Commutative algebra - Department of Mathematical Sciences - old ...

5.2. LOCALIZATION OF RINGS 67
5.2. Localization of rings
5.2.1. Definition. A ring R which contains precisely one maximal ideal P is a
local ring and denoted (R, P ). The residue field of R is R/P denoted by k(P ).
A ring homomorphism φ : R → S of local rings (R, P ), (S, Q) is a local ring
homomorphism if φ(P ) ⊂ Q.
5.2.2. Lemma. Let R be a ring and Q = ∅ a subset. Then R is a local ring with
maximal ideal Q if and only if R\Q is the set of units in R.
Proof. Use that an ideal I = R if and only if it contains a unit.
5.2.3. Lemma. A ring homomorphism φ : R → S of local rings (R, P ), (S, Q) is
a local ring homomorphism if the extended ideal P S ⊂ Q or the contracted ideal
Q ∩ R = P . The residue homomorphism k(P ) → k(Q) is a field extension.
Proof. The contraction Q ∩ R is a prime ideal containing P . The rest is clear.
5.2.4. Lemma. Let R be a ring and P a prime ideal. Then U = R\P is a multiplicative
subset. The ring of fractions U −1 R is a local ring. The maximal ideal is
the extended ideal P U −1 R. The residue field is U −1 R/P U −1 R which is canonical
isomorphic to the fraction field of R/P .
5.2.5. Definition. Let R be a ring, P a prime ideal and U = R\P . The localized
ring at P is the local ring RP = U −1 R. The residue field is denoted k(P ) =
RP /P RP .
5.2.6. Example. Let the ring be Z.
(1) The local ring at (0) is the fraction field Q = Z (0).
(2) The local ring Z (p) for a prime number p is identified with a subring of Q
Z (p) = { m
|p not dividing n}
n
The residue field Fp = Z (p)/(p).
(3) Any nonzero ideal in Z (p) is principal of the form (pn ) for some n.
5.2.7. Proposition. Let (R, P ) be a local ring. One of the following conditions is
satisfied:
(1) The characteristic char(R) = 0. P ∩ Z = (0) and Q ⊂ R is a subfield.
Q → k(P ) is a field extension.
(2) The characteristic char(R) = 0. P ∩ Z = (p), p a prime number. Z (p) ⊂ R
is a local subring. Fp → k(P ) is a field extension.
(3) The characteristic char(R) = p n , a power of a prime number. Z/(p n ) ⊂ R
is a local subring. Fp → k(P ) is a field extension.
Proof. (1) (2) are clear by 5.2.3 and 5.2.6. (3) If the characteristic is nonzero then
any prime ideal contracts Q ∩ Z = (p). So a prime number q = p gives a unit in
R. There is a local homomorphism Z (p) → R, 4.1.3. The nontrivial kernel is (p n )
by 5.2.6.
5.2.8. Example. A field K is a local ring with maximal ideal (0). The power series
ring K[[X]] is a local ring with maximal ideal (X) and residue field K.
5.2.9. Remark. Let R be a domain.
(1) The local ring at (0) is the fraction field K = R (0).