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

(3) FP is flat for all maximal ideals P .
5.5. FLAT RING HOMOMORPHISMS 71
Proof. Let 0 → M → N. Use 5.3.7 and 5.4.2 on M ⊗R F → N ⊗R F .
5.4.7. Proposition. Let R be a ring and M a module. Then there is an exact
sequence
0 → M →
P maximal
Proof. Let 0 = x ∈ M be given. Then Ann(x) ⊂ P is contained in a maximal
ideal, 5.1.2. Clearly 0 = x
1 ∈ MP .
5.4.8. Corollary. Let R be a ring. Then there is an injective ring homomorphism
R →
P maximal
5.4.9. Corollary. Let R be a ring. The following conditions are equivalent.
(1) R is reduced.
(2) RP is reduced for all prime ideals P .
(3) RP is reduced for all maximal ideals P .
Proof. Use 5.1.8 and 5.4.10.
5.4.10. Exercise. (1) Let R be a ring and
0
f
M
g
N
a split exact sequence. Show that the localized sequence is split exact for all prime
ideals P .
RP
MP
L
5.5. Flat ring homomorphisms
5.5.1. Definition. A flat R-module F is faithfully flat if for any M = 0 the tensor
product M ⊗R F = 0.
5.5.2. Example. A nonzero free module is faithfully flat.
5.5.3. Lemma. Let F be a faithfully flat R-module and M → N a homomorphism.
If M ⊗R F → N ⊗R F is injective, then M → N is injective.
Proof. Let f : M → N. Then 0 → Ker f ⊗R F → M ⊗R F → N ⊗R F is exact.
It follows, that Ker f = 0.
5.5.4. Definition. A ring homomorphism R → S is a flat ring homomorphism if
S is a flat R-module and a faithfully flat ring homomorphism if S is faithfully flat.
5.5.5. Proposition. Let R → S be a faithfully flat ring homomorphism. for any
ideal I ⊂ R the extended contracted returns I, i.e.
I = IS ∩ R
Proof. Tensor the homomorphism R/I → R/IS∩R with S. The induced R/I⊗R
S → R/IS ∩ R ⊗R S is canonically isomorphic to the identity S/IS → S/IS.
By 5.5.3 R/I → R/IS ∩ R is injective giving I = IS ∩ R.
5.5.6. Corollary. A faithfully flat ring homomorphism R → S is injective.
5.5.7. Proposition. Let φ : R → S be a ring homomorphism. The following
conditions are equivalent
0