User lars - MathOverflow

Answer by Lars for Albanese dual to the Picard scheme

2013-04-17T12:04:44Z

The nicest modern reference for the theory of the Albanese that I know of is the appendix to this article of S. Mochizuki.

Fundamental groups of topoi

2010-05-04T13:57:34Z

Just yesterday I heard of the notion of a fundamental group of a topos, so I looked it up on the nLab, where the following nice definition is given:

If $T$ is a Grothendieck topos arising as category of sheaves on a site $X$, then there is the notion of locally constant, locally finite objects in $T$ (which I presume just means that there is a cover $(U_i)$ in $X$ such that each restriction to $U_i$ is constant and finite). If $C$ is the subcategory of $T$ consisting of all the locally constant, locally finite objects of $T$, and if $F:C\rightarrow FinSets$ is a functor ("fiber functor"), satisfying certain unnamed properties which should imply prorepresentability, then one defines $\pi_1(T,F)=Aut(F)$.

Now, if $X_{et}$ is the small étale site of a connected scheme $X$, then it is well known the category of locally constant, locally finite sheaves on $X$ is equivalent to the category of finite étale coverings of $X$, and with the appropriate notion of fiber functor it surely follows that the étale fundamental group and the fundamental group of the topos on $X_{et}$ coincide.

Similarly, as the nlab entry mentions, if $X$ is a nice topological space, locally finite, locally constant sheaves correspond to finite covering spaces (via the "éspace étalé"), and we should recover the profinite completion of the usual topological fundamental group.

Before I come to my main question: Did I manage to summarize this correctly, or is there something wrong with the above?

My question:

Has the fundamental group of other topoi been studied, and in what context or disguise might we already know them? For example, what is known about the fundamental group of the category of fppf sheaves over a scheme $X$?

Answer by Lars for About "de-Rham" and "l-adic" local systems - comparison

2013-03-22T22:00:28Z

I cannot say much about the $\ell$-adic side. I will give "classical" answers to 1)-3):As you know, the Riemann-Hilbert correspondence says that on a smooth complex variety $X$ the category of $A$ of vector bundles with flat regular singular connection is equivalent to the category of representations of the topological fundamental group $\pi_1^{\text{top}}$ on finite dimensional complex vector spaces. Lets write this category $\operatorname{Repf}_{\mathbb{C}} \pi_1^{\text{top}}(X)$ (neglecting base points).

Since $\pi_1^{\operatorname{et}}(X)$ is the profinite completion of the abstract group $\pi_1^{\operatorname{top}}(X)$, a representation of $\pi_1^{\operatorname{top}}(X)\rightarrow GL(V)$ which factors through a finite quotient can be thought of a representation of $\pi_1^{\operatorname{et}}(X)\rightarrow GL(V)$ which is continuous with respect to the profinite topology on the left and the discrete topology on the right. Hence, given an etale covering of $f:Y\rightarrow X$, Galois theory associates with it a finite $\pi^{\operatorname{et}}(X)$-set, which we can linearize to and get a representation and then a $\mathcal{D}$-module. But what does this mean concretely? It is not difficult to check that ${f_*}\mathcal{O}_Y$ is a $\mathcal{O}_{X}$-coherent $\mathcal{D}_X$-module (hence a vector bundle), and it is a theorem that it is regular singular (Gauss-Manin).

About your third question: The pro-algebraic affine group scheme associated with the Tannaka category $\operatorname{Repf}_{\mathbb{C}} \pi_1^{\operatorname{top}}(X)$ is by definition the pro-algebraic completion of the finitely generated group $\pi_1^{\operatorname{top}}(X)$. The etale fundamental group is the profinite completion of this group. And amazingly, the profinite completion "controls" the pro-algebraic completion:

Theorem: Let $f:G\rightarrow H$ be a morphism of finitely generated (abstract) groups. Then $f$ induces an isomorphism on pro-algebraic completions if and only if it induces an isomorphism on profinite completions.

I am told that this was first discovered by Malcev, and then independently rediscovered by Grothendieck. Grothendieck precisely had the application the the Riemann-Hilbert correspondence in mind.

See: Grothendieck, Alexander Représentations linéaires et compactification profinie des groupes discrets. (French. English summary) Manuscripta Math. 2 1970 375–396.

Answer by Lars for A submodule of a constant D-module is constant

2013-03-04T17:07:48Z

If your base field $k$ is algebraically closed (or if $X$ has a rational point), then this follows directly from the fact that the category of $\mathcal{O}_X$-coherent flat connections is neutral Tannakian, i.e. equivalent to the category of finite dimensional $k$-representations of some affine $k$-group scheme $G$. For this fact the Riemann-Hilbert correspondence is not needed, it just gives you additional information about $G$.

