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. We will give lots of
motivation, examples, and applications.
Topics will include Grothendieck topologies, descent, algebraic spaces, fibered categories, algebraic stacks, quotient stacks, deformation theory, torsors and gerbes.
Additional topics will be included based on feedback from students.
There will be no exams or final. There will be lots of homework. N.B.
Instead, there will be an additional extra weekly meeting/office hours
(to be set once the semester begins) in which we will discuss the
homework problems and review background material.
The notes for this course are a modified copy of the notes that Anton
Geraschenko live-TeXed from Martin Olsson's Spring 2007 course on
stacks at UC Berkeley. We will not follow these notes exactly; I class
I am planning longer digressions on background material, and will omit
many of the topics of Martin's notes.
Please, whenever possible, do me (and classmates, rest of the world,
etc) a favor and send me a short email whenever you find an error in
the notes [and there are errors!]. (Better -- if you replace
`Stacks.pdf' in the url with, e.g., `StackLec12.tex', this will
produce the source file for that lecture. )
Highly recommended reading -- to get a sense of the various
perspectives on and roles played by stacks (and algebraic spaces), please take a look at each
of the following. (More to come.)
Introduction to Algebraic spaces by Donald Knutson
Preface to Kresch et al Wikipedia page and its external links
Ravi Vakil's second lecture (numbered 21/2) from Anton's notes
from the MSRI deformation theory workshop What is