stacky wiki - User contributions [en]

Course notes

2012-07-08T07:17:35Z

208.106.86.75:

This page collects math notes I've taken, mostly course notes. I've also written some [[TeXnical notes]] and [[scripts]].

I provide the source for most of my notes for a couple of reasons.
#It feels good for some reason.
#If you're curious about how to typeset something you've seen in these notes, you can download the source and have a look.
#Somebody might download all my notes, correct the errors and unclear presentation, and then send them back to me. It's not likely, but it could happen.
#If a bomb drops on my house and I lose all my stuff, maybe somebody can give me a copy of my notes.
#The source is much faster to download an compile if you're on a slow connection.

Since I've learned about it, I've started using [http://en.wikipedia.org/wiki/Subversion_(software) Subversion (svn)] for everything. If you actually make corrections in any of the notes, I recommend doing it through svn. If you don't yet know how to use svn, I wrote an [http://stacky.net/old-webpage/svnguide.html svn crash course] just for you.

{| class="wikitable"
|-
|My notes
|What?
|Who?
|When?
|Additional Resources
|-
|[http://stacky.net/files/written/GIT/GIT.pdf pdf][svn://sheafy.net/courses/git_fa2009 svn][http://stacky.net/files/written/GIT/GITSource.tgz tgz]
|Geometric Invariant Theory
|[http://math.berkeley.edu/~serganov/ Vera Serganova]
|Fall 2009
|
|-
|[http://stacky.net/files/written/Halg/Halg.pdf pdf][svn://sheafy.net/courses/halg_sp2008 svn][http://stacky.net/files/written/Halg/HalgSource.tgz tgz]
|Homological Algebra
|[http://math.berkeley.edu/~teichner Peter Teichner]
|Spring 2008
|
|-
|[http://stacky.net/files/written/defthy07/defthy07.pdf pdf][svn://sheafy.net/defthy_su07 svn][http://stacky.net/files/written/defthy07/DefThySource.tgz tgz]
|[http://www.msri.org/people/members/defthy07/ Deformation Theory Workshop, MSRI]
|[http://www.math.princeton.edu/~lieblich/ Max Lieblich] 
[http://math.berkeley.edu/%7Emolsson Martin Olsson] 
[http://math.berkeley.edu/~osserman/ Brian Osserman] 
[http://math.stanford.edu/~vakil/ Ravi Vakil]
|Summer 2007
|Other peoples' notes can be found [http://www.msri.org/people/members/defthy07/lectures.html here] and [http://math.stanford.edu/~vakil/defthy/ here].
|-
|[http://stacky.net/files/written/Stacks/Stacks.pdf pdf][svn://sheafy.net/courses/stacks_sp2007 svn][http://stacky.net/files/written/Stacks/FullStackSource.tgz tgz]
|Math 274, [http://math.berkeley.edu/%7Emolsson/274.html Stacks]
|[http://math.berkeley.edu/%7Emolsson Martin Olsson]
|Spring 2007
|
|-
|[http://stacky.net/files/written/AlgTopology/AlgebraicTopologyNotes.pdf pdf][http://stacky.net/files/written/AlgTopology/AlgebraicTopologyNotes.tex tex]
|Math 215A, Algebraic Topology
|[http://math.berkeley.edu/%7Eteichner/ Peter Teichner]
|Fall 2006
|
|-
|[http://stacky.net/files/written/CommRings/CommRing.pdf pdf][http://stacky.net/files/written/CommRings/CommutativeRingsSource.tgz tgz]
|Math 274, [http://stacky.net/files/written/CommRings/index.html Commutative Rings]
|[http://math.berkeley.edu/%7Elam/ Tsit Yuen Lam]
|Fall 2006
|
|-
|[http://stacky.net/files/written/LieGroups/LieGroups.pdf pdf][http://stacky.net/files/written/LieGroups/LieGroupsSource.tgz tgz]
|Math 261A, Lie Groups and Lie Algebras
|[http://math.berkeley.edu/%7Ereshetik/ Nicolai Reshetikhin] [http://math.berkeley.edu/%7Eserganov/ Vera Serganova] [http://math.berkeley.edu/%7Ereb/ Richard Borcherds]
|Spring 2006
|
|-
|[http://stacky.net/files/written/SymplecticGeometry/SymplecticGeomNotes.pdf pdf][http://stacky.net/files/written/SymplecticGeometry/SymplecticGeometrySource.tgz tgz]
|Math 242, Symplectic Geometry
|[http://math.berkeley.edu/%7Ealanw/ Alan Wienstein]
|Fall 2005
|
|-
|
|Math 252, [http://math.berkeley.edu/%7Eserganov/math252/index.html Representation Theory]
|[http://math.berkeley.edu/%7Eserganov/ Vera Serganova]
|Fall 2005
|[http://math.berkeley.edu/%7Eserganov/ Vera Serganova]'s [http://math.berkeley.edu/%7Eserganov/math252/index.html Representation Theorey notes] 

[http://www.maths.leeds.ac.uk/%7Epmtwc/ William Crawley-Boevey]'s lectures on [http://www.maths.leeds.ac.uk/%7Epmtwc/quivlecs.pdf representations of quivers]
|-
|[http://stacky.net/files/written/AlgebraicGeometry/AlgebraicGeometryNotes.pdf pdf] [http://stacky.net/files/written/AlgebraicGeometry/AlgebraicGeometryNotes.tgz tgz]
|Math 256B, Algebraic Geometry
|[http://math.berkeley.edu/~vojta/ Paul Vojta]
|Spring 2005
|[http://math.berkeley.edu/%7Evojta/ Paul Vojta's] [http://stacky.net/files/written/VojtaSolutions/ handouts and solutions] 

[http://modular.ucsd.edu/ William Stein's] [http://modular.ucsd.edu/AG.html notes and solutions] 

[http://math.berkeley.edu/%7Ereb/ Richard Borcherds's] selected [http://math.berkeley.edu/%7Ereb/courses/alggeom/ solutions] 

[http://math.arizona.edu/~cais/ Bryden Cais's] [http://math.arizona.edu/~cais/CourseNotes/AlgGeom04/ notes] and [http://math.arizona.edu/~cais/CourseNotes/AlgGeom04/Hartshorne_Solutions.pdf solutions] 

[http://mathsci.kaist.ac.kr/~jinhyun/ Jinhyun Park's] [http://mathsci.kaist.ac.kr/~jinhyun/sol2/hart.html solutions] 

[http://math.berkeley.edu/%7Emhaiman/index.html Mark Haiman]'s [http://math.berkeley.edu/%7Emhaiman/math256/index.html Math 256AB page]
|}

I gave a couple of talks on toric varieties in a student seminar. If [http://stacky.net/files/written/toric.pdf my notes] make sense to you, you're welcome to use them.

My notes ([http://stacky.net/files/written/AnalysisPrelim/AnalysisPrelim.pdf pdf], [http://stacky.net/files/written/AnalysisPrelim/AnalysisPrelim.tgz tgz]), made from [http://math.rice.edu/~av15/ Tony's] notes, of Yonathan's prelim workshop on analysis.

One of my favorite facts about right adjoint functors is that they [http://stacky.net/files/written/AdjointsCommuteLimits.pdf commute with limits].

[http://stacky.net/files/written/sestina.pdf Sestina's and primes], an easy problem Richard Dore and I worked out our first year in grad school, which I wrote up for some reason.

I talked about the Salamander Lemma in [http://math.berkeley.edu/~slofstra/mcf/ MCF]. My reference was [http://math.berkeley.edu/~gbergman/ George Bergman]'s preprint, [http://stacky.net/files/written/DiagramChasingBergman.pdf On diagram-chasing in double complexes].

==More moved notes==

[http://docs.google.com/View?docid=dsrd28g_14rgs6ptht Aspects of Moduli]:
{|class="wikitable"
|-
|Tom Bridgeland
| Stability in triangulated categories
| [http://stacky.net/files/written/AspectsModuli/TB.pdf pdf]
|-
|Kai Behrend
| Foundations of Donaldson-Thomas theory
| [http://stacky.net/files/written/AspectsModuli/KB.pdf pdf]
|-
|Alessio Corti
| Foundations of Donaldson-Thomas theory
| [http://stacky.net/files/written/AspectsModuli/AC.pdf pdf]
|-
|Valery Alexeev
| Moduli of higher-dimensional varieties
| [http://stacky.net/files/written/AspectsModuli/VA.pdf pdf]
|-
|Martin Olsson
| Logarithmic structures with a view towards moduli
| [http://stacky.net/files/written/AspectsModuli/MO.pdf pdf]
|-
|All in one (chronologically)
| [http://stacky.net/files/written/AspectsModuli/AspectsModuli.pdf pdf] [http://stacky.net/files/written/AspectsModuli/AspectsModuliSource.tgz tgz]
|}

Quantum Field Theory, my notes from
{|class="wikitable"
|-
|[http://math.berkeley.edu/~reshetik/ Nicolai Reshetikhin]'s [http://math.berkeley.edu/~reshetik/topics/CFT-TQFT.pdf course]
|[http://stacky.net/files/written/QFT/NR.pdf pub] [http://stacky.net/files/written/QFT/private/NR.pdf pri]
|-
|[http://math.berkeley.edu/~teichner Peter Teichner's] course
|[http://stacky.net/files/written/QFT/PT.pdf pdf]
|-
|[http://math.berkeley.edu/~reb/ Richard Borcherd's] course
|[http://stacky.net/files/written/QFT/RB.pdf pdf]
|-
|All in one (chronologically)
| [http://stacky.net/files/written/QFT/QFT.pdf pub] [http://stacky.net/files/written/QFT/private/QFT.pdf pri] [http://stacky.net/files/written/QFT/QFTSource.tgz tgz] [http://stacky.net/files/written/QFT/ dir]
|}

The public version is missing the last two weeks of Reshetikhin's class, and the private version is not. If you want the username and password for the private version, ask me for it and I'll give it to you.

For more QFT stuff, see [http://sites.google.com/site/chrisschommerpriesmath/ Chris Schommer-Pries'] QFT site for Fall 2007. (doesn't exist any more?)

For those who just want to get some notes, just download the pdf file. For those who want to look at the souce, download the tgz (<code>tar</code>-ed and <code>gzip</code>-ped) file, which has all the tex source in it. You'll find that the main file (<code>QFT.tex</code>) is quite boring. The actual lecture notes are in the other files and the top matter (like macro definitions) is in <code>QFTPreamble.tex</code>. As far as I know, the command <code>\input{abcdefg}</code> gives exactly the same result as pasting in the contents of the file <code>abcdefg.tex</code> (note that <code>abcdefg.tex</code> should be in the same directory as the main file).

If you have notes that I'm missing or if you have a correct/clear explanation for something which is incorrect/unclear, let me know (either tell me what you'd like to modify, give me some notes to go on, or update the tex yourself and send me a copy). Real (mathematical) errors should be fixed because it would be immoral to let them propagate (er ... that is, sit there), and typographical errors hardly take any time to fix, so you shouldn't be shy about telling me about them.

DZB

2011-11-18T05:16:55Z

208.106.86.75: /* chat with Tom - Anton 11:58, 16 November 2011 (PST) */

== chat with Tom - [[User:Anton|Anton]] 11:58, 16 November 2011 (PST) ==

Have toric $W=[\{x^ry^{r+1}=t\}/\GG_m]$ and $\X=[V/\GG_m^2]=\overline W\times_{\PP^1} \overline W\times_{\PP^1}\overline W$. So try to study integer points of $\overline W$. These are relatively prime pairs $[a:b]$ up to sign, together with a factorization of $a$ as an $r$-th power times an $(r+1)$-th power. Integer points of this thing essentially never miss the $B\GG_m$, so they don't factor through the "layers" of the non-separated line. If they did, then you'd have a generically degree 8 cover of so that integer points all lift. For each of the 8 curves, you could apply the machinery that says you have an finite etale cover by a finite number of higher genus curves, so you have finitely many integer points.

Instead, try to find nice covers of $\overline W$ by surfaces, with the idea of covering $\X$ by a finite number of surfaces so that all integral points lift to one of the guys in the cover. Lang's conjecture for surfaces is wide open, but we may at least be able to reduce to it. {{question|Are there smooth covers by toric surfaces or toric DM stacks?}}

Understand the case of $x^a+y^b+z^c=0$. How do you construct those finite etale curves ... Riemann existence gives you one, but the fact that there are a finite number which capture all the integral points has something to do with universal torsors and Cox rings ... see Bjorn's notes? In particular, something funny is going on between rational points and integer points. The stack has lots of rational points (since $\PP^1$ minus three points has lots of rational points), but not many integer points. The claim is that all the ''integer'' points lift to integer points of the higher genus curves. This is clearly false for rational points since the higher genus curves have finitely many rational points and the stack has infinitely many.

We already know that there are a finite number of rational points on $x^a+y^b+z^c=0$ when $a,b,c$ big. Showing that there are a finite number of rational points on $\X$ shows that there is a ''uniform'' upper bound on the number of solutions when $a,b,c$ are big..

[[Category:Note]]

Math 193a: Algebraic Stacks, Fall 2011

2011-10-24T08:28:49Z

208.106.86.75: /* Set 4 (due Nov. 4) */

The class meets 9–10am MWF in 257 Sloan. Office hours are Wednesday 2–4pm in 374 Sloan (but moving to the common room for tea at 3:30).

= Course Description =

Algebraic stacks arise naturally as solutions to classification (moduli) problems, so it is desirable to understand their geometry. In this course, we will assume a working knowledge of the geometry of schemes. We will extend the definitions and techniques used to study schemes to algebraic spaces and algebraic stacks. Topics will include Grothendieck topologies, descent, algebraic spaces, fibered categories, and algebraic stacks.

Specific topics will be included based on feedback from students.

=Exercises=

Try to do the following problems. If you get stuck, come to my office hours. If you're taking the class for a grade, make sure you hand in a substantial fraction of the exercises (or talk to me about doing a project of some sort). I know some of them are very tedious to write up, so you don't need to hand in everything, but please do attempt all of the problems. Also, please don't hand in solutions to problems that were assigned several weeks ago; for concreteness, let's not hand in solutions to problems in Set $n$ any later than week $n+2$.

<nowiki>$\def\C{\mathcal C}
\def\AA{\mathbb A}
\def\GG{\mathbb G}
\def\O{\mathcal O}$</nowiki>

==Set 1 (due Oct. 14)==
'''1.''' If you have never done so before, prove Yoneda's Lemma: for any category $\C$, taking an object $X\in \C$ to the functor $h_X\colon \C^{op}\to (Set)$ (defined by $h_X(T)=Hom_\C(T,X)$) defines a fully faithful functor $\C\to Func(\C^{op},(Set))$. ['''Edit:''' Actually, I'd like you to show a bit more. Show that for any functor $F:\C^{op}\to (Set)$, we have $Hom(h_X,F)\cong F(X)$.]

'''2.''' Play the game "find the representing object" whenever you get the chance. Determine if the following functors are representable. If they are, find the representing object.
:* The functor $(Top)\to(Set)$ taking a topological space $X$ to the set of open subsets of $X$.
:* The functor $(Top)\to(Set)$ taking a topological space $X$ to the set of closed subsets of $X$.
:* The functor $(Top)\to(Set)$ taking a topological space $X$ to the open subsets of $X$ whose complement is also open.
:* The functor $GL_n:(CommRing)\to(Set)$ taking a commutative ring $A$ to the set of invertible $n\times n$ matrices with entries in $A$. ['''Edit:''' In this case, try to find a ring $R$ so that $Hom(R,A)=GL_n(A)$, rather than $Hom(A,R)$.]
:* The functor $Nil:(CommRing)\to(Set)$ taking a commutative ring $A$ to $\{x\in A| x^n=0$ for some integer $n\}$. ['''Edit:''' In this case, try to find a ring $R$ so that $Hom(R,A)=Nil(A)$, rather than $Hom(A,R)$.]
:* The functor $\AA^n-\{0\}:(Sch)\to(Set)$ taking a scheme $T$ to $\{(f_1,\dots, f_n)\in \O_T(T)^n|$the $f_i$ do not all simultaneously vanish$\}$.
:* The functor $(\AA^n-\{0\})/\GG_m:(Sch)\to(Set)$ taking a scheme $T$ to $(\AA^n-\{0\})(T)/\sim$, where $\sim$ is the equivalence relation $(f_1,\dots, f_n)\sim (f_1',\dots, f_n')$ if there is a unit $u\in \O_T(T)$ such that $f_i'=uf_i$ for each $i$.

'''3.''' Let $A:\C\to \def\D{\mathcal D}$ and $B:\D\to \C$ be functors. Show that an adjunction $Hom(A-,-)\cong \hom(-,B-)$ is equivalent to a choice of natural transformations $\epsilon:id_\D\to BA$ (a unit) and $\eta:AB\to id_\C$ (a counit) such that the compositions $A\xrightarrow{A\epsilon} ABA\xrightarrow{\eta A}A$ and $B\xrightarrow{\epsilon B} BAB\xrightarrow{B\eta} B$ are $id_A$ and $id_B$, respectively.

'''4.''' With the notation in the previous problem, show that $A$ is fully faithful (i.e. $Hom(-,-)\to Hom(A-,A-)$ is an isomorphism) if and only if the unit of adjunction $\epsilon$ is an isomorphism. Similarly, show that $B$ is fully faithful if and only if $\eta$ is an isomorphism. (Hint: use Yoneda's Lemma)

== Set 2 (due Oct. 21) ==
'''1.''' Suppose $X$ and $X'$ are hausdorff topological spaces. Let $T$ and $T'$ denote the topoi of $X$ and $X'$, respectively, using the classical topology. Show that every morphism of topoi $T\to T'$ is induced by a continuous map $X\to X'$. (I'm pretty sure this is true, but I haven't done this exercise) ['''Edit:''' This follows from statement 4.2.3 in [http://ifile.it/ljgv9p/ebooksclub.org__SGA_4_I__Theorie_des_Topos_et_Cohomologie_Etale_des_Schemas__Seminaire_de_Geometrie_Algebrique_du_Bois_Marie_1963_1964__Tome_1.l_83xzo731xt7x6o.djvu SGA4] Expose IV, but I can't find the proof of that statement ... perhaps your French is better than mine.]

'''2.''' (How to pull back representable sheaves) Let $f:\C'\to \C$ be a continuous morphism of sites. Let $Y\in \C'$ be an object, and suppose the functor $h_Y$ is a sheaf. Show that $f^{-1}h_Y\cong h_{f(Y)}$. (Hint: use Yoneda's lemma.)

'''3.''' (Non-functoriality of the lisse-étale topos) For a scheme $X$, the lisse-étale site on $X$ is the category of smooth&dagger; schemes over $X$, where a collection of morphisms over $X$ $\{f_i:U_i\to Y\}$ is said to be a covering if each $f_i$ is étale and the $f_i$ are jointly surjective. We donote the lisse-étale topos of $X$ by $\def\liset{\text{lis-et}}X_\liset$.

Let $\def\O{\mathcal O}\O$ in $\AA^1_\liset$ be given by sending any object $U\to \AA^1$ to $\Gamma(U,\O_U)$ (we will see in class that this is a sheaf). Define a morphism $t\cdot -:\O\to \O$ by multiplication by the coordinate on $\AA^1$ (what does this do on each $U$?). Show that $t\cdot -$ has no kernel.

Let $f:Spec(k)\to \AA^1$ be the inclusion of the origin. We get a continuous morphism of lisse-étale sites $f:\liset(\AA^1)\to\liset(Spec(k))$ given by sending $U\to \AA^1$ to $U\times_{\AA^1}Spec(k)\to Spec(k)$. Show that $f^{-1}:\AA^1_\liset\to Spec(k)_\liset$ takes $t\cdot -$ to a morphism with a non-trivial kernel (Hint: use the previous exercise to compute $f^{-1}\O$). Conclude that $f^{-1}$ does not commute with finite limits.

&dagger;en français, «lisse»

'''Bonus:''' The above problem ($f^{-1}$ not commuting with finite limits) does not occur in the big étale topology. Where does the argument break down? We will see later that the lisse-étale topology has the advantage that $f_*$ usually respects quasi-coherence, a property not enjoyed by the big étale site.

'''4.''' Let $\C$ be a site, and let $X$ be an object in $\C$. Recall that the comma category $\C/X$ inherits the structure of a site.
:(a) Show that there is an equivalence of categories between $Sh(\C/X)$ and $Sh(\C)/h_X$.
:(b) Show that $j^*:Sh(\C)\to Sh(\C)/h_X$, given by $F\mapsto (F\times h_X\xrightarrow{p_2}h_X)$ commutes with finite projective limits and has a right adjoint $j_*$. Therefore, we have a morphism of topoi $Sh(\C)/h_X\to Sh(\C)$.

'''5.''' (Facts about representability) Recall that a morphism of sheaves $\phi:F\to G$ is ''representable'' if for every object $T\in \C$ and every morphism $T\to G$, the fiber product $T\times_G F$ is in $\C$.
:(a) Show that representability is stable under base change.
:(b) Show that a composition of representable morphisms is representable.
:(c) Suppose $F\xrightarrow\phi G\xrightarrow\psi H$ are morphisms of sheaves, where $\psi$ has representable diagonal. Show that if $\psi\circ\phi$ and $\psi$ are representable, then so is $\phi$. (Hint: use the "property P argument")

== Set 3 (due Oct. 28)==
1. Suppose $f:X\to Y$ is a morphism of schemes. If $f$ is surjective as a morphism of schemes, must it be surjective as a morphism of (zariski, étale, smooth, fppf, or fpqc) sheaves? Conversely, if $f$ is surjective as a morphism of sheaves (in one of our topologies), must it be surjective as a morphism of schemes?

2. Suppose $\def\D{\mathcal D}p:\D\to \C$ is a fibered category ''fibered in groupoids'' (i.e. for any object $X$ of $\C$, every morphism in $\D(X)$ is an isomorphism). Show that every arrow of $\D$ is cartesian.

3. The "real" definition of a quasi-coherent sheaf on a site is as follows.
:Let $\O$ be a sheaf of rings on a site $\C$, and let $F$ be an $\O$-module. We say $F$ is ''quasi-coherent'' if for every object $Y$ of $\C$, there is a cover $X\to Y$ so that $F|_{\C/X}$ has a presentation (i.e. $F|_{\C/X}$ is the cokernel of a module morphism $\O^J|_{\C/X}\to \O^I|_{\C/X}$ for some (possibly infinite) sets $I$ and $J$).
*(a) If you have never done so, show that this definition agrees with the other notion of quasi-coherence for the small Zariski topology ($F$ is quasi-coherent if for any open affine $U=Spec(A)$ and any regular function $f\in A$, $F(Spec(A_f))$ is the localization $F(U)_f$).
*(b) Show that the definition of a quasi-coherent big sheaf given in class is the same as the above notion of a quasi-coherent sheaf in the fpqc topology on $Sch$.

4. (Descent for affine morphisms) Suppose $f:X\to Y$ is an fpqc morphism of schemes. Suppose $Z\to X$ is an affine morphism, and there is an isomorphism $Z\times_{X,p_2}(X\times_Y X)\cong Z\times_{X,p_1}(X\times_Y X)$ satisfying the natural cocycle condition. Show that there is an affine morphism $Z_Y\to Y$ so that $Z\cong Z_Y\times_Y X$.

