HW due Wednesday May 30

For those who want practice with Hilbert series...

1. Let I in R=C[a,b,c,d] be the ideal < a,b > intersect < c,d >, which for your information happens to be generated by < ac,ad,bc,bd >. Compute the Hilbert function of R/I, and compute the Hilbert series as a rational function.

2. Let X be the space of rank 1 matrices with entries
[a b c]
[d e f].
a. What are the equations that say that X is rank at most 1?
b. Let I be the ideal they generate, and compute the Hilbert function/series for C[a,b,c,d,e,f]/I.

3. Define a quasipolynomial f:Z->Z to be a function s.t. there exists a d>0 s.t. f is a polynomial on {a in Z : a = k mod d}, for each k=1..d. For example f(n) = (-1)ⁿ is a quasipolynomial with d=2 (or any multiple).

Let R be a Noetherian graded ring, with R₀ a field F, and let M be a finitely generated graded R-module.

a. Show that the Hilbert function of M is eventually a quasipolynomial.
b. Show that the Hilbert series of M is a rational function.