A sample solution of Problem 2 in Problem Set 19 with Chris Peterson and Hirotachi Abo 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 finite-dimensional 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, Collections: Mathematics