Behrend K. Introduction to Algebraic Stacks

Lecture notes. — Behrend Kai, 2012. — 127 p.These are lecture notes based on a short course on stacks given at the Newton Institute in Cambridge in January 2011. They form a self-contained introduction to some of the basic ideas of stack theory. Stacks and algebraic stacks were invented by the Grothendieck school of algebraic geometry in the 1960s. One purpose, was to give geometric meaning to higher cohomology classes. The other, was to develop are more general framework for studying moduli problems. It is the latter aspect that interests us in these notes. Since the 1980s, stacks have become an increasingly important tool in geometry, topology and theoretical physics. Stack theory examines how mathematical objects can vary in families. For our examples, the mathematical objects will be the triangles familiar from Euclidean geometry, and closely related concepts. At least to begin with, we will let these vary in continuous families, parametrized by topological spaces. A surprising number of stacky phenomena can be seen in such simple cases. (In fact, one of the founders of the theory of algebraic stacks, M. Artin, is famously reputed to have said that one need only understand the stack of triangles to understand stacks.) These lecture notes are divided into three parts. The first is a very leisurely and elementary introduction to stacks, introducing the main ideas by considering a few elementary examples of topological stacks. The only prerequisites for this part are basic undergraduate courses in abstract algebra (groups and group actions) and topology (topological spaces, covering spaces, the fundamental group).
The second part introduces the basic formalism of stacks. The prerequisites are the same, although this part is more demanding than the previous.
The third part introduces algebraic stacks, culminating in the Riemann Roch theorem for stacky curves. The prerequisite here is some basic scheme theory.Topological stacks: Triangles
Families and their symmetry groupoids
Various types of symmetry groupoids
Continuous families
Gluing families
Classification
Pulling back families
The moduli map of a family
Fine moduli spaces
Coarse moduli spaces
Scalene triangles
Isosceles triangles
Equilateral triangles
Oriented triangles
Non-equilateral oriented triangles
Stacks
Versal families
Generalized moduli maps
Reconstructing a family from its generalized moduli map
Versal families: definition
Isosceles triangles
Equilateral triangles
Oriented triangles
Degenerate triangles
Lengths of sides viewpoint
Embedded viewpoint
Complex viewpoint
Oriented degenerate triangles
Change of versal family
Oriented triangles by projecting equilateral ones
Comparison
The comparison theorem
Weierstrass compactification
The j-plane