Let R = C[x,y].
Find all monomial ideals I such that there are exactly 4 monomials not in I.
(Note that you never need to have two generators where one divides the other.)
2. Let I,J be ideals in a ring R. Show that I intersect J is an ideal
(by checking each of conditions 1-3).
3. Let R = C[x,y], I = < x^2 y^20, x^15 y^4 >, J = < x^8 y^9 >.
Find generators for I intersect J.
4. Let Y = {0,1}, two elements of C. Find the ideal I_Y in C[x].
5. Compute the Hilbert function of C[x,y] / < x^a y^b >, for arbitrary a,b.
6. Let I be a monomial ideal in C[x,y]. Say that h_I(N)=0, for some large N. What can you say about the generators of I? (Assuming no redundant generators are used, meaning multiples of other generators.)
