MATH 737 -- Topics in Algebraic Geometry: An introduction to sheaf and scheme theory

Fall 2005

Instructor: Katia Consani
Office: 216 Krieger Hall
Phone: (410) 516-5116.
Email: kc@math.jhu.edu

Class Times: MW, 2:30-3:45 pm.
Room: GIL 48 (first class: Monday September 12).

References: There are no required textbooks for the course, but the following references may be useful.

S. Mac Lane, Categories for working mathematicians , Springer, 1998.

S.I. Gelfand, Y.I. Manin, Methods of Homological Algebra , Springer monographs in math. series , 2nd ed., 2003. ISBN 3540435832.

R. Godement, Topologie algebrique et theorie des fascieaux .

R. Hartshorne, Algebraic Geometry , Springer, 1977.

M. Kashiwara, P. Schapira, Sheaves on Manifolds , Grundlehren der math. Wissenschaften, 292 (1990). ISBN 3540518614 or 0387518614.

C. Weibel, An introduction to homological algebra , Cambridge studies in Adv. Math. 38.

Outline of the course: The concept of sheaf provides a systematic way of keeping track of local algebraic data on a topological space. The theory of schemes is the fundation of a modern approach to algebraic geometry and the basis for a unified view of number theory and algebraic geometry. The aim of the first part of this course is to introduce graduate students to the notion of a sheaf and its properties. The second part of the course will be dedicated to scheme theory.

I will start by reviewing the basics of categories and functors, in particular I will focus on the important notion of representable functor. In this first part I will also introduce the notion of limit, additive and abelian category, the definition of a complex in an abelian category, and that of resolution (in particular injective resolution) and derived functor. The second part of the course will focus on the notion of sheaf on a topological space: its stalks, sections and support. This introductory part will be followed by a section devoted to the study of the operations on sheaves, including direct and inverse images. More material on sheaf theory will be explained later on in the course. The third part of the course will be devoted to scheme theory, starting with the notion of affine scheme and its structural sheaf and related functorial properties. The general notion of a scheme will follow and particular emphasis will be given to the definition and the properties of a projective scheme. More material on sheaf theory will be reviewed at this point, in particular I will focus on the notion of sheaf of modules and the more refined one of (quasi) coherent sheaf. As application we will see the notion of Chech cohomology and that of de Rham cohomology. In the last part of the course, I plan to read and comment in class A. Grothendieck paper "On the de Rham cohomology of algebraic varieties".

Prerequisites: Basic point-set topology and some comfort with manifolds are probably essential. Also useful: some previous exposure to concepts of homological algebra such as categories, exact sequences, and complexes, and to the basic notions in ring theory and complex analysis.

Grading: Some exercises will be assigned regularly during lectures. The exercises will not be collected, and in fact there will be no written work required in this course. The final grade will be determined from two components: 1) class participation (i.e. did you basically attend all the lectures and try to ask questions now and then), and 2) a 15-minute oral exam that will be scheduled during the last two or three weeks of the course. During the oral exam you will be expected to explain to me 2 or 3 topics from a list of things we covered during the lectures. The list of possible topics will be determined later. You will know ahead of time exactly which topics you need to present to me, but you may or may not be allowed to choose the topics yourself (I haven't decided yet). Details will be announced later in the course.

Important Note: Class will be cancelled during the week of October 24-28, as I will be away.
Important Note: Class will be cancelled on Wednesday November 23, for ``turkey cooking technique class''