5. (Descent for immersions) Suppose $f:X\to Y$ is an fpqc morphism of schemes. Suppose $U\to X$ is an open immersion such that $U\times_{X,p_1}(X\times_Y X)=U\times_{X,p_2}(X\times_Y X)$. Show that $U$ is the pullback of an open immersion to $Y$. (Hint: consider the closed complement of $U$) Conclude descent for all immersions. (Hint: we showed descent for closed immersions in class)

['''Edit: Bonus.''' (Descent for quasi-affine morphisms) Suppose $f:X\to Y$ is an fpqc morphism of schemes. Suppose $Z\to X$ is a quasi-affine morphism, and there is an isomorphism $Z\times_{X,p_2}(X\times_Y X)\cong Z\times_{X,p_1}(X\times_Y X)$ satisfying the natural cocycle condition. Show that there is a quasi-affine morphism $Z_Y\to Y$ so that $Z\cong Z_Y\times_Y X$. (Hint: use the fact that quasi-affine morphisms have ''canonical'' factorizations as open immersions followed by affine morphisms, and that these factorizations commute with flat base change. Specifically, any quasi-affine morphism $f:Z\to X$ factors as $Z\to Spec_X(f_*\O_Z)\to X$, with the first morphism an open immersion.)]

== Set 4 (due Nov. 4) ==
1. Suppose $\C$ is a site, and $\D$ is a full subcategory of $\C$ which has all finite limits. Let $\D$ have the induced topology (i.e. $\{U_i\to X\}$ is a cover in $\D$ if and only if it is a cover in $\C$). Suppose that every object of $\C$ has a cover by objects in $\D$. Show that the morphism of topoi $Sh(\C)\to Sh(\D)$ induced by the inclusion $\D\to \C$ is an equivalence.

2. Suppose $Spec(A_1)\rightrightarrows Spec(A_0)$ is a finite flat equivalence relation. Let $A=Eq(A_0\rightrightarrows A_1)$. Show that $A_0$ is a finite $A$-module. This was a sticky point in lecture when we were proving that $Spec(A)$ is the quotient of $Spec(A_0)$ by $Spec(A_1)$ in the category of fpqc sheaves.

3. Let $Y$ is a noetherian algebraic space. Let $P$ be a property of algebraic spaces. Suppose that for any closed subspace $Z\subseteq Y$, if every proper closed subspace of $Z$ has $P$, then $Z$ has $P$. Prove that $Y$ has $P$.

4. Let $Y$ be an algebraic space. Prove that $Y$ is an fpqc sheaf. (Hint: we only know one way to show something is an fpqc sheaf.)

== Set 5 (in progress) ==

= Resources =

Jay Daigle is live-TeXing notes for the course. He's posted here [http://jaydaigle.net/math/classes/193a_2011/ here]. Thanks Jay!

<hr>

[http://www.math.columbia.edu/algebraic_geometry/stacks-git/ The stacks project] by Johan de Jong et. al. 
[http://books.google.com/books?id=75lgcAAACAAJ Algebraic spaces] by Donald Knutson 
[http://books.google.com.pe/books?id=RZOe_4CnWqMC&pg=PP1 Champs algébriques] by Gérard Laumon and Laurent Moret-Bailly 
[http://arxiv.org/abs/math/0412512 Notes on Grothendieck topologies, fibered categories and descent theory] by Angelo Vistoli 
[http://math.berkeley.edu/~anton/written/Stacks/Stacks.pdf my notes] from Martin Olsson's course at Berkeley (source available in an [svn://sheafy.net/courses/stacks_sp2007 svn repo]) 
[http://stacky.net/posted/Artin%20-%20Algebraization%20of%20Formal%20Moduli%20I.djvu Algebraization of Formal Moduli I] by Michael Artin

==Bjorn's table==

The following table is taken from page 179 of [http://www-math.mit.edu/~poonen/ Bjorn Poonen's] [http://math.mit.edu/~poonen/papers/Qpoints.pdf Rational points on varieties]. I copied it because it's slightly more accessible to me as a web page, and so that I can add information to it.

{|class="wikitable"
|-
|
|Definition
|Composition
|Base Change
|fpqc Descent
|Spreading Out
|-
|affine
|EGA II, 1.6.1
|EGA II, 1.6.2(ii)
|EGA II, 1.6.2(iii)
|EGA IV2, 2.7.1(xiii)
|EGA IV3, 8.10.5(viii)
|-
|bijective
|
|YES
| bgcolor="pink" | NO
|EGA IV2, 2.6.1(iv)
| bgcolor="pink" | NO
|-
|closed
|EGA I, 2.2.6
|EGA I, 2.2.7(i)
| bgcolor="pink" | NO
|EGA IV2, 2.6.2(ii)
| bgcolor="red" |
|-
|closed immersion
|EGA I, 4.2.1
|EGA I, 4.2.5
|EGA I, 4.3.2
|EGA IV2, 2.7.1(xii)
|EGA IV3, 8.10.5(iv)
|-
|dominant
|EGA I, 2.2.6
|EGA I, 2.2.7(i)
| bgcolor="pink" | NO
| bgcolor="red" |
| bgcolor="red" |
|-
|etale
|EGA IV4, 17.3.1
|EGA IV4, 17.3.3(ii)
|EGA IV4, 17.3.3(ii)
|EGA IV4, 17.7.4(vi)
|EGA IV4, 17.7.8(ii)
|-
|faithfully flat
|EGA I, 0:6.7.8
|YES
|YES
|YES
| bgcolor="red" |
|-
|finite
|EGA II, 6.1.1
|EGA II, 6.1.5(ii)
|EGA II, 6.1.5(iii)
|EGA IV2, 2.7.1(xv)
|EGA IV3, 8.10.5(x)
|-
|finite presentation
|EGA IV1, 1.6.1
|EGA IV1, 1.6.2(ii)
|EGA IV1, 1.6.2(iii)
|EGA IV2, 2.7.1(vi)
| bgcolor="red" |
|-
|finite type
|EGA I, 6.3.1
|EGA I, 6.3.4(ii)
|EGA I, 6.3.4(iv)
|EGA IV2, 2.7.1(v)
| bgcolor="red" |
|-
|flat
|EGA I, 0:6.7.1
|EGA IV2, 2.1.6
|EGA IV2, 2.1.4
|EGA IV2, 2.2.11(iv)
| bgcolor="red" |
|-
|formally etale
|EGA IV4, 17.1.1
|EGA IV4, 17.1.3(ii)
|EGA IV4, 17.1.3(iii)
| bgcolor="red" |
| bgcolor="red" |
|-
|formally smooth
|EGA IV4, 17.1.1
|EGA IV4, 17.1.3(ii)
|EGA IV4, 17.1.3(iii)
| bgcolor="red" |
| bgcolor="red" |
|-
|formally unram.
|EGA IV4, 17.1.1
|EGA IV4, 17.1.3(ii)
|EGA IV4, 17.1.3(iii)
| bgcolor="red" |
| bgcolor="red" |
|-
|fppf
|Definition 3.4.1
|YES
|YES
|YES
| bgcolor="red" |
|-
|fpqc
|[http://arxiv.org/abs/math/0412512 Vis05], 2.34
|[http://arxiv.org/abs/math/0412512 Vis05], 2.35(i)
|[http://arxiv.org/abs/math/0412512 Vis05], 2.35(v)
| bgcolor="red" |
| bgcolor="red" |
|-
|good moduli space
|[http://arxiv.org/abs/0804.2242 Alp08] 4.1
|YES
|[http://arxiv.org/abs/0804.2242 Alp08] 4.7(i)
|[http://arxiv.org/abs/0804.2242 Alp08] 4.7(ii)
| bgcolor="red" |
|-
|homeomorphism
|
|YES
| bgcolor="pink" | NO
|EGA IV2, 2.6.2(iv)
|-
|immersion
|EGA I, 4.2.1
|EGA I, 4.2.5
|EGA I, 4.3.2
| bgcolor="red" |
|EGA IV3, 8.10.5(iii)
|-
|injective
|EGA I, 3.5.11
|YES
| bgcolor="pink" | NO
|EGA IV2, 2.6.1(ii)
| bgcolor="pink" | NO
|-
|isomorphism
|EGA I, 2.2.2
|YES
|YES
|EGA IV2, 2.7.1(viii)
|EGA IV3, 8.10.5(i)
|-
|loc. immersion
|EGA I, 4.5.1
|EGA I, 4.5.5(i)
|EGA I, 4.5.5(iii)
|YES
| bgcolor="red" |
|-
|loc. isomorphism
|EGA I, 4.5.2
|EGA I, 4.5.5(i)
|EGA I, 4.5.5(iii)
| bgcolor="red" |
| bgcolor="red" |
|-
|loc. of finite pres.
|EGA IV1, 1.4.2
|EGA IV1, 1.4.2(ii)
|EGA IV1, 1.4.2(iii)
|EGA IV2, 2.7.1(iv)
| bgcolor="red" |
|-
|loc. of finite type
|EGA I, 6.6.2
|EGA I, 6.6.6(ii)
|EGA I, 6.6.6(iii)
|EGA IV2, 2.7.1(iii)
|-
|monomorphism
|EGA I, 0:4.1.1
|YES
|YES
|EGA IV2, 2.7.1(ix)
|EGA IV3, 8.10.5(ii)
|-
|open
|EGA I, 2.2.6
|EGA I, 2.2.7(i)
| bgcolor="pink" | NO
|EGA IV2, 2.6.2(i)
| bgcolor="red" |
|-
|open immersion
|EGA I, 4.2.1
|EGA I, 4.2.5
|EGA I, 4.3.2
|EGA IV2, 2.7.1(x)
|EGA IV3, 8.10.5(iii)
|-
|projective
|EGA II, 5.5.2
|EGA II, 5.5.5(ii).
|EGA II, 5.5.5(iii)
| bgcolor="pink" | NO.
|EGA IV3, 8.10.5(xiii)
|-
|proper
|EGA II, 5.4.1
|EGA II, 5.4.2(ii)
|EGA II, 5.4.2(iii)
|EGA IV2, 2.7.1(vii)
|EGA IV3, 8.10.5(xii)
|-
|quasi-affine
|EGA II, 5.1.1
|EGA II, 5.1.10(ii)
|EGA II, 5.1.10(iii)
|EGA IV2, 2.7.1(xiv)
|EGA IV3, 8.10.5(ix)
|-
|quasi-compact
|EGA I, 6.6.1
|EGA I, 6.6.4(ii)
|EGA I, 6.6.4(iii)
|EGA IV2, 2.6.4(v)
| bgcolor="red" |
|-
|quasi-finite
|EGA II, 6.2.3
|EGA II, 6.2.4(ii)
|EGA II, 6.2.4(iii)
|EGA IV2, 2.7.1(xvi)
|EGA IV3, 8.10.5(xi)
|-
|quasi-projective
|EGA II, 5.3.1
|EGA II, 5.3.4(ii)
|EGA II, 5.3.4(iii)
| bgcolor="pink" | NO
|EGA IV3, 8.10.5(xiv)
|-
|quasi-separated
|EGA IV1, 1.2.1
|EGA IV1, 1.2.2(ii)
|EGA IV1, 1.2.2(iii)
|EGA IV2, 2.7.1(ii)
| bgcolor="red" |
|-
|radicial
|EGA I, 3.5.4
|EGA I, 3.5.6(i)
|EGA I, 3.5.7(ii)
|EGA IV2, 2.6.1(v)
|EGA IV3, 8.10.5(vii)
|-
|sch.-th. dominant
|EGA IV3, 11.10.2
|YES
| bgcolor="pink" | NO
|EGA IV3, 11.10.5(i)
| bgcolor="red" |
|-
|separated
|EGA I, 5.4.1
|EGA I, 5.5.1(ii)
|EGA I, 5.5.1(iv)
|EGA IV2, 2.7.1(i)
|EGA IV3, 8.10.5(v)
|-
|smooth
|EGA IV4, 17.3.1
|EGA IV4, 17.3.3(ii)
|EGA IV4, 17.3.3(iii)
|EGA IV4, 17.7.4(v)
|EGA IV4, 17.7.8(ii)
|-
|surjective
|EGA I, 2.2.6
|EGA I, 2.2.7(i)
|EGA I, 3.5.2(ii)
|EGA IV2, 2.6.1(i)
|EGA IV3, 8.10.5(vi)
|-
|univ. bicontinuous
|EGA IV2, 2.4.2
|EGA IV2, 2.4.3(ii)
|EGA IV2, 2.4.3(iii)
| bgcolor="red" |
| bgcolor="red" |
|-
|univ. closed
|EGA II, 5.4.9
|EGA IV2, 2.4.3(ii)
|EGA IV2, 2.4.3(iii)
|EGA IV2, 2.6.4(ii)
| bgcolor="red" |
|-
|univ. homeom.
|EGA IV2, 2.4.2
|EGA IV2, 2.4.3(ii)
|EGA IV2, 2.4.3(iii)
|EGA IV2, 2.6.4(iv)
| bgcolor="red" |
|-
|univ. open
|EGA IV2, 2.4.2
|EGA IV2, 2.4.3(ii)
|EGA IV2, 2.4.3(iii)
|EGA IV2, 2.6.4(i)
| bgcolor="red" |
|-
|unramified
|EGA IV4, 17.3.1
|EGA IV4, 17.3.3(ii)
|EGA IV4, 17.3.3(iii)
|EGA IV4, 17.7.4(iv)
|EGA IV4, 17.7.8(ii)
|}