Here is a more direct argument: If $E$ is a $D$-module which is coherent as an $\mathcal{O}_X$-module, then $E$ is automatically locally free. In particular, if $E$ is a sub $D$-module of the "constant" $D$-module $\mathcal{O}_X^n$, then the short exact sequence of $D$-modules

$$0\rightarrow E \rightarrow \mathcal{O}^n_X \rightarrow \mathcal{O}^n_X/E\rightarrow 0$$ is locally split as a sequence of $\mathcal{O}_X$-modules.

We see that if $n=1$, then $E=0$ or $E=\mathcal{O}_X$. We proceed by induction.

Since a $D$-module is trivial if and only if there exists a dense open subset of $X$ on which it is trivial, we may work in the local ring of a closed point $x\in X$ (lets assume $X$ to be connected…).

If $e_1,\ldots, e_n$ is a horizontal basis, then the horizontal sections of $\mathcal{O}_X^n$ are precisely those sections which are in the $k$-span of $e_1,\ldots, e_n$. The group of $D$-automorphisms of $\mathcal{O}_X^n$ then identifies with $GL_n(k)$. Thus if $E$ contains a horizontal section, we are done by induction. The case $n=1$ implies that if $E\cap e_i\mathcal{O}_X\neq 0$ for some $i$, then $e_i\in E$. If $f_1e_1+\ldots+f_n e_n$ is a section of $E$, which is not horizontal, then some $f_i\in \mathcal{O}_{X,x}$ is nonconstant, say $f_1$. One finds a differential operator $\partial$, such that $\partial(f_1)\in \mathcal{O}_{X,x}^\times$, and concludes that $E$ intersects the $\mathcal{O}_X$-span of $e_2,\ldots, e_n$ nontrivially. Hence, either $e_1\in E$, or $E\subset \bigoplus_{i\geq 2} e_i\mathcal{O}_X$, and in both cases we are done.

Answer by Lars for What conjectures in anabelian geometry are false?

2012-11-10T22:17:06Z

Mochizuki proved that the anabelian conjecture for hyperbolic curves (the hom-form) only needs the maximal pro-$p$ quotient of the fundamental groups, for some prime $p$, which gives a stronger result than originally envisioned by Grothendieck.

Y. Hoshi (a student of Mochizuki) proved that an analogous generalization of the section conjecture is false:

Existence of nongeometric pro-p Galois sections of hyperbolic curves, Publications of the Research Institute for Mathematical Sciences (Publ. Res. Inst. Math. Sci.) 46 (2010), no. 4, 829-848.

You can get this paper from his homepage.

Existence of (smooth) models

2009-10-13T08:12:06Z

Hi everyone,

let X be a variety over a field k, S an integral scheme such that the function field K of S is contained in k. An S-scheme X is called model of X/k if X x_S k = X, i.e. if the generic fiber of X over S is isomorphic to X.

Are there general conditions on X, S, k, such that X exists?
If X is smooth and projective, what are the conditions, such that there is a smooth model X?
Any good references that go into models and reduction in general, and not only in the case of curves?

eBook readers for mathematics

2010-07-04T12:59:45Z

For a while I have been eying stand-alone eBook readers that use "electronic ink" displays, the most popular ones seem to be the Amazon Kindle readers.

My criteria are as follows: It should be able to display pdf's and math formulas in them just like they were printed and they should be able to handle "big files", say the numdam versions of EGA (even with my desktop browsing them is noticeably slower than browsing other pdf's). A nice but not strictly necessary feature would be the ability to read djvu files.

Does anyone here have experience with these kinds of devices? Does anyone use them in their mathematical work days? Do they make things easier?

Answer by Lars for References about the Grothendieck's way of algebraizing the notions of calculus and differential geometry

2012-06-27T14:51:47Z

Here are three references which were/are very helpful to me:

Berthelot, Pierre; Ogus, Arthur: Notes on crystalline cohomology.

Chapter 2 covers much of what you are looking for.

Berthelot, Pierre: Cohomologie cristalline des schémas de caractéristique p>0. (French) Lecture Notes in Mathematics, Vol. 407

Here, also chapter 2 contains many things you are looking for, in a more general setup.

Grothendieck, Alexander: Crystals and the de Rham cohomology of schemes. 1968 Dix Exposés sur la Cohomologie des Schémas pp. 306–358

This doesn't contain many details, but is still very interesting (certainly not only historically).

Answer by Lars for M - affine and H^1(M,Z)=H^2(M,Z) = 0 imply (?) Pic^algebraic(M) = 0. Note: in algebraic category NO exponential sequence

2012-05-18T13:47:43Z

Let us assume that we work over an algebraically closed base field $k$, and fix a prime $\ell$ different from the characteristic of $k$. First of all, the Kummer-sequence $$0\rightarrow \mu_{\ell^n}\rightarrow \mathbb{G}_m\xrightarrow{\ell^n}\mathbb{G}_m\rightarrow 0$$ for $\ell$ prime to the characteristic of $k$, shows that $Pic(X)$ contains no $\ell$-power torsion. In particular, if $char(k)=0$, then $Pic(X)$ is torsion free.

