Summary: Problem Set 12
For this entire problem set, R = k[x1, x2, . . . , xn] with k a eld.
Recall that R = k[x1, x2, . . . , xn] is a graded ring by the decomposition R =
d0 Rd where Rd is the space spanned by the monomials of degree d. Given a
graded R-module, M, we can dene the Hilbert function M of M by M (t) =
dimk(Mt) for each t Z. We have the following theorem:
Theorem 1. (Hilbert-Serre) Let M be a nitely generated graded R = k[x0, x1, . . . , xn]
module. There is a unique polynomial PM (t) Q[t] such that M (t) = PM (t) for
The polynomial that appears in the theorem above is called the Hilbert Poly-
nomial of M. In the near future, we will have tools that will enable us to compute
the Hilbert Polynomial. For now, we must be satised to compute the Hilbert
For now I will let dimension be dened intuitively, the empty set will have
dimension -1, points will have dimension 0, curves will have dimension 1, etc.
Later we will give a more precise denition of dimension.
Recall the following lemma from Problem Set 9:
Lemma 2. Let R = k[x1, x2, . . . , xn]. Let F R. There is a short exact sequence
of the form
0 R/(I : F) R/I R/(I, F) 0.