Math 582BA
Homological Algebraic Methods in Commutative Algebra and Algebraic Geometry
Sándor Kovács
Winter 2003, Monday/Wednesday/Friday 9:30-10:20
The goal of this course is to introduce the students to homological
algebraic techniques in Commutative Algebra and Algebraic Geometry. Derived
functors and cohomology will be introduced along with the notion of depth.
An explicit derived functor, Ext will be used in many different ways, for
instance describing depth in a very useful way. Connections with dimension
theory, Cohen-Macaulay and Gorenstein rings will be discussed as well as
(time permitting) local cohomology and duality. The algebraic methods and
results will be complemented by geometric examples and applications
throughout.
This is primarily an algebra course and one will be able to follow it
without reference to geometry. (Although, in that case one will have a
less exciting time.)
Prerequisites
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506 (Algebra), or instructor's permission.
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A working knowledge of commutative algebra is very important for this
course. Miles Reid's Undergraduate Commutative Algebra [London
Math. Soc. Student Texts #29, Cambridge Univ. Press, 1995] contains a good
introduction to the subject. Knowledge of the first 6 chapters will be
assumed and familiarity with Chapters 7-8 is also useful.
-
Basic knowledge of algebraic geometry (e.g., on the level of 507) is helpful
in order to appreciate the applications, but it is not mandatory.
Textbook
Bruns-Herzog, Cohen-Macaulay Rings, 2nd ed. Cambridge University
Press, 1997. ISBN: 0521566746.