Next, recall that for étale cohomology we have $H^1(X,\mathbb{Z}_\ell(1))=\hom^{cont}(\pi^{et}_1(X),\mathbb{Z}_{\ell})$ . Hence, if $H^1(X,\mathbb{Z}_\ell(1))=0$, then the maximal abelian pro-$\ell$-quotient of $\pi_1^{et}$ is $0$.

Now assume that we have a smooth proper scheme $X'$ containing $X$ as a dense open subset. In your characteristic $0$ situation this can always be achieved. Then the abelian maximal pro-$\ell$-quotient of $\pi_1^{et}(X)$ surjects onto the maximal abelian pro-$\ell$ quotient of $\pi_1(X')$, so this group is also trivial. Again, using the Kummer sequence, this implies that $Pic(X')$ contains no $\ell$-power torsion. Since $X'$ is proper, there is a Picard scheme $Pic_{X'/k}$, such that $Pic(X')=Pic_{X'/k}(k)$. In particular, if $Pic^0_{X'/k}$ denotes the connected component of the origin, then $Pic^0_{X'/k}(k)$ has no $\ell$-power torsion. But $Pic^0_{X'/k}$ (or rather its reduced closed subscheme, if $k$ has positive characteristic) is an abelian variety. It follows that $Pic^0_{X'/k}$ is a $0$-dimensional abelian variety (as otherwise there would be nontrivial $\ell$-torsion). This implies that $Pic(X')=NS(X')=Pic_{X'/k}(k)/Pic^0_{X'/k}(k)$, which is a finitely generated group without prime-to-$p$-torsion, i.e. if $char (k)=0$, then $Pic(X')$ is free of finite rank.

Since $X'$ was smooth, we have a surjection $Pic(X')\twoheadrightarrow Pic(X)$, so $Pic(X)$ is free of finite rank (because we already knew it is torsion free).

Next, the Kummer sequence gives an exact sequence $$0\rightarrow Pic(X)\xrightarrow{(-)^{\ell^n}}Pic(X)\xrightarrow{c_{1,\ell^n}}H^2(X,\mathbb{Z}/\ell^n\mathbb{Z}(1))$$ for every $n$, so $Pic(X)/\ell^n\subset H^2(X,\mathbb{Z}/\ell^n\mathbb{Z}(1))$, and passing to the limit gives $Pic(X)\otimes \mathbb{Z}_\ell\subset H^2(X,\mathbb{Z}_\ell(1))$ , because we know that $Pic(X)$ is finitely generated.

Thus, if we assume that $H^2(X,\mathbb{Z}_\ell(1))=0$, then $Pic(X)=0$.

Topological invariance for formally étale morphisms

2012-05-04T12:36:36Z

If $f:X_0\rightarrow X$ is a closed immersion of locally noetherian schemes such that the topological spaces of $X_0$ and $X$ are identical (or, more generally, if $f$ is a universal homeomorphism), then it is known (see e.g. SGA1, Thm I.8.2, or, more generally, SGA1, Thm. IX.4.10) that pullback along $f$ induces an equivalence between the categories of etale $X$-schemes and etale $X_0$-schemes.

The main property of étale morphisms is the "infinitesimal lifting criterion", after which the definition of formally étale morphisms is modeled.

Is a "topological invariance result" as above also true for the categories of formally étale $X_0$- and $X$-schemes?

Answer by Lars for Is there a really big ring of differential operators in characteristic p?

2012-03-08T15:28:05Z

Let me give some more details on Mariano's comment: The ring of differential operators a la EGA4 in this particular case will be a free $k[t]$-algebra generated by the following operators: We write $$\partial_t^{(n)}$$ for the operator which is defined by $$\partial_t^{(n)}(t^m)={m\choose n}t^{m-n}.$$ Because of this, sometimes the notation $$\partial_t^{(n)}=\frac{1}{n!}\frac{\partial^n}{\partial t^n}$$ is used.

Actually, to generate the ring, the operators $\partial_t^{(p^n)}$ suffice.

Now this ring is not noetherian, but it is an increasing union of noetherian subalgebras, lets denote them by $D^{(m)}$, which are the subalgebras generated by operators of degree $\leq p^m$.

Using partially divided powers, Berthelot abstractly defines rings $\mathcal{D}^{(m)}$ such that the full ring of differential operators $\mathcal{D}$ is the direct limit of the $\mathcal{D}^{(m)}$. The image of $\mathcal{D}^{(m)}$ in $\mathcal{D}$ is then precisely the $D^{(m)}$ that I defined ad-hoc above. The crystalline operators that you defined in the question correspond to Berthelot's $\mathcal{D}^{(0)}$.

