Summer school in commutative algebra:
Local cohomology and its interactions with algebra, geometry, and analysis

for graduate students

Monday June 20 -- Thursday June 30, 2005 in Snowbird, Utah

Lecture Notes

The lecture notes have been rewritten and are now open for inspection. Here is the current version. You are invited to give us comments regarding omissions, repetitions, and errors of all sorts. Please send them to Uli at, regardless of the author of the chapter in question. Of course, the sooner you send in corrections/comments, the earlier that we can address them. (Time is relevant since the notes will be published by the AMS.)

Thanks so much!

Organizing committee:

Anurag Singh (Georgia Tech,
Uli Walther (Purdue,

We wish to express our THANKS to Wayne Drady and the AMS for all their HELP and SUPPORT, and to MSRI at Berkeley for the much appreciated COSPONSORSHIP of this summer school.


This is one of the 2005 Joint Summer Research Conferences. (Look here for a lot of useful info about these conferences, and Snowbird in general.)

This summer school is intended for graduate students only. The application deadline has passed. Invitation letters for successful applicants should have arrived.

This page is under construction. Please send questions/remarks to -- but be patient...

Please note that Snowbird is not at sea level. If you have asthma or altitude sickness you might need to bring medication.

Basic Material

Here is a short script that gives an introduction to the theory of injective modules. These are important theoretical gadgets for the summer school and the properties and results discussed in the script are yoused throughout the summer school. Injective modules are covered in most courses on commutative algebra, but we realize that you may not have had such course. We expect you to have read the script before you come to Snowbird!


The topic of the summer school will be commutative algebra, with special emphasis on topics that are related to local cohomology and its interactions with other areas of mathematics such as algebraic geometry, topology, and D-modules. Some part of the workshop will be devoted to algorithmic questions and explicit computations.

Lecture notes for the first part will be developed by the speakers and they will be distributed before the start of the summer school.



We will have 7 days of lectures, followed by a conference during the last 3 days. During the lecture period, there will be 3 to 4 one hour classes per day, supplemented by problem sessions, computer demos, and discussions. These lectures, demos and sessions will be delivered by

Srikanth Iyengar, University of Nebraska
Graham Leuschke, Syracuse University
Anton Leykin, UI at Chicago
Claudia Miller, Syracuse University
Ezra Miller, University of Minnesota

and the organizers.

The speakers for the conference that makes the second part of the Summer School are expected to be

Markus Brodmann, Universitaet Zuerich
Ragnar-Olaf Buchweitz, University of Toronto
Phillippe Gimenez, Universidad de Valladolid
Gennady Lyubeznik, University of Minnesota
Paul Roberts, University of Utah
Peter Schenzel, Universitaet Halle
Rodney Sharp, University of Sheffield
Ngo Viet Trung, Institute of Mathematics, Hanoi
Keiichi Watanabe, Nihon University
Santiago Zarzuela, Universitat de Barcelona

The talks in this final part are intended to be expository and centered around the main themes of the summer school.

There will be one day of rest, June 25. We will have an excursion that day.


Here is the schedule for the summer school with a rough outline of the topics.

Travel and Lodging

Travel is most easily done through Salt Lake City, which is about 30 miles from Snowbird. The accomodations are entirely provided by the AMS, including the meals. As stated above, we will cover these expenses for the students. Rooms will be double occupancy. If you have someone in mind that you want as roommate, please point this out to the organizers.

For applicants:

The target population consists of graduate students in their early years, similar to the Barcelona summer school in 1996. The workshop will be designed to accommodate 30 to 40 graduate students
All persons who are interested in participating in one of the conferences should request an invitation by sending the following information to:

Summer Research Conferences Coordinator,
AMS, P.O. Box 6887,
Providence, RI 02940,

or by email to Wayne Drady at no later than March 3, 2005.

Please type or print the following:

1. Title and dates of conference. (See above.)
2. Full name.
3. Mailing address.
4. Phone numbers (including area code) for office, home, and fax.
5. Email address.
6. Your anticipated arrival/departure dates. (June 19 and July 1.)
7. Scientific background relevant to the conference topics; please indicate your student status, whether you have an advisor, and why you are interested in the conference.
8. The amount of financial assistance requested (or indicate if no support is required).
9. If you are a student at a sponsoring institution of the Mathematics Sciences Research Institute (MSRI) at Berkeley, please says so very explicitly in the application (to find out, ask your department chair).


Anticipated funding

We expect to cover the lodging and the food at Snowbird for all students. We may be able to cover some travel expenses. This will depend on luck.


The following comprises a list of words that will be used often during the talks. Attendants should be familiar with their meaning.
* ring, homomorphism, prime ideal, maximal ideal, field, zerodivisor, localization, radical, associated primes of ideals and modules, what are flat/projective/injective modules?
* topological space, topology, open and closed sets, subspace topology, continuity (in the general case when no metric is present), discrete spaces, manifolds, (co)homology
* (additive/linear) functor, category, morphism, natural transformation, complex of objects in a category, homology, left/right-exact functor

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