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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62 4. FRACTION CONSTRUCTIONS<br />
and get the exact sequence<br />
Conclusion from 4.2.6.<br />
0 → I → IU −1 R ∩ R → Ker(R/I → U −1 (R/I)) → 0<br />
4.3.8. Corollary. Let P ⊂ R be a prime ideal. Then U −1 P ⊂ U −1 R is either a<br />
prime ideal or the whole ring.<br />
Pro<strong>of</strong>. By 4.1.5 and 4.3.5 U −1 R/U −1 P = U −1 (R/P ) is either 0 or a domain.<br />
4.3.9. Corollary. Let R be a principal ideal domain. Then U −1 R is a principal<br />
ideal domain.<br />
Pro<strong>of</strong>. A restricted ideal is principal by hypothesis and the extension <strong>of</strong> a principal<br />
ideal is principal. Conclude by 4.3.6.<br />
4.3.10. Proposition. Let R be a unique factorization domain. Then U −1 R is a<br />
unique factorization domain.<br />
Pro<strong>of</strong>. By 4.3.7 the extension <strong>of</strong> an irreducible element is either a unit or an irreducible<br />
element. Since a principal ideal ( a a<br />
u ) = ( 1 ), a factorization into irreducibles<br />
in R gives a factorization in U −1R. Now if (a) = (p1) . . . (pn) is a factorization<br />
in R and ( a<br />
1 ) is irreducible in U −1R. Then all but one pi<br />
1 is a unit in U −1R so<br />
( a pi<br />
1 ) = ( 1 ) is a prime ideal 4.3.9. The conditions 1.5.3 are satisfied.<br />
4.3.11. Exercise. (1) Let U ⊂ R be multiplicative. Show that<br />
U −1 (R[X]) = (U −1 R)[X]<br />
(2) Let φ : R → S be a ring homomorphism and U ⊂ R a multiplicative subset. Show<br />
that<br />
U −1 S = φ(U) −1 S<br />
4.4. Tensor modules <strong>of</strong> fractions<br />
4.4.1. Proposition. Let R be a ring and U a multiplicative subset. For any module<br />
M, the homomorphism<br />
M ⊗R U −1 R → U −1 M, x ⊗ a ax<br />
↦→<br />
u u<br />
is a natural isomorphism <strong>of</strong> U −1R-modules. Pro<strong>of</strong>. By 2.6.3 there is an R-module homomorphism x ⊗ a<br />
u<br />
b a<br />
ba bax b ax<br />
(x ⊗ ) = x ⊗ ↦→ =<br />
v<br />
u<br />
vu<br />
vu<br />
map U −1 M → M ⊗R U −1 R, x<br />
u<br />
v<br />
ax → u . By definition<br />
u , so the this is a U −1 R-homomorphism. The<br />
↦→ x ⊗ 1<br />
u<br />
is an inverse.<br />
4.4.2. Remark. The two constructions, module change <strong>of</strong> ring to a fraction ring<br />
and fraction module are natural isomorphic functors from modules to modules over<br />
the fraction ring.<br />
4.4.3. Corollary. Let R be a ring and U a multiplicative subset. Then U −1 R is a<br />
flat R-module.<br />
Pro<strong>of</strong>. This follows from 4.4.1 and 4.3.1.<br />
4.4.4. Corollary. Let R be a ring and U a multiplicative subset and M, N modules.<br />
Then there is a natural isomorphism<br />
U −1 (M ⊗R N) U −1 M ⊗ U −1 R U −1 N