Answer by Lars for Good book on Riemann surfaces and Galois theory?

2012-02-17T17:39:59Z

There is a chapter on Riemann surfaces in Tamās Szamuely's book "Galois Groups and Fundamental Groups", which contains the facts that you are looking for.

Answer by Lars for Proofs in the same vein as Ax-Grothendieck

2012-02-11T16:45:39Z

Since in your edit you say that you don't insist on the use of model theory:

A very nice and famous example is Deligne-Illusie's proof of the degeneration of the Hodge to deRahm spectral sequence. Deligne, P.; Illusie, L. Relèvements modulo p^2 et décomposition du complexe de de Rham. Invent. Math. 89 (1987), no. 2, 247–270.

Answer by Lars for Have people successfully worked with the full ring of diferential operators in characteristic p?

2010-02-12T12:56:37Z

This might not be what you are looking for, as they use the actual full ring of differential operators (in Berthelot's theory, your "full ring" would be $D^{(0)}$, if I understand correctly), but the following papers are very beautiful in my opinion:

Gieseker, D. - Flat vector bundles and the fundamental group in non-zero characteristics.

dos Santos, João Pedro Pinto - Fundamental group schemes for stratified sheaves. J. Algebra 317 (2007), no. 2, 691--713.

Hélène Esnault, Vikram Mehta - Simply connected projective manifolds in characteristic $p>0$ have no nontrivial stratified bundles

You'll of course notice very quickly that in all cases, the D-Module flavor is lost, as a $O_X$-coherent D-module can be translated into the world of vector bundles thanks to Frobenius descent.

Global sections of lisse sheaf as invariants of $\pi_1$-action

2011-10-11T10:38:47Z

Let $k$ be an algebraically closed field of characteristic $p\geq 0$ and $\ell$ a prime different from $p$. For a connected scheme of finite type over $k$ with geometric point $x$, and a lisse $\overline{\mathbb{Q}}_{\ell}$-sheaf $F$ on $X$ one can compute the global sections as follows $$H^0(X,F)=F_x^{\pi_1(X,x)}$$ I don't know where a proof of this is written down, but it seems to me that it works like this: Consider $F$ as projective system $(F_n)$ of $\mathcal{O}_E/\mathbf{m}^n$-sheaves, for $E$ a finite extension of $\mathbb{Q}_{\ell}$ , and $\mathbf{m}$ the maximal ideal of $\mathcal{O}_E$. Each $F_n$ corresponds to a étale covering $X_n$ of $X$, and $(F_n)_x$ is the set of geometric points of $X_n$ over $x$. Now the sections of $X_n\rightarrow X$ correspond to points in $(F_n)_x$ which are fixed by the $\pi_1(X,x)$-action. Passing to the limit proves the formula for $H^0$.

My question is: Does this also hold for $k$ not algebraically closed?

I've never seen it stated like this, but my argument doesn't seem to use the fact that $k=\bar{k}$.

I think it is true for smooth curves, because a smooth curve $X\neq \mathbb{P}^1$ over any field of characteristic $p\geq 0$ is even an "étale $K(\pi_1(X,x),1)$ space", i.e. $$H^n(X,\mathbb{Z}_{\ell})=H^n(\pi_1(X,x), \mathbb{Z}_\ell)$$ for all $n\geq 0$.

Answer by Lars for Help motivating log-structures

2011-07-12T08:01:35Z

Regarding tame covers: For log-schemes there are the notions of log-étale and log-smooth morphisms, which behave very similarly to the classical notions of étaleness and smoothness.

If $X\subset \overline{X}$ and $Y\subset \overline{Y}$ are open immersions of smooth $k$-schemes, for, say, $k$ a field, such that the complements of $X$ and $Y$ are strict normal crossings divisors, then $\overline{X}$ and $\overline{Y}$ get canonical (fine, saturated) log-structures. Lets call the log-schemes $X^{\log}$ and $Y^{\log}$. If $f:X\rightarrow Y$ is a finite étale morphism, extending to a finite morphism $\bar{f}:\overline{X}\rightarrow \overline{Y}$, then $f$ induces a morphism of log-schemes $f^{\log}:X^{\log}\rightarrow Y^{\log}$, and $f^{\log}$ is log-étale if and only if $\bar{f}$ is a tame covering in the usual sense. So it "behaves" like an étale covering in the category of log-schemes. For example, one can develop a theory of log-fundamengal groups and so on. A very nice reference for this is Jakob Stix thesis, which can be found on his homepage.

