User peter toth - MathOverflow

Answer by Peter Toth for Good introductory references on algebraic stacks?

2011-10-29T05:51:55Z

You might take also a look at this: http://staff.science.uva.nl/~heinloth/SeminarStacks.html

especially the references and more especially the last two paper of them.

Answer by Peter Toth for Constructing invertible sheaves out of actions of finite groups

2011-04-13T05:30:29Z

This is classical, can be found in Mumford: Abelian varieties, II.7 (i think) group actions on varieties (however without the twist, you mention, but i think it is no poblem), and the main reason (again without the twist) is that $\pi^{*}\mathcal{F}\cong \mathcal{O}_{X}$ . You can use this to conclude...

Geometric abelian class field theory

2011-02-09T15:24:35Z

There is a very nice geometric proof of Deligne for the Artin Reciprocity in the geometric setting, namely for a smooth, projective, geometrically irreducible curve $C$ over a finite field $\mathbb{F}_{q}$ , with function field $K=k(C)$, and idele group $\mathbb{I}_{K}:=\prod^{'}_{p\in|C|}K^{*}_{p}$ there is a one-to-one correspondence between the finite quotients of the double quotient space $k(C)^{*}\backslash\mathbb{I}_{K}/\prod_{p \in |C|}\widehat{\mathcal{O}_{p}^{*}}$ (which is isomorphic to $Pic_{C}(\mathbb{F}_{q})$ ) and the finite quotients of $\pi^{ab}_{1}(C)$ .

Now on the other hand the Artin Reciprocity Law for function fields states (e.g. in Artin-Tate: Class field theory) that the group $\mathbb{I}^{0}_{K}/K^{*}$ of norm 1 idele classes is isomorphic via the Reciprocity map to $Gal(\bar{K}^{ab}/K\bar{k})$ .

My questions would be:

These two statements seem to me first as different statements, don´t they?
If we put aside Deligne´s geometric proof for the geometric statement (not seriously and not so for long :-)) then how could one prove the geometric statement using the "number theoretic" Reciprocity Law for function fields?

Stalks of etale sheaves

2011-03-09T16:30:11Z

Let $\pi:X \rightarrow Y$ be a finite morphism of schemes and $\mathfrak{F}$ be an etale sheaf on $X$ . Then for a $y \in Y$ we have the stalk $(\pi_{*}\mathfrak{F})_{y}=\prod_{\pi(x)=y}\mathfrak{F}_{x}^{d(x)}$ where $d(x)$ is the separable degree of the field extension $k(x)/k(y)$ (Corollary 3.5.(c) in Chapter II. in Milne: Etale cohomology). The proof in the book would be completely clear for me, if there were no separable degrees (and Milne does not mention either why are they there)....So my question is where from these separable degrees come into the picture?

Answer by Peter Toth for Picard group, Fundamental group, and deformation

2011-03-08T16:29:17Z

If your question concerns - as mentioned in one of your comments - if there is any relationship between them, then a very beautiful connection exists in what is called geometric class field theory:

namely classical number theoretic class field theory concentrates around what is called Artin Reciprocity, which establishes an isomorphism for a number field $K$ and its ring of integers $\mathcal{O}_{K}$ an isomorphism $Pic(Spec(\mathcal{O}_{K})) \cong \pi_{1}^{ab}(Spec(\mathcal{O}_{K}))$ between the Picard group and the abelianized etale fundamental group (it is a geometric reformulation of classical Artin reciprocity). We can see it as a special case of one-dimensional class field theory and the question arises naturally if we can extend somehow this correspondence for higher dimensions (and also for other one dimensional schemes). There are different approaches (K-theory, cycle theory, geometric Langlands) but the main cornerstones are the following:

Bloch-Kaito-Saito Theorem: Let $X$ be a regular, connected, projective scheme over $Spec(\mathbb{Z})$ , then there exists also a reciprocity map $Pic(X) \rightarrow \pi_{1}^{ab}(X)$ which is an isomorphism if in addition $X$ is flat over $Spec(\mathbb{Z})$ . If $X$ factors through a finite field $k=\mathbb{F}_{q}$ then the reciprocity map is injective and with cokernel $\widehat{\mathbb{Z}}/\mathbb{Z}$ .

