Revision as of 20:57, 27 September 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

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

Set 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 $\C\to Func(\C^{op},(Set))$.
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 $i$.

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

@@ Line 1: / Line 1: @@
 The class meets 9&ndash;10am MWF in 257 Sloan. Office hours are Wednesday 2&ndash;4pm in 374 Sloan (but moving to the common room for tea at 3:30).
-== Course Description ==
+= 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.
@@ Line 7: / Line 7: @@
 Specific topics will be included based on feedback from students.
-== Resources ==
+=Exercises=
+<nowiki>$\def\C{\mathcal C}
+\def\AA{\mathbb A}
+\def\GG{\mathbb G}
+\def\O{\mathcal O}$</nowiki>
+==Set 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 $\C\to Func(\C^{op},(Set))$.
+# 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 $i$.
+= Resources =
 [http://www.math.columbia.edu/algebraic_geometry/stacks-git/ The stacks project] by Johan de Jong et. al.<br>
@@ Line 16: / Line 32: @@
 [http://stacky.net/posted/Artin%20-%20Algebraization%20of%20Formal%20Moduli%20I.djvu Algebraization of Formal Moduli I] by Michael Artin
-== Possible Topics ==
+= Possible Topics =
 * Grothendieck topologies (using sieves?), topoi