 
Summary: Problem Set 20
Problem 1. Let F and G be two homogeneous polynomials in R = k[x0, x1, . . . , xn].
Assume that F and G do not share any common factors. Suppose that A, B R
and AF + BG = 0. Prove that there exists an H R such that A = HG
and B = HF. (In other words prove that every syzygy of F G is of the form
H
G
F
.)
Suppose the Hilbert Polynomial of R/I is F(X) = arXr
+ ar1Xr1
+ · · · +
a1X + a0. Associated to I is a scheme (not necessarily equidimensional). The
dimension of the scheme is equal to r. The degree of the scheme is ar/r!. If
F(X) = a1X + a0 then the scheme is one dimensional and has degree a1. Let
G = 1  a0. If the scheme is one dimensional then G is called the arithmetic
genus.
Problem 2. a) Use the previous problem to determine the Hilbert Polynomial of
k[x0, x1, x2, x3]/(F, G) where F is irreducible of degree 3 and G is irreducible of
degree 5.
