1. Let I be an ideal in C[x_1..x_n], and let sqrt(J) denote the radical of J.
Prove rad(I)^h = rad(I^h), i.e. taking radical commutes with homogenizing.
2. Show (A intersect B)^h = A^h intersect B^h, where A,B are ideals in C[x_1..x_n].
3. Show (A+B)^h contains A^h + B^h.
4a. Find an example where this containment is proper.
b. Explain geometrically why you might expect this to happen.
(If you can do (b) first, it might help you with (a); on the other hand it's easy to luck into an example of (a).)
5. How can you compute (A+B)^h from A^h + B^h? (Again, 4(b) should help.)
6. Let R = C[a,b,c,d], and p = a*(ad-bc)*d. Let I = < p >.
Decompose I as an intersection of prime ideals.
If you add some of those prime ideals together, you get new ideals; decompose all those.
Keep adding and decomposing until you can't find any new ideals this way.
Draw the poset of ideals, with each one labeled with its Hilbert polynomial.
If you use Macaulay 2 to do #6, so much the better, though it's not really so hard to do this little example directly. Anyway if you do turn in your code, too.