= Possible Topics =

* Grothendieck topologies (using sieves?), topoi
** Descent for sheaves in a site
** bootstrapping properties of objects/morphisms
* Algebraic spaces
** affine/(finite etale) = affine
** Artin's results on modifications of algebraic spaces
* Torsors and $H^1$, gerbes and $H^2$
* Fibered categories
** splittings
** 2-Yoneda lemma
* Algebraic stacks
** criteria for representability
** DM $\Leftrightarrow$ unramified diagonal
** coarse/good moduli space morphisms, Keel-Mori
** DM stacks etale locally quotients by stabilizer
** ZMT, Chow
** valuative criteria
** Artin representability

[[Category:Course Page]]

Math 193a: Algebraic Stacks, Fall 2011

2011-10-24T07:09:32Z

208.106.86.75: /* Set 3 (due Oct. 28) */

The class meets 9–10am MWF in 257 Sloan. Office hours are Wednesday 2–4pm in 374 Sloan (but moving to the common room for tea at 3:30).

= Course Description =

Algebraic stacks arise naturally as solutions to classification (moduli) problems, so it is desirable to understand their geometry. In this course, we will assume a working knowledge of the geometry of schemes. We will extend the definitions and techniques used to study schemes to algebraic spaces and algebraic stacks. Topics will include Grothendieck topologies, descent, algebraic spaces, fibered categories, and algebraic stacks.

Specific topics will be included based on feedback from students.

=Exercises=

Try to do the following problems. If you get stuck, come to my office hours. If you're taking the class for a grade, make sure you hand in a substantial fraction of the exercises (or talk to me about doing a project of some sort). I know some of them are very tedious to write up, so you don't need to hand in everything, but please do attempt all of the problems. Also, please don't hand in solutions to problems that were assigned several weeks ago; for concreteness, let's not hand in solutions to problems in Set $n$ any later than week $n+2$.

<nowiki>$\def\C{\mathcal C}
\def\AA{\mathbb A}
\def\GG{\mathbb G}
\def\O{\mathcal O}$</nowiki>

==Set 1 (due Oct. 14)==
'''1.''' If you have never done so before, prove Yoneda's Lemma: for any category $\C$, taking an object $X\in \C$ to the functor $h_X\colon \C^{op}\to (Set)$ (defined by $h_X(T)=Hom_\C(T,X)$) defines a fully faithful functor $\C\to Func(\C^{op},(Set))$. ['''Edit:''' Actually, I'd like you to show a bit more. Show that for any functor $F:\C^{op}\to (Set)$, we have $Hom(h_X,F)\cong F(X)$.]

'''2.''' Play the game "find the representing object" whenever you get the chance. Determine if the following functors are representable. If they are, find the representing object.
:* The functor $(Top)\to(Set)$ taking a topological space $X$ to the set of open subsets of $X$.
:* The functor $(Top)\to(Set)$ taking a topological space $X$ to the set of closed subsets of $X$.
:* The functor $(Top)\to(Set)$ taking a topological space $X$ to the open subsets of $X$ whose complement is also open.
:* The functor $GL_n:(CommRing)\to(Set)$ taking a commutative ring $A$ to the set of invertible $n\times n$ matrices with entries in $A$. ['''Edit:''' In this case, try to find a ring $R$ so that $Hom(R,A)=GL_n(A)$, rather than $Hom(A,R)$.]
:* The functor $Nil:(CommRing)\to(Set)$ taking a commutative ring $A$ to $\{x\in A| x^n=0$ for some integer $n\}$. ['''Edit:''' In this case, try to find a ring $R$ so that $Hom(R,A)=Nil(A)$, rather than $Hom(A,R)$.]
:* The functor $\AA^n-\{0\}:(Sch)\to(Set)$ taking a scheme $T$ to $\{(f_1,\dots, f_n)\in \O_T(T)^n|$the $f_i$ do not all simultaneously vanish$\}$.
:* The functor $(\AA^n-\{0\})/\GG_m:(Sch)\to(Set)$ taking a scheme $T$ to $(\AA^n-\{0\})(T)/\sim$, where $\sim$ is the equivalence relation $(f_1,\dots, f_n)\sim (f_1',\dots, f_n')$ if there is a unit $u\in \O_T(T)$ such that $f_i'=uf_i$ for each $i$.

'''3.''' Let $A:\C\to \def\D{\mathcal D}$ and $B:\D\to \C$ be functors. Show that an adjunction $Hom(A-,-)\cong \hom(-,B-)$ is equivalent to a choice of natural transformations $\epsilon:id_\D\to BA$ (a unit) and $\eta:AB\to id_\C$ (a counit) such that the compositions $A\xrightarrow{A\epsilon} ABA\xrightarrow{\eta A}A$ and $B\xrightarrow{\epsilon B} BAB\xrightarrow{B\eta} B$ are $id_A$ and $id_B$, respectively.

'''4.''' With the notation in the previous problem, show that $A$ is fully faithful (i.e. $Hom(-,-)\to Hom(A-,A-)$ is an isomorphism) if and only if the unit of adjunction $\epsilon$ is an isomorphism. Similarly, show that $B$ is fully faithful if and only if $\eta$ is an isomorphism. (Hint: use Yoneda's Lemma)

== Set 2 (due Oct. 21) ==
'''1.''' Suppose $X$ and $X'$ are hausdorff topological spaces. Let $T$ and $T'$ denote the topoi of $X$ and $X'$, respectively, using the classical topology. Show that every morphism of topoi $T\to T'$ is induced by a continuous map $X\to X'$. (I'm pretty sure this is true, but I haven't done this exercise) ['''Edit:''' This follows from statement 4.2.3 in [http://ifile.it/ljgv9p/ebooksclub.org__SGA_4_I__Theorie_des_Topos_et_Cohomologie_Etale_des_Schemas__Seminaire_de_Geometrie_Algebrique_du_Bois_Marie_1963_1964__Tome_1.l_83xzo731xt7x6o.djvu SGA4] Expose IV, but I can't find the proof of that statement ... perhaps your French is better than mine.]

'''2.''' (How to pull back representable sheaves) Let $f:\C'\to \C$ be a continuous morphism of sites. Let $Y\in \C'$ be an object, and suppose the functor $h_Y$ is a sheaf. Show that $f^{-1}h_Y\cong h_{f(Y)}$. (Hint: use Yoneda's lemma.)

'''3.''' (Non-functoriality of the lisse-étale topos) For a scheme $X$, the lisse-étale site on $X$ is the category of smooth&dagger; schemes over $X$, where a collection of morphisms over $X$ $\{f_i:U_i\to Y\}$ is said to be a covering if each $f_i$ is étale and the $f_i$ are jointly surjective. We donote the lisse-étale topos of $X$ by $\def\liset{\text{lis-et}}X_\liset$.

Let $\def\O{\mathcal O}\O$ in $\AA^1_\liset$ be given by sending any object $U\to \AA^1$ to $\Gamma(U,\O_U)$ (we will see in class that this is a sheaf). Define a morphism $t\cdot -:\O\to \O$ by multiplication by the coordinate on $\AA^1$ (what does this do on each $U$?). Show that $t\cdot -$ has no kernel.

Let $f:Spec(k)\to \AA^1$ be the inclusion of the origin. We get a continuous morphism of lisse-étale sites $f:\liset(\AA^1)\to\liset(Spec(k))$ given by sending $U\to \AA^1$ to $U\times_{\AA^1}Spec(k)\to Spec(k)$. Show that $f^{-1}:\AA^1_\liset\to Spec(k)_\liset$ takes $t\cdot -$ to a morphism with a non-trivial kernel (Hint: use the previous exercise to compute $f^{-1}\O$). Conclude that $f^{-1}$ does not commute with finite limits.

&dagger;en français, «lisse»

'''Bonus:''' The above problem ($f^{-1}$ not commuting with finite limits) does not occur in the big étale topology. Where does the argument break down? We will see later that the lisse-étale topology has the advantage that $f_*$ usually respects quasi-coherence, a property not enjoyed by the big étale site.

'''4.''' Let $\C$ be a site, and let $X$ be an object in $\C$. Recall that the comma category $\C/X$ inherits the structure of a site.
:(a) Show that there is an equivalence of categories between $Sh(\C/X)$ and $Sh(\C)/h_X$.
:(b) Show that $j^*:Sh(\C)\to Sh(\C)/h_X$, given by $F\mapsto (F\times h_X\xrightarrow{p_2}h_X)$ commutes with finite projective limits and has a right adjoint $j_*$. Therefore, we have a morphism of topoi $Sh(\C)/h_X\to Sh(\C)$.

'''5.''' (Facts about representability) Recall that a morphism of sheaves $\phi:F\to G$ is ''representable'' if for every object $T\in \C$ and every morphism $T\to G$, the fiber product $T\times_G F$ is in $\C$.
:(a) Show that representability is stable under base change.
:(b) Show that a composition of representable morphisms is representable.
:(c) Suppose $F\xrightarrow\phi G\xrightarrow\psi H$ are morphisms of sheaves, where $\psi$ has representable diagonal. Show that if $\psi\circ\phi$ and $\psi$ are representable, then so is $\phi$. (Hint: use the "property P argument")

== Set 3 (due Oct. 28)==
1. Suppose $f:X\to Y$ is a morphism of schemes. If $f$ is surjective as a morphism of schemes, must it be surjective as a morphism of (zariski, étale, smooth, fppf, or fpqc) sheaves? Conversely, if $f$ is surjective as a morphism of sheaves (in one of our topologies), must it be surjective as a morphism of schemes?

2. Suppose $\def\D{\mathcal D}p:\D\to \C$ is a fibered category ''fibered in groupoids'' (i.e. for any object $X$ of $\C$, every morphism in $\D(X)$ is an isomorphism). Show that every arrow of $\D$ is cartesian.