Also for curves over finite fields there exists a correspondence, namely if $C$ is a smooth, projective, geometrically irreducible curve over a finite field $k$ , then there exists a reciprocity homomorphism $Pic_{C}(k) \rightarrow \pi_{1}^{ab}(C)$ which induces an isomorphism on the degree zero parts $Pic_{C}^{0}(k) \rightarrow \pi_{1}^{ab}(C)^{0}$ , where the degree maps are the obvious maps to $\mathbb{Z}$ and $\widehat{\mathbb{Z}}$ resp.

Also if $S \subset C$ is a finite set of points of a smooth, projective, geom. irreducible curve $C$ over a finite field, then there is a ramified version of the previous reciprocity, namely between $Pic_{C,S}$ (which is isomorphism classes of line bundles together with fixed isomorphisms of the stalks at every point in $S$ ) and the abelianization of the tame fundamental group of $U:=C \setminus S$ .

Some reference: http://epub.uni-regensburg.de/13979/1/MP92.pdf

and then it gives many other references and so on...

Group action on sheaves

2011-02-24T15:41:21Z

I am currently reading D. Mumford´s Abelian Varieties and it came up the following question: let $X$ be an algebraic variety over an algebraically closed field $k$ and $G$ a finite group acting on $X$ . Assume we are in a situation that there exists a quotient $(Y, \pi:X \rightarrow Y)$ , he then proves a proposition that there is a one-to-one correspondence between coherent $\mathcal{O}_{Y}$ -modules and $G$ -equivariant coherent $\mathcal{O}_{X}$ -modules. In the course of the proof he shows that for such a $G$ -sheaf $\mathfrak{F}$ on $X$ the natural morphism $\pi^{-1}((\pi_{*}\mathfrak{F})^{G}) \rightarrow \mathfrak{F}$ is an isomorphism.

What about other kind of sheaves, especially locally constant sheaves (regarding whether this natural morphism is an isomorphism)?

Answer by Peter Toth for geometric Langlands for GL(1)

2011-02-09T19:02:23Z

The main idea of the proof is the following: the 1-dimensional representations $\{\pi_{1}^{ab}(X) \rightarrow \bar{\mathbb{Q}}_{l}^{*}\}$ are in 1-1 correspondence with 1-dim local systems $L$ on $X$ , on the other hand 1-dimensional representations $\{Pic_{X} \rightarrow \bar{\mathbb{Q}}_{l}^{*}\}$ are in 1-1 correspondence with 1-dimensional local systems $A_{L}$ on $Pic_{X}$ together with a rigidification, i.e. a fixed isomorphism $A_{L}|_{0} \cong \bar{\mathbb{Q}}_{l}$ (this is the famous faisceaux-fonctions correspondence of Grothendieck). Now consider the d-symmetric product $X^{(d)}$ of $X$ (which is just the effective divisors of degree $d$ on $X$ ) which maps in an obvious way to $Pic_{X}^{d}$ , the degree $d$ component of the Picard. If there is given a local system $L$ on $X$ then we can produce a local system $L^{(d)}$ on $X^{(d)}$ by having this local system fibres $\bigotimes_{i} Sym^{d_{i}}(L_{x_{i}})$ over a point $\sum_{i} d_{i}x_{i}$ . Now by Riemann-Roch if the degree $d$ is greater than $2g(X)-2$ then it follows that this map has fibre over a degree- $d$ line bundle $\mathfrak{L}$ the $d-g(X)$ -dimensional projective space $\mathbb{P}(H^{0}(X,\mathfrak{L}))$ . As projective spaces are simply connected, it follows that this locally constant sheaf $L^{(d)}$ is actually constant on these fibres, so descends to a local system $A_{L}$ on $Pic_{X}^{d}$ . There is also a way to extend these construction to the remaining components using the natural action $X \times Pic \rightarrow Pic$ given by $(x,L) \mapsto L(x)$ . It is the idea of the proof, which is actually of Deligne!

