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

72 5. LOCALIZATION
(1) φ is flat.
(2) φP : RP → SP is flat for all prime ideals P ⊂ R.
(3) φP : RP → SP is flat for all maximal ideals P ⊂ R.
Proof. Use 5.4.6.
5.5.8. Proposition. A flat local homomorphism (R, P ) → (S, Q) is faithfully flat.
Proof. Let 0 = x ∈ M be an R-module. R/ Ann(x) Rx and I = Ann(x) ⊂ P ,
so IS ⊂ Q. Then R/I ⊗R S S/IS = 0 and therefore M ⊗R S = 0 giving the
claim.
5.5.9. Proposition (going-down). Let φ : R → S be a flat ring homomorphism
and Q ⊂ S a prime ideal. For any prime ideal P ′ ⊂ P = Q ∩ R there is a prime
ideal Q ′ ⊂ Q contracting to P ′ = Q ′ ∩ R.
Proof. The local homomorphism RP → SQ is faithfully flat. The ring k(P ′ ) ⊗R
SQ is nonzero and therefor contains a maximal ideal Q ′′ . The contraction to S,
Q ′ = Q ′′ ∩ S contracts to P ′ = R ∩ Q ′ .
5.5.10. Proposition. Let R → S be a flat homomorphism. The following conditions
are equivalent.
(1) R → S is faithfully flat.
(2) Any prime ideal P ⊂ R is the contraction P = Q ∩ R of a prime ideal
Q ⊂ S.
Proof. Assume (2). Let M = 0 be an R-module. Then MP = 0 for some P . Let
P = Q ∩ R, then (M ⊗R S)Q MP ⊗RP SQ = 0 as RP → SQ is faithfully flat.
So (1) is true.
5.5.11. Proposition. The inclusion R → R[X1, . . . , Xn] is a faithfully flat homomorphism
Proof. The R-module R[X1, . . . , Xn] is free.
5.5.12. Exercise. (1) Show that a free module is faithfully flat.
(2) Let R → S and S → T be flat homomorphisms. Show that the composite R → S is
flat.
(3) Show that Q is a flat but not faithfully flat Z-module.
(4) Let R be a ring and I = √ 0 the nilradical. Show that IR[X] is the nilradical of
R[X].