3. The "real" definition of a quasi-coherent sheaf on a site is as follows.
:Let $\O$ be a sheaf of rings on a site $\C$, and let $F$ be an $\O$-module. We say $F$ is ''quasi-coherent'' if for every object $Y$ of $\C$, there is a cover $X\to Y$ so that $F|_{\C/X}$ has a presentation (i.e. $F|_{\C/X}$ is the cokernel of a module morphism $\O^J|_{\C/X}\to \O^I|_{\C/X}$ for some (possibly infinite) sets $I$ and $J$).
*(a) If you have never done so, show that this definition agrees with the other notion of quasi-coherence for the small Zariski topology ($F$ is quasi-coherent if for any open affine $U=Spec(A)$ and any regular function $f\in A$, $F(Spec(A_f))$ is the localization $F(U)_f$).
*(b) Show that the definition of a quasi-coherent big sheaf given in class is the same as the above notion of a quasi-coherent sheaf in the fpqc topology on $Sch$.

4. (Descent for affine morphisms) Suppose $f:X\to Y$ is an fpqc morphism of schemes. Suppose $Z\to X$ is an affine morphism, and there is an isomorphism $Z\times_{X,p_2}(X\times_Y X)\cong Z\times_{X,p_1}(X\times_Y X)$ satisfying the natural cocycle condition. Show that there is an affine morphism $Z_Y\to Y$ so that $Z\cong Z_Y\times_Y X$.

5. (Descent for immersions) Suppose $f:X\to Y$ is an fpqc morphism of schemes. Suppose $U\to X$ is an open immersion such that $U\times_{X,p_1}(X\times_Y X)=U\times_{X,p_2}(X\times_Y X)$. Show that $U$ is the pullback of an open immersion to $Y$. (Hint: consider the closed complement of $U$) Conclude descent for all immersions. (Hint: we showed descent for closed immersions in class)

['''Edit: Bonus.''' (Descent for quasi-affine morphisms) Suppose $f:X\to Y$ is an fpqc morphism of schemes. Suppose $Z\to X$ is a quasi-affine morphism, and there is an isomorphism $Z\times_{X,p_2}(X\times_Y X)\cong Z\times_{X,p_1}(X\times_Y X)$ satisfying the natural cocycle condition. Show that there is a quasi-affine morphism $Z_Y\to Y$ so that $Z\cong Z_Y\times_Y X$. (Hint: use the fact that quasi-affine morphisms have ''canonical'' factorizations as open immersions followed by affine morphisms, and that these factorizations commute with flat base change. Specifically, any quasi-affine morphism $f:Z\to X$ factors as $Z\to Spec_X(f_*\O_Z)\to X$, with the first morphism an open immersion.)]

== Set 4 (due Nov. 4) ==
1. Suppose $\C$ is a site, and $\D$ is a full subcategory of $\C$ which has all finite limits. Let $\D$ have the induced topology (i.e. $\{U_i\to X\}$ is a cover in $\D$ if and only if it is a cover in $\C$). Suppose that every object of $\C$ has a cover by objects in $\D$. Show that the morphism of topoi $Sh(\C)\to Sh(\D)$ induced by the inclusion $\D\to \C$ is an equivalence.

2. Suppose $Spec(A_1)\rightrightarrows Spec(A_0)$ is a finite flat equivalence relation. Let $A=Eq(A_0\rightrightarrows A_1)$. Show that $A_0$ is a finite $A$-module. This was a sticky point in lecture when we were proving that $Spec(A)$ is the quotient of $Spec(A_0)$ by $Spec(A_1)$ in the category of fpqc sheaves.

3. Let $X$ is a noetherian algebraic space. Let $P$ be a property of algebraic spaces. Suppose that for any closed subspace $Z\subseteq X$, if every proper closed subspace of $Z$ has $P$, then $Z$ has $P$. Prove that $X$ has $P$.

== Set 5 (in progress) ==

= Resources =

Jay Daigle is live-TeXing notes for the course. He's posted here [http://jaydaigle.net/math/classes/193a_2011/ here]. Thanks Jay!

<hr>

[http://www.math.columbia.edu/algebraic_geometry/stacks-git/ The stacks project] by Johan de Jong et. al. 
[http://books.google.com/books?id=75lgcAAACAAJ Algebraic spaces] by Donald Knutson 
[http://books.google.com.pe/books?id=RZOe_4CnWqMC&pg=PP1 Champs algébriques] by Gérard Laumon and Laurent Moret-Bailly 
[http://arxiv.org/abs/math/0412512 Notes on Grothendieck topologies, fibered categories and descent theory] by Angelo Vistoli 
[http://math.berkeley.edu/~anton/written/Stacks/Stacks.pdf my notes] from Martin Olsson's course at Berkeley (source available in an [svn://sheafy.net/courses/stacks_sp2007 svn repo]) 
[http://stacky.net/posted/Artin%20-%20Algebraization%20of%20Formal%20Moduli%20I.djvu Algebraization of Formal Moduli I] by Michael Artin

==Bjorn's table==

The following table is taken from page 179 of [http://www-math.mit.edu/~poonen/ Bjorn Poonen's] [http://math.mit.edu/~poonen/papers/Qpoints.pdf Rational points on varieties]. I copied it because it's slightly more accessible to me as a web page, and so that I can add information to it.

{|class="wikitable"
|-
|
|Definition
|Composition
|Base Change
|fpqc Descent
|Spreading Out
|-
|affine
|EGA II, 1.6.1
|EGA II, 1.6.2(ii)
|EGA II, 1.6.2(iii)
|EGA IV2, 2.7.1(xiii)
|EGA IV3, 8.10.5(viii)
|-
|bijective
|
|YES
| bgcolor="pink" | NO
|EGA IV2, 2.6.1(iv)
| bgcolor="pink" | NO
|-
|closed
|EGA I, 2.2.6
|EGA I, 2.2.7(i)
| bgcolor="pink" | NO
|EGA IV2, 2.6.2(ii)
| bgcolor="red" |
|-
|closed immersion
|EGA I, 4.2.1
|EGA I, 4.2.5
|EGA I, 4.3.2
|EGA IV2, 2.7.1(xii)
|EGA IV3, 8.10.5(iv)
|-
|dominant
|EGA I, 2.2.6
|EGA I, 2.2.7(i)
| bgcolor="pink" | NO
| bgcolor="red" |
| bgcolor="red" |
|-
|etale
|EGA IV4, 17.3.1
|EGA IV4, 17.3.3(ii)
|EGA IV4, 17.3.3(ii)
|EGA IV4, 17.7.4(vi)
|EGA IV4, 17.7.8(ii)
|-
|faithfully flat
|EGA I, 0:6.7.8
|YES
|YES
|YES
| bgcolor="red" |
|-
|finite
|EGA II, 6.1.1
|EGA II, 6.1.5(ii)
|EGA II, 6.1.5(iii)
|EGA IV2, 2.7.1(xv)
|EGA IV3, 8.10.5(x)
|-
|finite presentation
|EGA IV1, 1.6.1
|EGA IV1, 1.6.2(ii)
|EGA IV1, 1.6.2(iii)
|EGA IV2, 2.7.1(vi)
| bgcolor="red" |
|-
|finite type
|EGA I, 6.3.1
|EGA I, 6.3.4(ii)
|EGA I, 6.3.4(iv)
|EGA IV2, 2.7.1(v)
| bgcolor="red" |
|-
|flat
|EGA I, 0:6.7.1
|EGA IV2, 2.1.6
|EGA IV2, 2.1.4
|EGA IV2, 2.2.11(iv)
| bgcolor="red" |
|-
|formally etale
|EGA IV4, 17.1.1
|EGA IV4, 17.1.3(ii)
|EGA IV4, 17.1.3(iii)
| bgcolor="red" |
| bgcolor="red" |
|-
|formally smooth
|EGA IV4, 17.1.1
|EGA IV4, 17.1.3(ii)
|EGA IV4, 17.1.3(iii)
| bgcolor="red" |
| bgcolor="red" |
|-
|formally unram.
|EGA IV4, 17.1.1
|EGA IV4, 17.1.3(ii)
|EGA IV4, 17.1.3(iii)
| bgcolor="red" |
| bgcolor="red" |
|-
|fppf
|Definition 3.4.1
|YES
|YES
|YES
| bgcolor="red" |
|-
|fpqc
|[http://arxiv.org/abs/math/0412512 Vis05], 2.34
|[http://arxiv.org/abs/math/0412512 Vis05], 2.35(i)
|[http://arxiv.org/abs/math/0412512 Vis05], 2.35(v)
| bgcolor="red" |
| bgcolor="red" |
|-
|good moduli space
|[http://arxiv.org/abs/0804.2242 Alp08] 4.1
|YES
|[http://arxiv.org/abs/0804.2242 Alp08] 4.7(i)
|[http://arxiv.org/abs/0804.2242 Alp08] 4.7(ii)
| bgcolor="red" |
|-
|homeomorphism
|
|YES
| bgcolor="pink" | NO
|EGA IV2, 2.6.2(iv)
|-
|immersion
|EGA I, 4.2.1
|EGA I, 4.2.5
|EGA I, 4.3.2
| bgcolor="red" |
|EGA IV3, 8.10.5(iii)
|-
|injective
|EGA I, 3.5.11
|YES
| bgcolor="pink" | NO
|EGA IV2, 2.6.1(ii)
| bgcolor="pink" | NO
|-
|isomorphism
|EGA I, 2.2.2
|YES
|YES
|EGA IV2, 2.7.1(viii)
|EGA IV3, 8.10.5(i)
|-
|loc. immersion
|EGA I, 4.5.1
|EGA I, 4.5.5(i)
|EGA I, 4.5.5(iii)
|YES
| bgcolor="red" |
|-
|loc. isomorphism
|EGA I, 4.5.2
|EGA I, 4.5.5(i)
|EGA I, 4.5.5(iii)
| bgcolor="red" |
| bgcolor="red" |
|-
|loc. of finite pres.
|EGA IV1, 1.4.2
|EGA IV1, 1.4.2(ii)
|EGA IV1, 1.4.2(iii)
|EGA IV2, 2.7.1(iv)
| bgcolor="red" |
|-
|loc. of finite type
|EGA I, 6.6.2
|EGA I, 6.6.6(ii)
|EGA I, 6.6.6(iii)
|EGA IV2, 2.7.1(iii)
|-
|monomorphism
|EGA I, 0:4.1.1
|YES
|YES
|EGA IV2, 2.7.1(ix)
|EGA IV3, 8.10.5(ii)
|-
|open
|EGA I, 2.2.6
|EGA I, 2.2.7(i)
| bgcolor="pink" | NO
|EGA IV2, 2.6.2(i)
| bgcolor="red" |
|-
|open immersion
|EGA I, 4.2.1
|EGA I, 4.2.5
|EGA I, 4.3.2
|EGA IV2, 2.7.1(x)
|EGA IV3, 8.10.5(iii)
|-
|projective
|EGA II, 5.5.2
|EGA II, 5.5.5(ii).
|EGA II, 5.5.5(iii)
| bgcolor="pink" | NO.
|EGA IV3, 8.10.5(xiii)
|-
|proper
|EGA II, 5.4.1
|EGA II, 5.4.2(ii)
|EGA II, 5.4.2(iii)
|EGA IV2, 2.7.1(vii)
|EGA IV3, 8.10.5(xii)
|-
|quasi-affine
|EGA II, 5.1.1
|EGA II, 5.1.10(ii)
|EGA II, 5.1.10(iii)
|EGA IV2, 2.7.1(xiv)
|EGA IV3, 8.10.5(ix)
|-
|quasi-compact
|EGA I, 6.6.1
|EGA I, 6.6.4(ii)
|EGA I, 6.6.4(iii)
|EGA IV2, 2.6.4(v)
| bgcolor="red" |
|-
|quasi-finite
|EGA II, 6.2.3
|EGA II, 6.2.4(ii)
|EGA II, 6.2.4(iii)
|EGA IV2, 2.7.1(xvi)
|EGA IV3, 8.10.5(xi)
|-
|quasi-projective
|EGA II, 5.3.1
|EGA II, 5.3.4(ii)
|EGA II, 5.3.4(iii)
| bgcolor="pink" | NO
|EGA IV3, 8.10.5(xiv)
|-
|quasi-separated
|EGA IV1, 1.2.1
|EGA IV1, 1.2.2(ii)
|EGA IV1, 1.2.2(iii)
|EGA IV2, 2.7.1(ii)
| bgcolor="red" |
|-
|radicial
|EGA I, 3.5.4
|EGA I, 3.5.6(i)
|EGA I, 3.5.7(ii)
|EGA IV2, 2.6.1(v)
|EGA IV3, 8.10.5(vii)
|-
|sch.-th. dominant
|EGA IV3, 11.10.2
|YES
| bgcolor="pink" | NO
|EGA IV3, 11.10.5(i)
| bgcolor="red" |
|-
|separated
|EGA I, 5.4.1
|EGA I, 5.5.1(ii)
|EGA I, 5.5.1(iv)
|EGA IV2, 2.7.1(i)
|EGA IV3, 8.10.5(v)
|-
|smooth
|EGA IV4, 17.3.1
|EGA IV4, 17.3.3(ii)
|EGA IV4, 17.3.3(iii)
|EGA IV4, 17.7.4(v)
|EGA IV4, 17.7.8(ii)
|-
|surjective
|EGA I, 2.2.6
|EGA I, 2.2.7(i)
|EGA I, 3.5.2(ii)
|EGA IV2, 2.6.1(i)
|EGA IV3, 8.10.5(vi)
|-
|univ. bicontinuous
|EGA IV2, 2.4.2
|EGA IV2, 2.4.3(ii)
|EGA IV2, 2.4.3(iii)
| bgcolor="red" |
| bgcolor="red" |
|-
|univ. closed
|EGA II, 5.4.9
|EGA IV2, 2.4.3(ii)
|EGA IV2, 2.4.3(iii)
|EGA IV2, 2.6.4(ii)
| bgcolor="red" |
|-
|univ. homeom.
|EGA IV2, 2.4.2
|EGA IV2, 2.4.3(ii)
|EGA IV2, 2.4.3(iii)
|EGA IV2, 2.6.4(iv)
| bgcolor="red" |
|-
|univ. open
|EGA IV2, 2.4.2
|EGA IV2, 2.4.3(ii)
|EGA IV2, 2.4.3(iii)
|EGA IV2, 2.6.4(i)
| bgcolor="red" |
|-
|unramified
|EGA IV4, 17.3.1
|EGA IV4, 17.3.3(ii)
|EGA IV4, 17.3.3(iii)
|EGA IV4, 17.7.4(iv)
|EGA IV4, 17.7.8(ii)
|}

