Cellular resolutions of monomial ideals

Prerequisites: MAT220, MAT221, and MAT224. The course MAT242 can also be useful, but the project may well be done without this.

Description: For an ideal I in a polynomial ring S = k[x₁, ..., x_n], one can make a resolution. This is a long exact sequence

(1) 0 ← I ← ⊕_i∈ℤ S(−i)^β_0,i ←^d₁ ⊕_i∈ℤ S(−i)^β_1,i ←^d₂ ··· ←^d_n ⊕_i∈ℤ S(−i)^β_n,i

where all the terms except I are free modules. This is called minimal if the matrices that represent the mapping d_p only have positions where the elements are either zero of have positive degree. For minimal resolutions, the numbers β_p,i are uniquely defined. They are important invariants of I and are called the graded Betti numbers of I. If I is now a monomial ideal, i.e. the generators of I are monomials x^a₁·x^a_2···x^a_n, then such a resolution can often be written as a cellular resolution. I.e. there is a topological cell complex C built out of points C₀, lines C₁, polygons C₂, three-dimensional cells C₃, etc., where the monomial generators of the ideal I are in a 1-1 correspondence with the points in C₀. Based on this, one can make a complex of S-modules (1) where ∑_iβ_p,i is equal to the number of cells of dimension p. What is challenging is to find a topological cell complex for a given set of monomial generators such that the associated complex of S-modules (1) is an exact sequence.

We are interested in computing resolutions for different monomial ideals in two, three and four variables (this requires good computer programs), and then examine whether we can find topological cell complexes that can be describe this cellular resolution.

References:

[1] D.Bayer, I.Peeva, and B.Sturmfels, Monomial resolutions Math.Res.Lett. 5 (1998) no.1-2, pp. 31-46.
[2] E.Miller, B.Sturmfels, Combinatorial commutative algebra, GTM 2005, Springer-Verlag, 2005.
[3] R.Stanley, Combinatorics and Commutative Algebra, Second Edition, Birkhäuser 1996.