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

4.3. EXACTNESS OF FRACTIONS 61
Proof. Represent the statements using short exact sequences. (1) The kernel is
determined by the exact sequence 0 → Ker f → M → N, 3.1.4. (3) The cokernel
is determined by the exact sequence M → N → Cok f → 0, 3.1.5. (2) The
image is determined by the exact sequence 0 → Ker f → M → Im f → 0, 2.3.5,
3.1.4.
4.3.4. Corollary. For submodules N, L ⊂ M there are natural identifications of
U −1 R-submodules and factor modules.
(1) U −1 (M/N) = U −1 M/U −1 N.
(2) U −1 (N + L) = U −1 N + U −1 L.
(3) U −1 (N ∩ L) = U −1 N ∩ U −1 L.
Proof. Represent the statements using short exact sequences. (1) 0 → N → M →
M/N → 0 is short exact giving 0 → U −1 N → U −1 M → U −1 (M/N) → 0. The
wanted isomorphism follows form 3.1.5. (2) N + L is the image of N ⊕ L → M
so conclude by 4.3.3. (3) N ∩ L is the kernel of N ⊕ L → M so conclude by
4.3.3.
4.3.5. Corollary. For ideals I, J ⊂ R there are natural identifications in U −1 R.
(1) U −1 (R/I) = U −1 R/U −1 I.
(2) U −1 (I + J) = U −1 I + U −1 J.
(3) U −1 (I ∩ J) = U −1 I ∩ U −1 J.
(4) U −1 (IJ) = U −1 IU −1 J.
Proof. (1) (2) (3) These are special cases of 4.3.4. (4) Both sides have the same
, a ∈ I, b ∈ J, u ∈ U.
generators ab
u
4.3.6. Proposition. Let R → U −1 R be the canonical homomorphism.
(1) For an ideal I ⊂ R the extended ideal
IU −1 R = U −1 I
(2) For an ideal J ⊂ U −1 R the extended contracted ideal
U −1 (J ∩ R) = J
(3) For an ideal I ⊂ R the contracted extended ideal
I ⊂ IU −1 R ∩ R
Proof. (1) This is clear. (2) U −1 (J ∩ R) ⊂ J is true for any ring homomorphism
1.2.6. If b
b
1 b
u ∈ J then 1 ∈ J giving b ∈ J ∩ R and ub = u ∈ U −1 (J ∩ R). (3) This
is true for any ring homomorphism 1.2.6.
4.3.7. Proposition. Let R → U −1 R be the canonical homomorphism. For an
ideal I ⊂ R the contracted extended ideal
I = IU −1 R ∩ R
if and only if each u ∈ U is a nonzero divisor on R/I.
Proof. Apply the snake lemma 3.2.4 to
0
0
I
0
R
U −1R/U −1I U −1R/U −1I
0
R/I
0