1
(a) If (A, m) is a Noetherian local ring, give the definition of an ideal of definition
of A.
For a finitely generated A-module M and ideal of definition I, state the relationship
between the Hilbert function
χ(M, I; n) = `(M/InM)
and dim M.
(b) Let k be a field. Suppose f ∈ (x, y) ⊆ k[x, y] is a polynomial. Let
A = (k[x, y]/(f))(x,y)
.
Calculate the Hilbert function χ(A, m; n) of A.
Calculate the Hilbert polynomial of A, i.e., the polynomial function in n which
agrees with χ(A, m; n) for large n.
What is a simple way of describing the coefficient of the leading term of the Hilbert
polynomial in terms of f?
(c) Let k be a field, and consider the graded ring S = k[x, y] where the degree of
x is 1 and the degree of y is 2, so that, e.g., x
2y + y
2
is a homogeneous element of S of
degree 4. Calculate the Hilbert function
FS(n) = `(Sn).
Does this agree with a polynomial for large n? If not, why does this not contradict a result
from lectures?
2
Let p ∈ Z be a prime number, and denote by Zp the completion of Z at the prime
ideal (p).
(a) Give a brief description of Zp.
(b) For a ∈ Zp, denote ord(a) = sup{m | a ∈ (p
m)} (so that ord(p
2
) = 2 and
ord(0) = +∞). Set
ZphTi =
(X∞
n=0
anT
n
| an ∈ Zp, ord(an) → +∞ when n → +∞
)
.
Note Z[T] ⊆ ZphTi ⊂ Zp[[T]] as sets, where the latter is the ring of formal power series in
T with coefficients in Zp.
Show that ZphTi is a subring of Zp[[T]].
Show that ZphTi is the completion of Z[T] with respect to the ideal (p) ⊆ Z[T].
Part III, Paper 101
3
3
Let S be a Noetherian graded ring. If I ⊆ S is an arbitrary ideal, denote by I
∗
the
ideal contained in I generated by all homogeneous elements of I.
(a) If p is prime, show that p
∗
is also prime.
(b) If p is a homogeneous prime ideal and q is a p-primary ideal, show that q
∗
is
also p-primary.
(c) If p is an inhomogenenous prime ideal, show that there are no primes contained
between p and p
∗
. Show that ht(p) = ht(p
∗
) + 1. [Hint: it may be useful to consider
the following construction. Let R be a graded domain, and U ⊆ R the set of non-zero
homogeneous elements. Consider the ring U
−1R.]
4
Let A be a Noetherian domain, M an A-module. We say M is torsion-free if
Ann(m) = 0 for all m ∈ M. We denote by M∗
the A-module HomA(M, A).
Define a natural map M → M∗∗, by a 7→ (f 7→ f(a)). We say M is reflexive if it is
finitely generated as an A-module and the natural map M → M∗∗ is an isomorphism.
(a) Prove that if M is a finitely generated A-module, then there is an exact sequence
0 −→ M∗ −→ N −→ P → 0
where N is a finitely generated free A-module and P is torsion-free.
(b) Prove that an A-module M is reflexive if and only if it can be included in an
exact sequence
0 −→ M −→ N −→ P → 0
where N is a finitely generated free A-module and P is torsion-free. Thus M∗
is reflexive
whenever M is a finitely generated A-module. [Hint: You might find it useful to localize
at the zero ideal of A in the course of the proof.]