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

3.7. FLAT MODULES 55
3.7.4. Proposition. Let Fα be family of flat modules, then the direct sum
α Fα
is a flat module.
Proof. Let M → N be injective. Then M ⊗R ( Fα) → N ⊗R ( Fα) is
injective by 2.6.11 and 3.1.6, so the product is flat.
3.7.5. Example. A free module is flat.
3.7.6. Corollary. A projective module is flat.
Proof. By 3.5.7 a projective module is a direct summand in a free. Conclusion by
3.7.4.
3.7.7. Proposition. Let F, F ′ be flat modules. Then F ⊗R F ′ is flat.
Proof. Let M → N be injective. Then by 2.6.10 M ⊗R (F ⊗R F ′ ) → N ⊗R
(F ⊗R F ′ ) is (M ⊗R F ) ⊗R F ′ → (N ⊗R F ) ⊗R F ′ being injective by using the
definition twice.
3.7.8. Proposition. Let R → S be a ring homomorphism and F a flat R-module.
The change of ring module F ⊗R S is a flat S-module.
Proof. Let M → N be an injective homomorphism of S-modules. Then by 2.7.5
M ⊗S (F ⊗R S ′ ) → N ⊗S (F ⊗R S) is M ⊗R F → N ⊗R F being injective since
F is a flat R-module.
3.7.9. Proposition. Let R be a ring and F a module. The following are equivalent.
(1) F is a flat module.
(2) HomR(F, E) is an injective module for any injective module E.
Proof. Let M → N be injective. By 3.6.13 M ⊗R F → N ⊗R F is injective if and
only if HomR(N ⊗R F, E) → HomR(M ⊗R F, E) is surjective for any injective
module E. By 2.6.13 this is HomR(N, HomR(F, E)) → HomR(M, HomR(F, E)).
3.7.10. Corollary. Given a short exact sequence
0
M
N
F
where F is a flat module. Then M is flat if and only if N is flat.
Proof. Let E be an injective module. By 3.7.9 and 3.6.3 the sequence
0
HomR(F, E)
HomR(N, E)
0
HomR(M, E)
is split exact. By 3.6.5 and 3.6.6 HomR(N, E) is injective if and only if HomR(M, E)
is so. Conclusion by 3.7.9.
3.7.11. Corollary. Given a short exact sequence
0
M
where F is a flat module. For any module L there is a short exact sequence
0
L ⊗R M
N
L ⊗R N
F
0
L ⊗R F
0
0