In fact log-étaleness is more general: For $f^{\log}$ to be log-étale, $\bar{f}$ does not have to be a finite morphism; certain non-finite ones are allowed, for example so called "log-blowups". As far as I understand they play a crucial role in developing log-étale cohomology. A good reference for this and much much more is

Illusie, Luc. An overview of the work of K. Fujiwara, K. Kato, and C. Nakayama on logarithmic étale cohomology. (English summary) Cohomologies p-adiques et applications arithmétiques, II. Astérisque No. 279 (2002), 271–322.

Answer by Lars for For quasi-coherent D-Modules

2011-05-08T12:02:42Z

Hi Lei, I think your second statement is false, even in char. $0$: Let $X=\mathbb{A}_k^1$ for some field $k$, and $U:=X\setminus {0}$. Denote the open immersion by $i$. Then $i_*\mathcal{O}_U$ is $\mathcal{O}_X$ -quasi-coherent and not coherent. Consider the canonical connection on $i_*\mathcal{O}_U$ : If $x$ is a coordinate on $\mathbb{A}^1_k$, then $\nabla(x):=dx$. This is a connection on $i_*\mathcal{O}_U$ : if we plug in $\frac{1}{x}$ we get $-\frac{1}{x^2}dx\in i_*\mathcal{O}_U\otimes \Omega^1_{X/k}$ . But the section $\frac{1}{x}$ is not contained in a $\mathcal{O}_X$ -coherent sub $D_X$-module: The smallest sub $D_X$-module containing $\frac{1}{x}$ contains $\frac{1}{x^n}$ for all $n\in \mathbb{Z}$. This is not finitely generated over $\mathcal{O}_X$.

What is true however, is that on a smooth variety any $\mathcal{O}_X$-quasi-coherent $D_X$-module is the union of its $D_X$-coherent submodules, but these do not need to be $\mathcal{O}_X$-coherent. A reference for this is this:

D-Modules, Perverse Sheaves, and Representation Theory: Cor. 1.4.17

http://books.google.com/books?id=8ewkW5SC7DcC&lpg=PP1&dq=hotta%20takeuchi&pg=PA29#v=onepage&q&f=false)

Answer by Lars for $\mathbb{P}^n$ is simply connected

2011-04-20T14:45:06Z

Let me give another answer, even though it does not fit into Hartshorne's context:

Show that $\pi_1(\mathbb{P}^n)$ has to be abelian.

Use Kummer-Theory to relate coverings to torsion in $Pic (\mathbb{P}^n)=\mathbb{Z}$, see e.g. Milne's Etale Cohomology, Prop 4.11. This implies that there are no nontrivial étale coverings of degree prime to the base characteristic.

Then use Artin-Schreier theory to relate the rest of the coverings to $\Gamma(\mathbb{P}^n,\mathcal{O}_{\mathbb{P}^n})/(F-1)\Gamma(\mathbb{P}^n,\mathcal{O}_{\mathbb{P}^n})=0$ , and $H^1(\mathbb{P}^n,\mathcal{O}_{\mathbb{P}^n})^F=0$, where $F$ is the Frobenius, see e.g. Milne's, Prop 4.12.

Answer by Lars for Examples of great mathematical writing

2009-10-12T20:16:17Z

One of the math books I enjoy reading in most is Neukirch's book "Algebraic Number Theory". In my opinion, he presents the material beautifully and with a good degree of generality for a text book. Also, he manages to use language beautifully without losing mathematical rigor and without compromising clarity (this holds for the German version as well as for the English translation). When I have to look up some fact from algebraic number theory, Neukirch is usually the first book I try.

Cohomological Flatness in degree 0

2011-04-11T12:11:13Z

Let $f:X\rightarrow S$ be a proper, flat, finite type morphism of noetherian schemes. $f$ is called cohomologically flat in degree 0, if formation of $f_*\mathcal{O}_X$ commutes with base change along any morphism $T\rightarrow S$.

Now let $f$ be a proper, flat, finite type morphism of noetherian schemes, and additionally require that $f$ has reduced geometric fibers. Then it follows from rather complicated results in EGAIII.2 (specifically 7.8.6), that $f$ is cohomologically flat in degree $0$.

My first question is: Is there are more elementary proof of this fact, possibly bypassing heavy cohomological machinery?

If in addition $f$ also has integral geometric fibers, then for every $T\rightarrow S$, the canonical map $$\mathcal{O}_T\rightarrow f_{T,*}\mathcal{O}_{X_T}$$ is an isomorphism, where the subscript $T$ denotes the objects base changed along $T$. For a proof, see e.g. Kleiman "The Picard Scheme", Solution to Ex. 3.11, which also uses heavy cohomological machinery.

My second question: Is there are more elementary proof of this fact?