= Possible Topics =

* Grothendieck topologies (using sieves?), topoi
** Descent for sheaves in a site
** bootstrapping properties of objects/morphisms
* Algebraic spaces
** affine/(finite etale) = affine
** Artin's results on modifications of algebraic spaces
* Torsors and $H^1$, gerbes and $H^2$
* Fibered categories
** splittings
** 2-Yoneda lemma
* Algebraic stacks
** criteria for representability
** DM $\Leftrightarrow$ unramified diagonal
** coarse/good moduli space morphisms, Keel-Mori
** DM stacks etale locally quotients by stabilizer
** ZMT, Chow
** valuative criteria
** Artin representability

[[Category:Course Page]]

Math 193a: Algebraic Stacks, Fall 2011

2011-10-24T07:01:09Z

208.106.86.75: /* Set 4 (in progress) */

The class meets 9–10am MWF in 257 Sloan. Office hours are Wednesday 2–4pm in 374 Sloan (but moving to the common room for tea at 3:30).

= Course Description =

Algebraic stacks arise naturally as solutions to classification (moduli) problems, so it is desirable to understand their geometry. In this course, we will assume a working knowledge of the geometry of schemes. We will extend the definitions and techniques used to study schemes to algebraic spaces and algebraic stacks. Topics will include Grothendieck topologies, descent, algebraic spaces, fibered categories, and algebraic stacks.

Specific topics will be included based on feedback from students.

=Exercises=

Try to do the following problems. If you get stuck, come to my office hours. If you're taking the class for a grade, make sure you hand in a substantial fraction of the exercises (or talk to me about doing a project of some sort). I know some of them are very tedious to write up, so you don't need to hand in everything, but please do attempt all of the problems. Also, please don't hand in solutions to problems that were assigned several weeks ago; for concreteness, let's not hand in solutions to problems in Set $n$ any later than week $n+2$.

<nowiki>$\def\C{\mathcal C}
\def\AA{\mathbb A}
\def\GG{\mathbb G}
\def\O{\mathcal O}$</nowiki>

==Set 1 (due Oct. 14)==
'''1.''' If you have never done so before, prove Yoneda's Lemma: for any category $\C$, taking an object $X\in \C$ to the functor $h_X\colon \C^{op}\to (Set)$ (defined by $h_X(T)=Hom_\C(T,X)$) defines a fully faithful functor $\C\to Func(\C^{op},(Set))$. ['''Edit:''' Actually, I'd like you to show a bit more. Show that for any functor $F:\C^{op}\to (Set)$, we have $Hom(h_X,F)\cong F(X)$.]

'''2.''' Play the game "find the representing object" whenever you get the chance. Determine if the following functors are representable. If they are, find the representing object.
:* The functor $(Top)\to(Set)$ taking a topological space $X$ to the set of open subsets of $X$.
:* The functor $(Top)\to(Set)$ taking a topological space $X$ to the set of closed subsets of $X$.
:* The functor $(Top)\to(Set)$ taking a topological space $X$ to the open subsets of $X$ whose complement is also open.
:* The functor $GL_n:(CommRing)\to(Set)$ taking a commutative ring $A$ to the set of invertible $n\times n$ matrices with entries in $A$. ['''Edit:''' In this case, try to find a ring $R$ so that $Hom(R,A)=GL_n(A)$, rather than $Hom(A,R)$.]
:* The functor $Nil:(CommRing)\to(Set)$ taking a commutative ring $A$ to $\{x\in A| x^n=0$ for some integer $n\}$. ['''Edit:''' In this case, try to find a ring $R$ so that $Hom(R,A)=Nil(A)$, rather than $Hom(A,R)$.]
:* The functor $\AA^n-\{0\}:(Sch)\to(Set)$ taking a scheme $T$ to $\{(f_1,\dots, f_n)\in \O_T(T)^n|$the $f_i$ do not all simultaneously vanish$\}$.
:* The functor $(\AA^n-\{0\})/\GG_m:(Sch)\to(Set)$ taking a scheme $T$ to $(\AA^n-\{0\})(T)/\sim$, where $\sim$ is the equivalence relation $(f_1,\dots, f_n)\sim (f_1',\dots, f_n')$ if there is a unit $u\in \O_T(T)$ such that $f_i'=uf_i$ for each $i$.

'''3.''' Let $A:\C\to \def\D{\mathcal D}$ and $B:\D\to \C$ be functors. Show that an adjunction $Hom(A-,-)\cong \hom(-,B-)$ is equivalent to a choice of natural transformations $\epsilon:id_\D\to BA$ (a unit) and $\eta:AB\to id_\C$ (a counit) such that the compositions $A\xrightarrow{A\epsilon} ABA\xrightarrow{\eta A}A$ and $B\xrightarrow{\epsilon B} BAB\xrightarrow{B\eta} B$ are $id_A$ and $id_B$, respectively.

'''4.''' With the notation in the previous problem, show that $A$ is fully faithful (i.e. $Hom(-,-)\to Hom(A-,A-)$ is an isomorphism) if and only if the unit of adjunction $\epsilon$ is an isomorphism. Similarly, show that $B$ is fully faithful if and only if $\eta$ is an isomorphism. (Hint: use Yoneda's Lemma)

== Set 2 (due Oct. 21) ==
'''1.''' Suppose $X$ and $X'$ are hausdorff topological spaces. Let $T$ and $T'$ denote the topoi of $X$ and $X'$, respectively, using the classical topology. Show that every morphism of topoi $T\to T'$ is induced by a continuous map $X\to X'$. (I'm pretty sure this is true, but I haven't done this exercise) ['''Edit:''' This follows from statement 4.2.3 in [http://ifile.it/ljgv9p/ebooksclub.org__SGA_4_I__Theorie_des_Topos_et_Cohomologie_Etale_des_Schemas__Seminaire_de_Geometrie_Algebrique_du_Bois_Marie_1963_1964__Tome_1.l_83xzo731xt7x6o.djvu SGA4] Expose IV, but I can't find the proof of that statement ... perhaps your French is better than mine.]

'''2.''' (How to pull back representable sheaves) Let $f:\C'\to \C$ be a continuous morphism of sites. Let $Y\in \C'$ be an object, and suppose the functor $h_Y$ is a sheaf. Show that $f^{-1}h_Y\cong h_{f(Y)}$. (Hint: use Yoneda's lemma.)

'''3.''' (Non-functoriality of the lisse-étale topos) For a scheme $X$, the lisse-étale site on $X$ is the category of smooth&dagger; schemes over $X$, where a collection of morphisms over $X$ $\{f_i:U_i\to Y\}$ is said to be a covering if each $f_i$ is étale and the $f_i$ are jointly surjective. We donote the lisse-étale topos of $X$ by $\def\liset{\text{lis-et}}X_\liset$.

Let $\def\O{\mathcal O}\O$ in $\AA^1_\liset$ be given by sending any object $U\to \AA^1$ to $\Gamma(U,\O_U)$ (we will see in class that this is a sheaf). Define a morphism $t\cdot -:\O\to \O$ by multiplication by the coordinate on $\AA^1$ (what does this do on each $U$?). Show that $t\cdot -$ has no kernel.

Let $f:Spec(k)\to \AA^1$ be the inclusion of the origin. We get a continuous morphism of lisse-étale sites $f:\liset(\AA^1)\to\liset(Spec(k))$ given by sending $U\to \AA^1$ to $U\times_{\AA^1}Spec(k)\to Spec(k)$. Show that $f^{-1}:\AA^1_\liset\to Spec(k)_\liset$ takes $t\cdot -$ to a morphism with a non-trivial kernel (Hint: use the previous exercise to compute $f^{-1}\O$). Conclude that $f^{-1}$ does not commute with finite limits.

&dagger;en français, «lisse»

'''Bonus:''' The above problem ($f^{-1}$ not commuting with finite limits) does not occur in the big étale topology. Where does the argument break down? We will see later that the lisse-étale topology has the advantage that $f_*$ usually respects quasi-coherence, a property not enjoyed by the big étale site.

'''4.''' Let $\C$ be a site, and let $X$ be an object in $\C$. Recall that the comma category $\C/X$ inherits the structure of a site.
:(a) Show that there is an equivalence of categories between $Sh(\C/X)$ and $Sh(\C)/h_X$.
:(b) Show that $j^*:Sh(\C)\to Sh(\C)/h_X$, given by $F\mapsto (F\times h_X\xrightarrow{p_2}h_X)$ commutes with finite projective limits and has a right adjoint $j_*$. Therefore, we have a morphism of topoi $Sh(\C)/h_X\to Sh(\C)$.

