i}$ of $A$, such that $A=lim_{i} A_{i}$ and where $A_{i}$ are excellent strict henselian DVR.

the algebraic closure of strict henselian DVR

2012-05-20T05:48:37Z

Let $A$ be a strict henselian DVR, and $\hat A$ is completion of $A$,

is $K(A)^{alg} \longrightarrow {K(\hat A)}^{alg}$ a isomorphism?

where $K(A)$ and $K(\hat A)$ are quotient fields.

Is strict Henselian ring a excellent ring?

2012-05-18T23:48:12Z

Hi, everyone, I want to ask following problem:

Is strict Henselian ring a excellent ring?

If not, could you give me a example?

generic etale morphism from curve to projective line

2012-05-01T11:02:35Z

Let $X$ be a smooth projective curve over $k$, ch$k=p>0$, dose there exist a generic etale morphism from $X$ to projective line ?

why find a good field extension such that the curve has semi-stable model is important?

2012-04-24T06:19:17Z

Hello everyone,

I'd like to ask some question about semi-stable reduction of curves.

The Deligne-Mumford theorem tell us "Let $A$ be an Dedekind domain, $K=K(A)$, for any smooth curve $X$ over $K$ and $g(X)>1$, there exist a separable extension $L$ of $K$, such that $X_{L}$ has semi-stable model."

But this theorem does not tell us any information about how to find the extension field $L$, but it seems is very important for arithmetic geometers.

My question is, in arithmetic geometry, why find the good extension field $L$ (for example:tame extension) such that $X_{L}$ has semi-stable model is so important?

why we need rigid geometry?

2012-01-07T08:34:34Z

Hello, everyone.

I want to ask some questions about rigid geometry.

1.what is the motivation of rigid geometry?

2.what is the applications of rigid geometry for solving arithmetic problems, especially for studying the fundamental groups of algebraic curves? what the beautiful theorems which were first proved by rigid geometry method?

thank you very much

Comment by kiseki

2013-04-29T13:26:39Z

@Jérôme Poineau: the global section of $SpaA$ is $\hat A$, so there is $B \longrightarrow \hat A$.

Comment by kiseki

2013-03-30T10:16:54Z

Simon, Thank you so much !

Comment by kiseki

2013-03-07T15:16:45Z

@Lars: I want to say pro-unipotent fundamental group.

Comment by kiseki

2013-01-29T01:16:02Z

@pranavk: If we assume $R=C[[t]]$, where $C$ is complex number field, how to find a deformation which make the generic fiber is smooth?

Comment by kiseki

2013-01-27T12:08:42Z

@ayanta: If we assume that $R$ is a complete DVR, is this deformation unique ?

Comment by kiseki

2012-09-11T09:57:10Z

Thank you very much, Pof.Liu. I have found the result in your book.

Comment by kiseki

2012-07-11T10:55:11Z

mathoverflow.net/questions/98385 showed a conterexamples for my problem. But if the dual graph of a stable curves is the graph of mathoverflow.net/questions/98385, we could construct a abelian covering of the stable curve directly. So maybe we should assume for any vertex $a$ such that ♯Ea>3

Comment by kiseki

2012-07-11T10:20:11Z

Thanks for your comment, darij. In fact, this problem arises from arithmetic geometry, I want to construct a special abelian covering of a stable curve which irreducible components are all projective line.

Comment by kiseki

2012-06-27T14:57:04Z

Thanks so much Lars!

Comment by kiseki

2012-06-25T12:36:02Z

Thanks! Thomas.

Comment by kiseki

2012-06-06T09:14:57Z

sorry, I modified the problem.

Comment by kiseki

2012-05-19T10:37:06Z

Thank you for answering!

Comment by kiseki

2012-05-01T13:14:41Z

Thank you for answering !

Comment by kiseki

2012-03-14T00:44:53Z

Thank you for your answer.

Comment by kiseki

2012-03-13T14:49:34Z

Sorry Dan, could you please explain in detial? If there is a finite morphism from the irreducible component which containing p, say $X_i$, to projective line, this morpism will not be flat since $X_i$ maybe singluar, in this case why this morphism is a isomorphim? On the other hand, the stable curve maybe has a component which isomorphic to projective line.