Representability on the big étale site and base change

2011-01-13T17:27:00Z

I am reading M. Artin's treatment of the proper base change theorem for étale cohomology in his "Théorèms de représentabilité pur les espaces algébriques", and I have trouble understanding the following remark on page 222:

If $f:X\rightarrow S$ and $g:S'\rightarrow S$ are morphisms of algebraic spaces (or schemes, if you prefer), and if $f':X'\rightarrow S'$, $g':X'\rightarrow X$ denote the base changes of $f$ and $g$, then one can construct for any abelian sheaf $F$ on the big étale site of $X$ the base change morphism $g^*R^qf_*F\rightarrow R^q f'_*(g'^*F)$ (the higher direct images also computed on the big sites). If I understand correctly, Artin claims that if $F$, $R^q f_*F$ and $R^qf'_*(g'^*F)$ are representable on the big étale site of $X$, resp. $S$, resp. $S'$ (i.e. locally constructable), then the base change morphism is an isomorphism.

Why is that? Is that an easy fact?

Answer by Lars for Representability on the big étale site and base change

2011-02-11T22:24:40Z

With help from Milne's book on étale cohomology, I figured out how answer the question, although I am not sure that this argument is what Artin had in mind, and I still think that there's an easier argument.

There are morphisms of topoi $\pi_X: X_{ET}\rightarrow X_{et}$ from the topos associated to the big étale site of $X$ to the topos of the small étale site. Similarly for $S$, and $f:X\rightarrow S$ induces $f^s:X_{et}\rightarrow S_{et}$ and $f^b:X_{ET}\rightarrow S_{ET}$, and the obvious diagram commutes, i.e. $f^s\pi_X=\pi_Sf^b$. Given a sheaf $F$ in $S_{ET}$ we get a base change morphism \[ \pi_S^*R^qf^s_*F\rightarrow R^qf_*^b\pi_X^*F\] Milne calls this "universal base change morphism", for good reasons: Given any morphism $g:S'\rightarrow S$, you also get morphisms of topoi $g^b:S'_{ET}\rightarrow S_{ET}$ and ${g'}^b:X'_{ET}\rightarrow X_{ET}$. Using this to restrict the universal base change morphism to $S'_{ET}$, we get the usual base change morphism for $g$ and $F$. (For this one has to check the commutativity of a few diagrams. All the ingredients can be found, e.g., in great detail in the Stacks Project)

Now, if $F$ is locally constructible, i.e. if the adjunction map $F\rightarrow \pi_X^*\pi_{X,*} F$ is an isomorphism, then it is not hard to check that the universal base change morphism is an isomorphism, and thus every base change morphism is an isorphism.

Answer by Lars for Does the etale fundamental group of the projective line minus a finite number of points over a finite field depend on the points?

2010-12-28T11:12:07Z

It is a result of Tamagawa that for two affine curves $C_1, C_2$ over finite fields $k_1,k_2$ any continuous isomorphism $\pi_1(C_1)\rightarrow \pi_1(C_2)$ arises from an isomorphism of schemes $C_1\rightarrow C_2$. Hence, if $\pi_1( \mathbb{P}^1\setminus{a_1,\ldots, a_r})$ were independent of the choice of the $a_i$, then the isomorphism class of the schemes $\mathbb{P}^1\setminus{a_1,\ldots, a_r}$ would be independent of the choice of $a_1,\ldots,a_r$.

Tamagawa's result is Theorem 0.6 in this paper:

The Grothendieck conjecture for affine curves, A Tamagawa - Compositio Mathematica, 1997 http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=298922

In the case of an algebraically closed field, the answer is also that the fundamental group depends on the choice of the points that are being removed. Again by a theorem by Tamagawa: If $k$ is the algebraic closure of $\mathbb{F}_p$, and $G$ a profinite group not isomorphic to $(\hat{\mathbb{Z}}^{(p')})^2\times \mathbb{Z}_p$, then there are only finitely many $k$-isomorphism classes of smooth curves $C$ with fundamental group $G$ (the restriction on $G$ excludes ordinary elliptic curves).

This can be found in

Finiteness of isomorphism classes of curves in positive characteristic with prescribed fundamental groups, A Tamagawa - Journal of Algebraic Geometry, 2004

Compactifications of varieties with small complement

2010-10-10T10:06:50Z

Let $X$ be a smooth variety over an algebraically closed field $k$. If it makes things easier, $X$ may be assumed to be quasi-projective. By Nagata (or quasi-projectivity) there exists a proper variety $\bar{X}$ which contains $X$ as a dense open subvariety, and by the smoothness of $X$ we may assume $\bar{X}$ to be normal. Are there any criteria/theorems which give information about the codimensio of the $\overline{X}\setminus X$?

The same question can be asked if we assume resolution of singularities, such that we may assume $\overline{X}$ to be smooth. Under which conditions can a smooth compactification $\overline{X}$ be found such that $X$ has complement of codimension $>1$ in $\overline{X}$?

Finally, to rephrase, how can one detect whether a given smooth variety $X$ arises by removing a codimemsion $>1$ closed subvariety from some proper variety?

