1. Let R = F_p[x,y], and I generated by y(y-x^2).
Show there is no Frobenius splitting on R that compatibly splits I.
(Hint: if I were compatibly split, then so would various other ideals be, including one that isn't radical.)
2. Same problem, but I is generated by xy(x+y).
3. Let R be a graded ring, and phi a Frobenius splitting of it.
a) Show that phi is determined by its values on homogeneous elements.
b) For r homogeneous of degree k, define phi'(r) = the degree k/p part of phi(r), or 0 if k/p is not an integer.
For r not homogeneous, define phi'(r) = the sum of phi'(its homogeneous pieces).
Show that phi' is a Frobenius splitting.
c) Give a reasonable definition of a "graded Frobenius splitting".
4. Let R be the subring of F_p[x] generated by x^2 and x^3, i.e. polynomials with no linear term. Show that R has no Frobenius splitting.
5. Let R = C[a,b,c] / {ac} stupid blogger.com.
Let I be generated by {a,b}, as an ideal in R not just C[a,b,c].
a) Compute the Hilbert series H_R and H_{R/I}.
b) Show that H_{R/I} is not H_R times a polynomial.
c) Prove that the Hilbert Syzygy Theorem fails for this R and I; there is no finite graded resolution.
6. Here is a simple program in Macaulay 2, a program to do (mostly) ring theory calculations.
Figure out what it's computing. Here's a comprehensive index of M2 commands. If you want to actually run M2 (so e.g. you can play with the code), here's how to get started.