As what the references concerns: there is a paper of Laumon: Faisceaux automorphes lies aux series Eisenstein, where he discusses this proof of Deligne. Also here http://www.cims.nyu.edu/~tschinke/publications.html the 3rd book (Mathematisches Institut, Seminars 2003/04, Universitätsverlag Göttingen, (2004) ) from page 145, but it is in german. And also the quoted paper of Frenkel is very good.

Comment by Peter Toth

2011-05-19T04:58:27Z

did you check the book, McDuff-Solomon: J-holomorphic curves and symplectic topology (or quantum cohomology...it is a shorter book)?

Comment by Peter Toth

2011-04-13T05:48:07Z

i mean $\mathcal{O}_{X}$ can be substituted by any coherent $\mathcal{O}_{X}$ module...

Comment by Peter Toth

2011-04-13T05:44:27Z

yes it is trivial without the twist, but i forgot to say that $\mathcal{F}$ can be any quasi-coherent $\mathcal{O}_{X}$ module...

Comment by Peter Toth

2011-03-13T17:47:21Z

yes, thanks....

Comment by Peter Toth

2011-03-08T15:50:15Z

Over what kind of field and which fundamental group? Also affine and projective spaces are not simply connected over arbitrary fields when using the etale fundamental group...

Comment by Peter Toth

2011-02-28T17:45:25Z

to André: if $char(k)=0$ then $\pi_{1}(\mathbb{A}_{k}^{1}=\pi_{1}(Spec(k))$. If $char(k)=p>0$ then the question is more delicate. Anyway it still exists a map $\pi_{1}(\mathbb{A}_{k}^{1} \rightarrow \pi_{1}(Spec(k))$ which is surjective, but not injective. So for example if the field k is of non-zero characteristic and separably closed, then $\pi_{1}(\mathbb{A}_{k}^{1}$ is non-trivial.

Comment by Peter Toth

2011-02-25T09:14:55Z

thanks a lot. this was my main concern even if a did not write it exactly...

Comment by Peter Toth

2011-02-25T09:11:38Z

thanks for the answer

Comment by Peter Toth

2011-02-24T17:13:05Z

hi....thanks for the very wide range answer (you seem to know also my hidden intentions also:-)...what i want to do is to prove local constancy of the G-invariants of the push forward on the quotient by proving that there is an isomorphism written in my question.

Comment by Peter Toth

2011-02-10T14:57:22Z

say representations into some alg. closed l-adic field of the Jac(k)? Anyway my question arised because i wanted to somehow relate the proofs regarding generalized Jacobians in Serre´s book (Alg. groups and class fields) to some possible adaptation of Deligne´s proof for geometric class field theory to the ramified case....

Comment by Peter Toth

2011-02-10T11:54:03Z

let´s say we make no restrictions and J^{'} is an algebraic group.

Comment by Peter Toth

2011-02-10T09:01:34Z

ok, so as i understand for a modulus m we can define the generalized jacobian J_m (which in my notation is basically the Pic^0_C,m and then every abelian, etale covering of U=C-m comes from an isogeny over J_m. So my question transforms to: how can one relate these isogenies to 1-dimensional representations?

Comment by Peter Toth

2011-02-10T06:16:20Z

yes, i am trying now to prove the ramified case, namely for a finite subset S of C i consider 1-dimensional representations of Pic_C,S:= isomorphism classes of line bundles on C with fixed isomorphisms of the stalks at the points in S. On the other hand there is the 1-dimensional representations of the tame fundamental group of the open U:=C-S. But i do not understand, how could i then prove with it the number theoretic Artin Reciprocity? (I think this you mean by the other way...so how does it go?)