Picard groups of non-projective varieties

2010-07-29T17:33:13Z

As far as I know, the main representability result for the relative Picard functor $Pic_{X/k}$, for a noeth. sep. scheme of finite type over a field $k$ is:

If $X$ is proper then $Pic_{X/k}$ is representable by a $k$-scheme loc. of finite type. (This is attributed to Murre and Oort in Bosch-Lüttkebohmert-Raynaud)

I am interested in what can be said once the requirement of properness is dropped, e.g. what can be said for quasi-projective varieties?

Representability is probably to much to ask for (even as an algebraic space), but do you have references or know of examples where the Picard functor of a non-projective quasi-projective variety is representable?

Is there a weaker sense of representability in which sense the "open" Picard functor is representable?

Is the group somehow controlled by (the group of $k$-points of) representable objects. (I have the naive impression that if $X$ is my quasi-projective variety, then a proper hypercovering of $X$ should be able to compute $H^1(X,\mathcal{O}_X^*)$, and that then one might be able to use representability theorems for proper/projective maps, but I know nearly nothing about the involved technical requirements.)

Edit: I should have added that I do not want to assume resolution of singularities.

References for logarithmic geometry

2010-01-21T21:20:37Z

Hi everyone,

I'm looking for a systematical introduction to (or treatment of) logarithmic structures on schemes. I am reading Kato's article ("Logarithmic structures of Fontaine-Illusie") at the moment, but I would like to have a more detailed source that goes through or gives an overview of the constructions of classical scheme theory that have analogs in the log-setup. Are there any articles/books that in your opinion are required reading if I want to learn about log-geometry? What are beautiful examples of applications of this machinery?

Sheaves of Principal parts

2010-04-10T17:24:08Z

In EGA IV, Sec. 16, Grothendieck defines the sheaf of principal parts as follows: Let $f:X\rightarrow S$ be a morphism of schemes and $\Delta:X\rightarrow X\times_S X$ the diagonal morphism associated to $f$. $\Delta$ is an immersion, so the corresponding morphism $\Delta^{-1}\mathcal{O}_{X\times_S X}\rightarrow\mathcal{O}_X$ is surjective. Let $\mathcal{I}$ denote its kernel and define the sheaves of principal parts as

\[\mathcal{P}_{X/S}^n:=\Delta^{-1}( \mathcal{O}_{X\times_S X}) / \mathcal{I}^{n+1}\]

In their book on Crystalline cohomology, Berthelot and Ogus define the sheaf of principal parts $\mathcal{P}^n_{X/S}$ as

\[(\mathcal{O}_X\otimes_{f^{-1}\mathcal{O}_S}\mathcal{O}_X)/\mathbf{I}^{n+1},\] where $\mathbf{I}$ is the kernel of the multiplication map from the tensor product to $\mathcal{O}_X$.

My question is probably simple, but I don't know how to see it: Why are those definitions equivalent if $X$ and $S$ are not affine and $n>0$?

I've not seen the second definition anywhere else, although it seems somewhat nicer than the first one...

Answer by Lars for Covers of the projective line over Z and arithmetic Grauert-Remmert

2010-07-02T20:17:27Z

Regarding Q3: For any scheme $X$ of finite type over $\mathbb{C}$ the Riemann-Existence Theorem (See SGA1 XII.5) says that the category of finite étale coverings of $X$ is equivalent to the category of finite covering spaces of the associated analytic space $X^{an}$. This implies that the finite quotients of the topological fundametal group of $X^{an}$ are the same as the finite quotients of the étale fundamental group, and one obtains that the étale fundamental group of $X$ is isomorphic to the profinite completion of the topological fundamental group of $X^{an}$.

Q4: The same short exact sequence that I mentioned in your other question is still valid.

As in your other question, I cannot say anything about the situation over $\mathbb{Z}$ :)

Tannaka formalism and the étale fundamental group

2010-05-07T14:37:08Z

For quite a while, I have been wondering if there is a general principle/theory that has both Tannaka fundamental groups and étale fundamental groups as a special case.

To elaborate: The theory of the étale fundamental group (more generally of Grothendieck's Galois categories from SGA1, or similarly of the fundamental group of a topos) works like this: Take a set valued functor from the category of finite étale coverings of a scheme satisfying certain axioms, let $\pi_1$ be its automorphism group and you will get an equivalence of categories ( (pro-)finite étale coverings) <-> ( (pro-)finite cont. $\pi_1$-sets).

The Tannaka formalism goes like this: Take a $k$-linear abelian tensor category $\mathbb{T}$ satisfying certain axioms (e.g. the category of finite dim. $k$-representations of an abstract group), and a $k$-Vector space valued tensor functor $F$ (the category with this functor is then called neutral Tannakian), and let $Aut$ be its tensor-automorphism functor, that is the functor that assigns to a $k$-algebra $R$ the set of $R$-linear tensor-automorphisms $F(-)\otimes R$. This will of course be a group-valued functor, and the theory says it's representable by a group scheme $\Pi_1$, such that there is a tensor equivalence of categories $Rep_k(\Pi_1)\cong \mathbb{T}$.

