HW #9

1. Let I be an ideal in C[x_1..x_n]. Let G_I be the sum of all the standard monomials. For example, if I = < x-y^2 > in C[x,y] with leading term x, then G_I = 1/(1-y). Whereas if the leading term is y^2, then G_I = (1+y)/(1-x). (Right?)

a. If I is homogeneous, what is the relation of G_I and the Hilbert series H_I?

b. Show that G_I is always a rational function.

c. If F is a subset of {1..n}, let t_F = product_{f in F} x_f / (1-x_f).
If I is generated by squarefree monomials, show G_I = sum_F t_F, where the sum is over those F such that product_{f in F} x_f is a standard monomial.

2. Consider pairs {(A,B) nxn matrices : AB, BA upper triangular}.

a. Use the "codim" command in Macaulay 2 to check that, for n=2,3, this is a complete intersection.

b. Assuming this for general n, what's its degree?

c. Decompose it (for n=2,3) and determine the degrees of the components. Check that they add up to the answer in (b).

3. In C[a,b,c,d], let I be the ideal of the ab-plane union the cd-plane.

a. What is that ideal?

b. Let J be the ideal < a + c - 1, b + d - 2 >. Describe V(I+J). Is it radical?

c. Let K be the ideal < a + c, b + d >. Describe V(I+J) and compute its degree.

d. Do you find (b) vs. (c) disturbing?

HW #8, due April 19

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.

HW #7, due 4/11

Let P be the set of polynomials {p(d)} in one variable, d, such that p(d) is an integer for all integer d. Of course this includes Z[d], but is larger, e.g. p(d) = d(d-1)/2. In particular, this includes Hilbert polynomials.

1. For each k, show c_k(d) := d(d-1)...(d-k+1)/k! is in P.

2. Show that the {c_k} are a basis of C[d].

3. Define Delta : P -> P by (Delta p)(d) = p(d) - p(d+1). Compute Delta(c_k).

4. Show that the {c_k} are a Z-basis of P, when we expand an element of P in {c_k}, the coefficients are in Z.

5. Define the _Hilbert series_ H_I(t) of a homogeneous ideal I as the series \sum_d h_I(d) t^d. Show that if I is the 0 ideal in C[x_0,...,x_n], then H_I(t) = 1/(1-t)^{n+1}.

6. Recall that r is a _zero divisor_ in R/I if r is not zero, and there exists s not zero such that rs = 0. Assume that I is graded and r is homogeneous of degree k and not a zero divisor.
It turns out that H_{I + < r >} is something times H_I; figure out what.

7. Assume that I is graded, r is homogeneous of degree k, and the formula you figured out in #6 holds. Show that, conversely, this implies r is not a zero divisor.