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

80 6. FINITE MODULES
6.5. Finite Presented Modules
6.5.1. Definition. Let R be a ring. A finite presented module is a module M
having an exact sequence
R n → R m → M → 0
or equivalently there is a short exact sequence
with N finite.
0 → N → R m → M → 0
6.5.2. Example. A finite projective module is finite presented, 6.1.10.
6.5.3. Lemma. For a short exact sequence
0
f
M
g
N
the following hold:
(1) If M, L are finite presented, then N is finite presented.
(2) If L is finite presented and N is finite , then M is finite.
(3) If N is finite presented and M is finite , then L is finite presented.
Proof. (1) Choose u : R n → M → 0 and v : R m → L → 0 exact with finite
kernels. By 3.5.4 there is w : R m → N such that g ◦ w = v. There is a diagram
0
0
Rn
u
M
f
L
Rn ⊕ Rm
f◦u+w
g
N
0
Rm
v
L
By the snake lemma 3.2.4 the sequence 0 → Ker u → Ker f ◦u+w → Ker v → 0
is exact. By 6.1.5 Ker f ◦ u + w is finite. (2) Choose v : R m → L → 0 exact with
finite kernel and w : R m → N such that g ◦ w = v. There is a diagram
0
0
0
f
M
R m
w
N
g
Rm
v
L
By the snake lemma 3.2.4 the sequence 0 → Ker w → Ker v → M → Cok w →
0 is exact. By 6.1.5 M is finite. (3) Choose w : R m → N → 0 exact with finite
kernel. Then v = g ◦ w : R m → L → 0 is exact and there is a diagram
0
0
0
f
M
R m
w
N
g
Rm
v
L
By the snake lemma 3.2.4 the sequence 0 → Ker w → Ker v → M → 0 is exact.
By 6.1.5 Ker v is finite and L is finite presented.
6.5.4. Corollary. Let
0
f
M
g
N
be a split exact sequence. Then N is finite presented if and only if M, L are finite
presented.
L
0
0
0
0
0
0
0