Both theories "describe" under which conditions a given category is the (tensor) category of representations of a group scheme (considering $\pi_1$-sets as "representations on sets" and $\pi_1$ as constant group scheme). Hence the question:

Are both theories special cases of some general concept? (Maybe, inspired by recent questions, the first theory can be thought of as "Tannaka formalism for $k=\mathbb{F}_1$"? :-))

Answer by Lars for What should be learned in a first serious schemes course?

2010-06-17T18:12:22Z

To prepare well for what comes after a first course, a more extensive discussion of étale morphisms than Hartshorne gives should be part of such a course, in my opinion.

Comment by Lars

2013-04-25T12:01:14Z

Also have a look here for an example: math.stanford.edu/~vakil/files/nonfg.pdf

Comment by Lars

2013-03-07T14:18:37Z

Hi, what do you mean by "tannakian etale fundamental group" in your last line?

Comment by Lars

2012-05-20T08:08:41Z

Hi, since $X$ is smooth over $\mathbb{C}$, it is true that there is a connical ismomorphism between etale cohomology with coefficients in $\mathbb{Z}/\ell^n\mathbb{Z}$ and classical cohomology with coefficients in $\mathbb{Z}/\ell^n\mathbb{Z}$.

Comment by Lars

2012-05-09T12:34:10Z

There is an overview paper on the arxiv: arxiv.org/abs/0909.0069 "Mapping F_1 land".

Comment by Lars

2012-05-05T15:31:18Z

So if you have a nice "formal" argument for essential surjectivity, I would be very interested.

Comment by Lars

2012-05-05T15:29:51Z

Martin, that's what I thought at first. I agree that the fully faithfulness follows more or less directly from the definition of a formally etale morphism. The essential surjectivity (for $X_0\rightarrow X$ a nilp. thickening) is usually proven using the reduction to "standard etale" morphisms, which to my knowledge does not work for formally etale morphisms. The more general case ($X_0\rightarrow X$ a universal homeomorphism) is an application of faithfully flat descent (that's why it is in Exp. IX of SGA1).

Comment by Lars

2012-05-04T18:15:55Z

Not precisely: etale=formally etale + locally of finite presentation.

Comment by Lars

2012-04-22T13:59:58Z

Luc Illusie gives an argument via derived lim's in his "FGA explained" chapter "Grothendieck's existence theorem of formal geometry"

Comment by Lars

2012-03-09T14:13:47Z

Ah, I wonder which of Bertholot's rings he means, and what precisely is $\delta^{[p]}$. In general, if $\delta$ is an operator of order 1, then $\delta^p=0$ in $\mathcal{D}^{(n)}$ for $n>1$, but not necessarily $\mathcal{D}^{(0)}$ . Modules over $\mathcal{D}^{(0)}$ are connections. Now if by `$\delta^{[p]}$ you mean what I wrote as $\delta^{(p}_t$`,then that′s an operator of order $p$, and cannot be considered as an elementof $\mathcal{D}^{(0)}$ , but if I remember correctly, the same reasoning applies.

Comment by Lars

2012-02-20T20:19:47Z

If you are using linux (or have cygwin installed), then you can use the program "pdfcrop". It automatically crops the whitespaces of a pdf document. Unfortunately this only works for pdf's which are not scanned. But for Arxiv papers its fine!

Comment by Lars

2012-02-14T22:25:15Z

Hi Timo, regarding the Albanese, I suggest you look at mathoverflow.net/questions/2548/…

Comment by Lars

2011-11-12T11:03:11Z

I remember reading somewhere (perhaps in Faltings' Bourbaki report on Mochizuki's Theorem?) that Mostow's Rigidity theorem (en.wikipedia.org/wiki/Mostow_rigidity_theorem) was one of the motivations.

Comment by Lars

2011-11-07T22:39:14Z

As a side note: if the base has positive characteristic $p$, and if one replaces the notion of "line bundle with flat connection" by "line bundle with operation of the ring of differential operators of $X/k$", the line bundle will still lie in $Pic^\tau$, at least on smooth $X/k$.

Comment by Lars

2011-10-18T08:10:15Z

Hey Ivan, here's a "solution" (actually a cheat): Don't put the arxiv links in the references of the papers, but a link to a website of yours (say a list of your publications) from which you link to your arxiv'ed papers...

Comment by Lars

2011-10-12T09:53:03Z

Thanks Alex! It seems that it's just unraveling of definitions, but I assumed that Deligne would have written it like this in Weil II if it was true. He explicitly writes the formula for $X\times_k \bar{k}$...