 
Summary: A sample solution of Problem 2 in Problem Set 19 with
Macaulay2
Chris Peterson and Hirotachi Abo
Partial differentiations.
Let k be an algebraically closed field, and let R denote the polynomial ring
k[x, y, z]. Assume that I is an ideal with V (I) = . Then the quotient ring
Q of R modulo I is a finitedimensional vector space over k. In other words,
Q is artinian. In this case, Q has a free resolution of the following type (this
is not trivial):
0 F3 F2 F1 R Q 0,
where Fi are free modules for each i = 1, 2 and 3. This ring is called arith
metically Gorenstein if F3 can be written as R(l) for some positive integer
l (i.e. F3 has rank 1).
Remark 1. Let Q be an artinian arithmetically Gorenstein ring over R. Then
there is a positive integer d such that dimk(Qd) = 1 and dimk(Qi) = 0 for all
i > d.
Let S be the polynomial ring k[X, Y, Z], and let S act on R by partial
differentiation:
X(x) := x(x), Y (y) := y(y) and Z(z) := z(z).
Let F be a single homogeneous polynomial of degree d in R. For this F,
