Math 615 is a second course in Commutative Algebra. The focus this semester will be on local cohomology.
The first part of the class will be on homological Commutative Algebra material: depth,
properties of free resolutions, Ext, and Tor, and the structure of injective modules. The second part
of the class is on local cohomology and its basic applications: its various definitions,
Grothendieck (non)vanishing, arithmetic
rank, canonical modules, local duality, Mayer-Vietoris, Hartshorne-Lichtenbaum,
and connectedness results. In the last part of the class, we will discuss some of the structural
results on local cohomology using differential operators and the Frobenius map.
We will not follow any particular textbook, but recommended sources include:
Local cohomology, Hartshorne’s lecture notes of a seminar given by A. Grothendieck.
Mel’s and Craig’s notes are the most self-contained. Twenty-four covers the most connections and applications of local cohomology,
while Brodmann and Sharp is the most encyclopedic in some aspects of structure.
Bruns and Herzog and Twenty-four cover the aspects of the class that are not properly local cohomology. The elegant Grothendieck/Hartshorne notes
remain very readable 57 years later! The lecture notes for this class owe a major intellectual debt to the sources above, and anything good in these notes is almost certainly taken from one of them. The lecture notes for this class should not be used as a reference, but rather an invitation to check out the sources listed above.
There will be half a dozen or so problem sets throughout the semester. We will also do worksheets in class sometimes.
Knowledge of basic Commutative Algebra (614) and basic algebraic topology (592) is assumed, but I will be
happy to go over or give references to anything unfamiliar.