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