'''5.''' (Facts about representability) Recall that a morphism of sheaves $\phi:F\to G$ is ''representable'' if for every object $T\in \C$ and every morphism $T\to G$, the fiber product $T\times_G F$ is in $\C$.
:(a) Show that representability is stable under base change.
:(b) Show that a composition of representable morphisms is representable.
:(c) Suppose $F\xrightarrow\phi G\xrightarrow\psi H$ are morphisms of sheaves, where $\psi$ has representable diagonal. Show that if $\psi\circ\phi$ and $\psi$ are representable, then so is $\phi$. (Hint: use the "property P argument")

== Set 3 (due Oct. 28)==
1. Suppose $f:X\to Y$ is a morphism of schemes. If $f$ is surjective as a morphism of schemes, must it be surjective as a morphism of (zariski, étale, smooth, fppf, or fpqc) sheaves? Conversely, if $f$ is surjective as a morphism of sheaves (in one of our topologies), must it be surjective as a morphism of schemes?

2. Suppose $\def\D{\mathcal D}p:\D\to \C$ is a fibered category ''fibered in groupoids'' (i.e. for any object $X$ of $\C$, every morphism in $\D(X)$ is an isomorphism). Show that every arrow of $\D$ is cartesian.

3. The "real" definition of a quasi-coherent sheaf on a site is as follows.
:Let $\O$ be a sheaf of rings on a site $\C$, and let $F$ be an $\O$-module. We say $F$ is ''quasi-coherent'' if for every object $Y$ of $\C$, there is a cover $X\to Y$ so that $F|_{\C/X}$ has a presentation (i.e. $F|_{\C/X}$ is the cokernel of a module morphism $\O^J|_{\C/X}\to \O^I|_{\C/X}$ for some (possibly infinite) sets $I$ and $J$).
*(a) If you have never done so, show that this definition agrees with the other notion of quasi-coherence for the small Zariski topology ($F$ is quasi-coherent if for any open affine $U=Spec(A)$ and any regular function $f\in A$, $F(Spec(A_f))$ is the localization $F(U)_f$).
*(b) Show that the definition of a quasi-coherent big sheaf given in class is the same as the above notion of a quasi-coherent sheaf in the fpqc topology on $Sch$.

4. (Descent for affine morphisms) Suppose $f:X\to Y$ is an fpqc morphism of schemes. Suppose $Z\to X$ is an affine morphism, and there is an isomorphism $Z\times_{X,p_2}(X\times_Y X)\cong Z\times_{X,p_1}(X\times_Y X)$ satisfying the natural cocycle condition. Show that there is an affine morphism $Z_Y\to Y$ so that $Z\cong Z_Y\times_Y X$.

5. (Descent for immersions) Suppose $f:X\to Y$ is an fpqc morphism of schemes. Suppose $U\to X$ is an open immersion such that $U\times_{X,p_1}(X\times_Y X)=U\times_{X,p_2}(X\times_Y X)$. Show that $U$ is the pullback of an open immersion to $Y$. (Hint: consider the closed complement of $U$) Conclude descent for all immersions. (Hint: we showed descent for closed immersions in class)

== Set 4 (due Nov. 4) ==
1. Suppose $\C$ is a site, and $\D$ is a full subcategory of $\C$ which has all finite limits. Let $\D$ have the induced topology (i.e. $\{U_i\to X\}$ is a cover in $\D$ if and only if it is a cover in $\C$). Suppose that every object of $\C$ has a cover by objects in $\D$. Show that the morphism of topoi $Sh(\C)\to Sh(\D)$ induced by the inclusion $\D\to \C$ is an equivalence.

2. Suppose $Spec(A_1)\rightrightarrows Spec(A_0)$ is a finite flat equivalence relation. Let $A=Eq(A_0\rightrightarrows A_1)$. Show that $A_0$ is a finite $A$-module. This was a sticky point in lecture when we were proving that $Spec(A)$ is the quotient of $Spec(A_0)$ by $Spec(A_1)$ in the category of fpqc sheaves.

3. Let $X$ is a noetherian algebraic space. Let $P$ be a property of algebraic spaces. Suppose that for any closed subspace $Z\subseteq X$, if every proper closed subspace of $Z$ has $P$, then $Z$ has $P$. Prove that $X$ has $P$.

== Set 5 (in progress) ==

= Resources =

Jay Daigle is live-TeXing notes for the course. He's posted here [http://jaydaigle.net/math/classes/193a_2011/ here]. Thanks Jay!

<hr>

[http://www.math.columbia.edu/algebraic_geometry/stacks-git/ The stacks project] by Johan de Jong et. al. 
[http://books.google.com/books?id=75lgcAAACAAJ Algebraic spaces] by Donald Knutson 
[http://books.google.com.pe/books?id=RZOe_4CnWqMC&pg=PP1 Champs algébriques] by Gérard Laumon and Laurent Moret-Bailly 
[http://arxiv.org/abs/math/0412512 Notes on Grothendieck topologies, fibered categories and descent theory] by Angelo Vistoli 
[http://math.berkeley.edu/~anton/written/Stacks/Stacks.pdf my notes] from Martin Olsson's course at Berkeley (source available in an [svn://sheafy.net/courses/stacks_sp2007 svn repo]) 
[http://stacky.net/posted/Artin%20-%20Algebraization%20of%20Formal%20Moduli%20I.djvu Algebraization of Formal Moduli I] by Michael Artin

==Bjorn's table==

The following table is taken from page 179 of [http://www-math.mit.edu/~poonen/ Bjorn Poonen's] [http://math.mit.edu/~poonen/papers/Qpoints.pdf Rational points on varieties]. I copied it because it's slightly more accessible to me as a web page, and so that I can add information to it.

{|class="wikitable"
|-
|
|Definition
|Composition
|Base Change
|fpqc Descent
|Spreading Out
|-
|affine
|EGA II, 1.6.1
|EGA II, 1.6.2(ii)
|EGA II, 1.6.2(iii)
|EGA IV2, 2.7.1(xiii)
|EGA IV3, 8.10.5(viii)
|-
|bijective
|
|YES
| bgcolor="pink" | NO
|EGA IV2, 2.6.1(iv)
| bgcolor="pink" | NO
|-
|closed
|EGA I, 2.2.6
|EGA I, 2.2.7(i)
| bgcolor="pink" | NO
|EGA IV2, 2.6.2(ii)
| bgcolor="red" |
|-
|closed immersion
|EGA I, 4.2.1
|EGA I, 4.2.5
|EGA I, 4.3.2
|EGA IV2, 2.7.1(xii)
|EGA IV3, 8.10.5(iv)
|-
|dominant
|EGA I, 2.2.6
|EGA I, 2.2.7(i)
| bgcolor="pink" | NO
| bgcolor="red" |
| bgcolor="red" |
|-
|etale
|EGA IV4, 17.3.1
|EGA IV4, 17.3.3(ii)
|EGA IV4, 17.3.3(ii)
|EGA IV4, 17.7.4(vi)
|EGA IV4, 17.7.8(ii)
|-
|faithfully flat
|EGA I, 0:6.7.8
|YES
|YES
|YES
| bgcolor="red" |
|-
|finite
|EGA II, 6.1.1
|EGA II, 6.1.5(ii)
|EGA II, 6.1.5(iii)
|EGA IV2, 2.7.1(xv)
|EGA IV3, 8.10.5(x)
|-
|finite presentation
|EGA IV1, 1.6.1
|EGA IV1, 1.6.2(ii)
|EGA IV1, 1.6.2(iii)
|EGA IV2, 2.7.1(vi)
| bgcolor="red" |
|-
|finite type
|EGA I, 6.3.1
|EGA I, 6.3.4(ii)
|EGA I, 6.3.4(iv)
|EGA IV2, 2.7.1(v)
| bgcolor="red" |
|-
|flat
|EGA I, 0:6.7.1
|EGA IV2, 2.1.6
|EGA IV2, 2.1.4
|EGA IV2, 2.2.11(iv)
| bgcolor="red" |
|-
|formally etale
|EGA IV4, 17.1.1
|EGA IV4, 17.1.3(ii)
|EGA IV4, 17.1.3(iii)
| bgcolor="red" |
| bgcolor="red" |
|-
|formally smooth
|EGA IV4, 17.1.1
|EGA IV4, 17.1.3(ii)
|EGA IV4, 17.1.3(iii)
| bgcolor="red" |
| bgcolor="red" |
|-
|formally unram.
|EGA IV4, 17.1.1
|EGA IV4, 17.1.3(ii)
|EGA IV4, 17.1.3(iii)
| bgcolor="red" |
| bgcolor="red" |
|-
|fppf
|Definition 3.4.1
|YES
|YES
|YES
| bgcolor="red" |
|-
|fpqc
|[http://arxiv.org/abs/math/0412512 Vis05], 2.34
|[http://arxiv.org/abs/math/0412512 Vis05], 2.35(i)
|[http://arxiv.org/abs/math/0412512 Vis05], 2.35(v)
| bgcolor="red" |
| bgcolor="red" |
|-
|good moduli space
|[http://arxiv.org/abs/0804.2242 Alp08] 4.1
|YES
|[http://arxiv.org/abs/0804.2242 Alp08] 4.7(i)
|[http://arxiv.org/abs/0804.2242 Alp08] 4.7(ii)
| bgcolor="red" |
|-
|homeomorphism
|
|YES
| bgcolor="pink" | NO
|EGA IV2, 2.6.2(iv)
|-
|immersion
|EGA I, 4.2.1
|EGA I, 4.2.5
|EGA I, 4.3.2
| bgcolor="red" |
|EGA IV3, 8.10.5(iii)
|-
|injective
|EGA I, 3.5.11
|YES
| bgcolor="pink" | NO
|EGA IV2, 2.6.1(ii)
| bgcolor="pink" | NO
|-
|isomorphism
|EGA I, 2.2.2
|YES
|YES
|EGA IV2, 2.7.1(viii)
|EGA IV3, 8.10.5(i)
|-
|loc. immersion
|EGA I, 4.5.1
|EGA I, 4.5.5(i)
|EGA I, 4.5.5(iii)
|YES
| bgcolor="red" |
|-
|loc. isomorphism
|EGA I, 4.5.2
|EGA I, 4.5.5(i)
|EGA I, 4.5.5(iii)
| bgcolor="red" |
| bgcolor="red" |
|-
|loc. of finite pres.
|EGA IV1, 1.4.2
|EGA IV1, 1.4.2(ii)
|EGA IV1, 1.4.2(iii)
|EGA IV2, 2.7.1(iv)
| bgcolor="red" |
|-
|loc. of finite type
|EGA I, 6.6.2
|EGA I, 6.6.6(ii)
|EGA I, 6.6.6(iii)
|EGA IV2, 2.7.1(iii)
|-
|monomorphism
|EGA I, 0:4.1.1
|YES
|YES
|EGA IV2, 2.7.1(ix)
|EGA IV3, 8.10.5(ii)
|-
|open
|EGA I, 2.2.6
|EGA I, 2.2.7(i)
| bgcolor="pink" | NO
|EGA IV2, 2.6.2(i)
| bgcolor="red" |
|-
|open immersion
|EGA I, 4.2.1
|EGA I, 4.2.5
|EGA I, 4.3.2
|EGA IV2, 2.7.1(x)
|EGA IV3, 8.10.5(iii)
|-
|projective
|EGA II, 5.5.2
|EGA II, 5.5.5(ii).
|EGA II, 5.5.5(iii)
| bgcolor="pink" | NO.
|EGA IV3, 8.10.5(xiii)
|-
|proper
|EGA II, 5.4.1
|EGA II, 5.4.2(ii)
|EGA II, 5.4.2(iii)
|EGA IV2, 2.7.1(vii)
|EGA IV3, 8.10.5(xii)
|-
|quasi-affine
|EGA II, 5.1.1
|EGA II, 5.1.10(ii)
|EGA II, 5.1.10(iii)
|EGA IV2, 2.7.1(xiv)
|EGA IV3, 8.10.5(ix)
|-
|quasi-compact
|EGA I, 6.6.1
|EGA I, 6.6.4(ii)
|EGA I, 6.6.4(iii)
|EGA IV2, 2.6.4(v)
| bgcolor="red" |
|-
|quasi-finite
|EGA II, 6.2.3
|EGA II, 6.2.4(ii)
|EGA II, 6.2.4(iii)
|EGA IV2, 2.7.1(xvi)
|EGA IV3, 8.10.5(xi)
|-
|quasi-projective
|EGA II, 5.3.1
|EGA II, 5.3.4(ii)
|EGA II, 5.3.4(iii)
| bgcolor="pink" | NO
|EGA IV3, 8.10.5(xiv)
|-
|quasi-separated
|EGA IV1, 1.2.1
|EGA IV1, 1.2.2(ii)
|EGA IV1, 1.2.2(iii)
|EGA IV2, 2.7.1(ii)
| bgcolor="red" |
|-
|radicial
|EGA I, 3.5.4
|EGA I, 3.5.6(i)
|EGA I, 3.5.7(ii)
|EGA IV2, 2.6.1(v)
|EGA IV3, 8.10.5(vii)
|-
|sch.-th. dominant
|EGA IV3, 11.10.2
|YES
| bgcolor="pink" | NO
|EGA IV3, 11.10.5(i)
| bgcolor="red" |
|-
|separated
|EGA I, 5.4.1
|EGA I, 5.5.1(ii)
|EGA I, 5.5.1(iv)
|EGA IV2, 2.7.1(i)
|EGA IV3, 8.10.5(v)
|-
|smooth
|EGA IV4, 17.3.1
|EGA IV4, 17.3.3(ii)
|EGA IV4, 17.3.3(iii)
|EGA IV4, 17.7.4(v)
|EGA IV4, 17.7.8(ii)
|-
|surjective
|EGA I, 2.2.6
|EGA I, 2.2.7(i)
|EGA I, 3.5.2(ii)
|EGA IV2, 2.6.1(i)
|EGA IV3, 8.10.5(vi)
|-
|univ. bicontinuous
|EGA IV2, 2.4.2
|EGA IV2, 2.4.3(ii)
|EGA IV2, 2.4.3(iii)
| bgcolor="red" |
| bgcolor="red" |
|-
|univ. closed
|EGA II, 5.4.9
|EGA IV2, 2.4.3(ii)
|EGA IV2, 2.4.3(iii)
|EGA IV2, 2.6.4(ii)
| bgcolor="red" |
|-
|univ. homeom.
|EGA IV2, 2.4.2
|EGA IV2, 2.4.3(ii)
|EGA IV2, 2.4.3(iii)
|EGA IV2, 2.6.4(iv)
| bgcolor="red" |
|-
|univ. open
|EGA IV2, 2.4.2
|EGA IV2, 2.4.3(ii)
|EGA IV2, 2.4.3(iii)
|EGA IV2, 2.6.4(i)
| bgcolor="red" |
|-
|unramified
|EGA IV4, 17.3.1
|EGA IV4, 17.3.3(ii)
|EGA IV4, 17.3.3(iii)
|EGA IV4, 17.7.4(iv)
|EGA IV4, 17.7.8(ii)
|}

