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

68 5. LOCALIZATION
(2) Any local ring RP is identified with a subring of the fraction field K.
(3) The intersection
R =
RP , P a maximal ideal
P
5.2.10. Remark. Let R × S be a product of rings.
(1) A prime ideal is of the form P × S or R × Q for uniquely determined prime
ideals P ⊂ R or Q ⊂ S.
(2) The local ring at P ×S is identified with RP through the projection R ×S →
R.
(3) The local ring at R × Q is identified with SQ through the projection R × S →
S.
5.2.11. Proposition. Let P be a prime ideal and R → RP the canonical homomorphism.
Extension and contraction gives a bijective correspondence between
prime ideals in R contained in P and all prime ideals in RP .
(1) For a prime ideal Q ⊂ P the extended ideal QRP is a prime ideal in RP and
the contracted QRP ∩ R = Q.
(2) For a prime ideal Q ′ ⊂ RP the contracted ideal Q ′ ∩ R ⊂ P is a prime ideal
and the extended (Q ′ ∩ R)RP = Q ′
Proof. This is a special case of 5.1.5.
5.2.12. Lemma. Let Q ⊂ P ⊂ R be prime ideals. Then QRP is a prime ideal in
RP and canonically
Proof. By fraction rules a w av
u / v = uw .
RQ = (RP )QRP
5.2.13. Definition. The intersection of all maximal ideals in a ring is the Jacobson
radical.
5.2.14. Remark. The Jacobson radical contains the nilradical. In a local ring the
Jacobson radical is the maximal ideal.
5.2.15. Exercise. (1) Show that a local ring is never a product of two nonzero rings.
(2) Show that a ∈ R is in the Jacobson radical if and only if 1+ab is a unit for all b ∈ R.
(3) Let p be a prime number. Describe the prime ideals in the ring Z (p).
(4) Let P be a prime ideal. Show that k(P ) is the fraction field of R/P .
(5) Let (R, P ) be a local ring. Show that (R[[X]], (P, X) is a local ring and the canonical
homomorphism R → R[[X]] is a local homomorphism.
5.3. Localization of modules
5.3.1. Definition. Let R be a ring, P a prime ideal and U = R\P . For a module
M, the localized module at P is the module MP = U −1 M over the local ring RP .
For a homomorphism f : M → N the localized homomorphism is fP : MP →
NP and the residue homomorphism is f(P ) : M ⊗R k(P ) → N ⊗R k(P ).
The constructions are functors, 2.7.4, 4.2.4.
5.3.2. Lemma. Let Q ⊂ P ⊂ R be prime ideals and M an R-module Then QRP
is a prime ideal in RP and canonically
Proof. See proof of 5.2.12.
MQ = (MP )QRP