COCOA VI COmputational COmmutative Algebra and International School on Computer Algebra Villa Gualino, Torino, Italy May 31 - June 5, 1999

problem sheets (4 pages plain TEX)

Dear participants of the COCOA school,

Ezra Miller and I look forward to discussing the topic of "Monomial Ideals" with you. Please expect my eight lectures and the accompanying exercises with the COCOA system to be a very intense experience; it may well be the most intense class you have ever attended. Therefore I recommend that take a few days between now and May 31, to prepare yourselves for this course.

Basic references for combinatorial/computational/commutative algebra are:

[CLO] D. Cox, J. Little, D. O'Shea: "Ideals, Varieties and Algorithms",
      Springer Undergraduate Texts in Math, 2nd edition, 1997.

[Eis] D. Eisenbud: "Commutative Algebra with a view toward Algebraic Geometry"
      Springer Graduate Texts in Math, 1994

[Sta] R. Stanley: "Combinatorics and Commutative Algebra",
      Birkh"auser, Boston, Progress in Math 41, 2nd edition,  1996

I will assume that you have some acquaintance with these books. Specifically, please review carefully Chapter 9 in [CLO], Exercises 3.6-3.11 and 17.11 and in [Eis], and Sections 0.1, 0.3, II.1 in [Sta].

Next please surf

This JAVA applet due to David Perkinson and Justin Campbell draws monomial ideals in three variables. Try the fourth power of the maximal ideal and see what happens as you remove one generator at a time. (Note: the applet may not run if your browser is not up-to-date.)

Now, you are ready to dive into the literature on monomial ideals. Please take a peek at the following six papers, and try to read as far as you can:

(1)  J. Eagon and V. Reiner: Resolutions of Stanley-Reisner rings and Alexander
     duality, Journal of Pure and Applied Algebra, 130 (1998) 265-275.

(2)  S. Eliahou and M. Kervaire: Minimal Resolutions of Some Monomial Ideals,
     Journal of Algebra 129 (1990) 1-25

(3)  A. Bigatti: Upper bounds for the Betti numbers of a given Hilbert
     function. Communications in  Algebra 21 (1993), no. 7, 2317-2334.

(4)  D. Bayer, I. Peeva and B. Sturmfels: Monomial resolutions,
     Mathematical Research Letters  5  (1998) 31-46.

(5)  E. Miller: Alexander Duality for Monomial Ideals and Their Resolutions,
     please download from  http://math.berkeley.edu/~enmiller/

(6)  D. Bayer and B. Sturmfels: Cellular Resolution of Monomial Modules,
     Journal f"ur die Reine und Angewandte Mathematik 502 (1998) 123-140.

That's more than enough to read, isn't it ?

Don't worry. Staircases are actually fun, and we'll have lots of fun in Torino.

Bernd Sturmfels