You may use the following result (very similar to one proved in class):
Let J be a homogeneous ideal. If for every homogeneous f one
has f^2 in J => f in J, then J is radical.
1. Let I be a radical ideal in R, and homog(I) its homogenization in R[y].
Show that homog(I) is radical.
2. Let I be a radical homogeneous ideal in R[y].
Show that dehomog(I) is radical.
3. Let I be an ideal in R such that homog(I) is radical. Show that I is radical.
4. Give an example of I in R[y], homogeneous but not radical, such that dehomog(I) is radical.
5. Let I be generated by xy, x+y-1. Find a term order such that init(homog(I)) is radical.
Conclude that I is radical.
6. Draw the simplicial complex associated to the ideal generated by {e, af, bd} in C[a,b,c,d,e,f].
(Grr. Blogger is eating my less-than and more-than symbols with which to generate ideals.)
7. Draw the solid abc triangle, and put a vertex d in the middle, with new edges connected to a,b,c. (The red, but not blue, lines in this picture.) What's the corresponding Stanley-Reisner ideal, and its Hilbert series?
