Math 193a: Algebraic Stacks, Fall 2011

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 either 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 $n+2$.

$\def\C{\mathcal C} \def\AA{\mathbb A} \def\GG{\mathbb G} \def\O{\mathcal O}$

Set 1

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 $A$.
• The functor $Nil:(CommRing)\to(Set)$ taking a commutative ring $A$ to $\{x\in A| x^n=0$ for some integer $n\}$.
• 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)

likely problems for the next set (in progress)

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)

2. (example showing non-functoriality of the lisse-etale topos)

Resources

The stacks project by Johan de Jong et. al.
Algebraic spaces by Donald Knutson
Champs algébriques by Gérard Laumon and Laurent Moret-Bailly
Notes on Grothendieck topologies, fibered categories and descent theory by Angelo Vistoli
my notes from Martin Olsson's course at Berkeley (source available in an svn repo)
Algebraization of Formal Moduli I by Michael Artin

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