= Possible Topics =

* Grothendieck topologies (using sieves?), topoi
** Descent for sheaves in a site
** bootstrapping properties of objects/morphisms
* Algebraic spaces
** affine/(finite etale) = affine
** Artin's results on modifications of algebraic spaces
* Torsors and $H^1$, gerbes and $H^2$
* Fibered categories
** splittings
** 2-Yoneda lemma
* Algebraic stacks
** criteria for representability
** DM $\Leftrightarrow$ unramified diagonal
** coarse/good moduli space morphisms, Keel-Mori
** DM stacks etale locally quotients by stabilizer
** ZMT, Chow
** valuative criteria
** Artin representability

[[Category:Course Page]]

TeXnical notes

2011-09-25T23:06:05Z

208.106.86.75: /* How to draw the Fox-Artin wild arc with pstricks */

Sometimes I find a solution to a TeXnical problem, and I think everybody should know about it. Some of these things are totally obvious, but (as far as I know) non-standard.

==BibTeX without a separate .bib file==

[http://stacky.net/files/selfcontainedBibTeX.dvi dvi] [http://stacky.net/files/selfcontainedBibTeX.tex tex]

==How to draw the Fox-Artin wild arc with pstricks==

<code><pre>
\documentclass{article}
\usepackage{pstricks}
\usepackage{multido}
\begin{document}
\[\begin{pspicture}(-7.6,-.5)(7.32,2.5)
\newdimen\totaljump % This measures where the orgin is
\newdimen\jumpinterval % This measures how much the origin moves each time
%%%%% First we draw the left hand side. Because we are utilizing borders, we have %%%%%
%%%%% to draw from left to right, so compute by hand what the unit should be. %%%%%
\psset{unit=0.17293822569mm,border=.05,linewidth=.03}
\totaljump=-75mm
\jumpinterval=0.17293822569mm
\multido{}{20}{
% Move the origin to the appropriate place %%%%
\psset{origin={\totaljump,0}, unit= 1.25, border=.05, linewidth=.03}
% Draw a piece of the curve
\pscurve(-.1,.1)(-.2,.3)(0,1.5)(1,.7)(2.3,.3)(2,-.05)(1.1,-.2)(1,-.1)
% Scale the jumpinterval by .8 and increment totaljump
\multiply\jumpinterval by 5 \divide\jumpinterval by 4
\advance\totaljump by \jumpinterval
}
%%%% Now we draw one piece of the curve in the middle %%%%
%%%% to get the two ends to match up nicely. %%%%
\psset{unit=1.5cm,border=.05,linewidth=.03}
\pscurve(-.15,.12)(-.2,.3)(0,1.7)(1,.7)(1.8,.3)(1.6,-.05)(1.1,-.2)(.95,-.15)
%%%% Now draw the right hand side %%%%
\psset{origin={12mm,0}}
\totaljump=12mm
\jumpinterval=12mm
\multido{}{20}{
\advance\totaljump by \jumpinterval
\multiply\jumpinterval by 4 \divide\jumpinterval by 5
\pscurve(-.1,.1)(-.2,.3)(0,1.5)(1,.7)(1.6,.3)(1.4,-.05)(1.1,-.2)(.9,-.1)
\psset{origin={\totaljump,0}, unit= .8, border=.05, linewidth=.03}
}
\end{pspicture}\]
\end{document}
</pre></code>
[[File:Fox-artin.png]]

==Placing labels on arrows in XY-pic==
(I learned this trick from [http://www.math.columbia.edu/~lauda/ Aaron Lauda]; I haven't seen it documented anywhere)
<pre>\xymatrix{asdfasdf \ar[r]^{f} & a }</pre>
produces the label "f" in a stupid place, half-way between the centers of the two entries, instead of where you'd like it to be, half-way along the arrow. One way to handle this is to do something like
<pre>\ar[r]^(.7){f}</pre>
but that is fairly unsatisfying because you have to calibrate the (.7) by eye. There is another way, which is to use
<pre>\ar[r]^-{f}</pre>
which will automatically place the label "f" half-way along the arrow. If you want the label .7 of the way ''along the arrow'', you can do
<pre>\ar[r]^-(.7){f}</pre>

Working with MediaWiki

2011-09-25T01:14:21Z

208.106.86.75:

Edit sidebar by visiting the [[Mediawiki:Sidebar]] page.

== Getting MathJax Working ==

I got MathJax: <code>git clone git://github.com/mathjax/MathJax.git MathJax</code>

I installed the JavaScript extension[http://www.mediawiki.org/wiki/Extension:Javascript] for MediaWiki.

I added a file <code>mathjax.js</code> (name doesn't matter) to <code>extensions/JavaScript</code>. It's content:
<code><pre>
var e = document.createElement('script');
e.type = "text/javascript";
e.src = "http://stacky.net/MathJax/MathJax.js?config=MOconfig";
document.getElementsByTagName('head')[0].appendChild(e);
</pre></code>

(I copied the MathJax configuration file used at MO)

== (Not) Getting Blahtex Working ==

It'd be much nicer to get MediaWiki to serve MathML and use MathJax to convert only if it has to, but I haven't been able to get Blahtex[http://www.mediawiki.org/wiki/Blahtex] to compile. I've been trying to follow the instructions here [http://www.mediawiki.org/wiki/Extension:Blahtex/Embedding_Blahtex_in_MediaWiki]. I'm able to checkout blahtex and to get texvc working, but if I try <code>make</code> or <code>make linux</code>, I get
<code><pre>
g++ -O3 -c -o source/main.o source/main.cpp
source/main.cpp: In function ‘void ShowUsage()’:
source/main.cpp:108: error: ‘exit’ was not declared in this scope
make: *** [source/main.o] Error 1
</pre></code>

Toric Geometry Seminar

2011-09-14T20:03:34Z

208.106.86.75:

Sign up for the mailing list [https://hermes.ugcs.caltech.edu/cgi-bin/mailman/listinfo/toric here].

We meet at TBA in Sloan TBA.

== About Toric Varieties ==

A toric variety is a normal variety $X$ with a dense open algebraic torus $T\subseteq X$ so that $T$ acts on $X$ (in a way that extends $T$'s action on itself). People often imagine fixing $T$ (i.e. fixing a dimension), and then considering various ways of "partially compactifying" it to a toric variety. $T$ can be used to "sniff out" properties of $X$. For example, a regular function on $X$ restricts to a regular function on $T$. We (will) understand regular functions on $T$ very well, so a natural question to ask is, "which regular functions on $T$ come from regular functions on $X$?" By asking and answering these types of questions, one can show that toric varieties are determined by a combinatorial wigit, called a fan.

Much of toric geometry is devoted to building the dictionary between the combinatorics of fans (which are easy to think about) and the geometry of the corresponding toric varieties (which, at least initially, are hard to think about). The advantage of having such a dictionary at your command is that it allows you to quickly generate examples and perform calculations. Even if you are thinking about problems on non-toric varieties, working a few toric examples often reveals key insights. This approach is used to crack hard problems in algebraic geometry, but can also be used when learning the basics of the field.

==Schedule==

{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"
|-
| '''Date'''
| '''Speaker'''
| '''Topic'''
|-
| The Future
| Speaker
| Topic
|}

== Possible Topics ==

* building a fan out of a toric variety
* building a toric variety out of a fan
* Weil divisors, computing the class group
* Cartier divisors, line bundles
* cohomology

== Questions ==

* Is Chow's lemma for toric varieties clear? That is, given a fan, can it always be made dual to a polytope by subdividing some of the cones?

== References ==

* Cox, Little, and Schenk: [http://www.cs.amherst.edu/~dac/toric.html Toric Varieties]
* Fulton: [http://www.amazon.com/Introduction-Varieties-AM-131-William-Fulton/dp/0691000492 Introduction to Toric Varieties]

[http://ifile.it/anbie1/ebooksclub.org__toric_varieties.pdf .]

Computing

2011-09-14T17:09:23Z

208.106.86.75: /* xmodmap */

==[http://www.debian.org/doc/manuals/apt-howto/ch-search.en.html#s-apt-file Apt]==
Includes handy things like
* find which package contains <code>empheq.sty</code> with <code>apt-file search empheq.sty</code>

==xmodmap==
To remap the left and right arrow of Lacra's pointer to the down and right arrows, respectively, did 
<code>
xmodmap -e "keycode 117 = Right"
xmodmap -e "keycode 112 = Down"
</code>
found the keycodes useing <code>xev</code>. This list of symbol names may be useful: [http://wiki.linuxquestions.org/wiki/List_of_Keysyms_Recognised_by_Xmodmap]

Computing

2011-09-14T03:40:23Z

208.106.86.75: