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

6
Finite modules
6.1. Finite Modules
6.1.1. Definition. Let R be a ring. A finite module is generated by finitely many
elements. The finite free module with standard basis e1, . . . , en is denoted R n .
6.1.2. Lemma. Let R be a ring and M a module. The following are equivalent.
(1) M is generated by n elements x1, . . . , xn.
(2) There is a surjective homomorphism R n → M → 0, ei ↦→ xi.
Proof. See 2.4.12.
6.1.3. Proposition. Let R → S be a ring homomorphism. If an R-module M is
generated by n elements x1, . . . , xn. Then the change of rings S-module M ⊗R S
is generated by x1 ⊗ 1, . . . , xn ⊗ 1 over S.
Proof. Follows from 6.1.2 and 3.4.1
6.1.4. Corollary. Let R be a ring and U a multiplicative subset. If M is a finite
R-module, then U −1 M is a finite U −1 R-module.
6.1.5. Proposition. For a short exact sequence
0
f
M
the following hold
(1) If N is finite, then L is finite.
(2) If M, L are finite, then N is finite.
g
N
Proof. (1) If y1, . . . , yn generates N, then g(y1), . . . , g(yn) generates L. (2) Choose
u : R n → M → 0 and v : R m → L → 0 exact. By 3.5.4 there is w : R m → N
such that g ◦ w = v. There is a diagram
0
0
Rn
u
M
Conclusion by the snake lemma 3.2.4.
6.1.6. Corollary. Let
0
f
f
M
L
Rn ⊕ Rm
f◦u+w
g
N
g
N
0
Rm
v
L
be a split exact sequence. Then N is finite if and only if M, L are finite.
Proof. Let u be a retraction of f. By 6.1.5 Im u = M is finite. The rest is contained
in 6.1.5.
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L
0
0
0