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