This is a topics course for Mathematics graduate students at the University of Missouri-Columbia for Fall 2012.
Instructor: Jason Lo
Office location: Room 320A, Math Sci Building
Department webpage: www.math.missouri.edu/~locc
Class times: 3-3.50pm, Mon/Wed/Fri; Room 110 of Math Sci Building
Office hours: 4-5pm, Mon/Wed or by appointment
Course description (formal): We will start with the basics of abelian categories and homological algebra, and gradually move towards defining the derived category of an abelian category, which is an example of a triangulated category. We will also consider t-structures and tilting.
The second part of the course will be on applications of derived categories. We will consider derived categories of coherent sheaves and stability conditions, which is motivated by mirror symmetry in physics.
This course will be self-contained, and the only pre-requisites are familiarity with graduate-level commutative algebra (groups, rings, modules, localisation of rings) and basic point-set topology. All the notions on varieties, sheaves and quivers that are needed will be defined in this course when necessary.
Course description (informal): The goal of this course is for you to get a taste of what it is like to work with derived categories, rather than to know everything there is to know about the basics of derived categories (which are all written down in many good textbooks anyway).
What this means is that we will not have time to cover every single important detail a standard textbook would cover. In particular:
- For certain things, I might explain them in different ways, at different points during the course. This way, you have more than one way to think of them.
- Sometimes, I might focus on examples for a while so that we have time to let the new concepts sink in.
- In general, I will not be introducing concepts or results in a linear fashion as a textbook would. For instance, when I introduce the axioms of a triangulated category, I might do examples that make use of results which will be explained later in the course.
Tentative course plan: I will start by telling you what an abelian category is (you actually know it already: just look at the category R-mod of modules over a ring R), and then talk about chain complexes, homotopy equivalence, etc. This will bring us to the homotopy category K(A) and the derived category D(A) of an abelian category A.
Morphisms in the derived category can seem rather odd at first, but we will get to know them. Both K(A) and D(A) are examples of triangulated categories, and we will look at the axioms of a triangulated category. In particular, I will make the point that the octahedral axiom is fun to draw and is more useful than some people might have you believe.
In order to have even richer examples we can work with, we will spend some time learning about sheaves, schemes and varieties, and quasi-coherent and coherent sheaves. (We will focus on concrete examples of varieties and sheaves, so this will be a non-rigorous discussion of these things. And so, there is no need to be deterred by the mention of these words.) By the end of this, we will know a very important example of an abelian category in algebraic geometry, the category Coh(X) of coherent sheaves on a variety X.
By now, having the two important abelian categories R-mod and Coh(X) as examples to work with, we will think about truncations, t-structures and tilting. This will set the ground for the final part of the course, where we can talk about stability conditions and moduli spaces, which is a very active current area of research, with a busy fusion of ideas, both new and old.
Grading: We will aim to have five assignments, each worth 20% of the total grade.
Reference materials: The course is not based on any particular textbook or article (perhaps a mixture of them). Books and articles that you might find useful include:
- Books:
- Fourier-Mukai Transforms in Algebraic Geometry, by Daniel Huybrechts
- Fourier-Mukai and Nahm Transforms in Geometry and Mathematical Physics, by Claudio Bartocci, Ugo Bruzzo and Daniel Hernandez Ruiperez
- An Introduction to Homological Algebra, by Charles Weibel
- Articles:
