Homological Algebra

        Homological methods arose in the 19th century in the work of
Riemann and Poincare, who were studying the shape of various kinds of space.  
To spaces they attached numbers that reflected certain aspects of the shape
of the space, such as the number of ``holes'', e.g. one for a circle or
sphere, two for a torus. By 1925 their attempts had evolved into the much
more powerful idea of attaching groups to spaces. The next 20 years were
spent developing basic methods of calculating these groups, a field now
called Algebraic Topology. The algebraic methods for doing this by 1945 had
become a subject in their own right --- Homological Algebra. Homological
methods have important applications in many other fields besides Algebraic
Topology, such as Algebraic Geometry, Commutative Rings, Group Theory,
Invariant Theory, Number Theory, and Combinatorics.


Text: An Introduction to Homological Algebra, by Charles Weibel


We will cover chapters 1-3 in some detail. This is the most basic part of
the subject (derived functors, Tor and Ext). These methods will be
applied primarily to Homological Dimension of Rings (chapter 4), with a
brief mention of Group Homology (chapter 6) and Simplicial Methods
(chapter 8).


Prerequisite: A course in abstract algebra, up to quotient structures
(such as a group mod a normal subgroup or a ring mod an ideal). Mathematics
310 is more than sufficient, so the course will be accessible to
undergraduates as well as graduate students.