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

(2) P MP = MP for all prime ideals P .
(3) P MP = MP for all maximal ideals P .
(4) M ⊗R k(P ) = 0 for all prime ideals P .
(5) M ⊗R k(P ) = 0 for all maximal ideals P .
Proof. Combine 6.4.1 with 5.4.1.
6.4. NAKAYAMA’S LEMMA 79
6.4.5. Corollary. Let (R, P ) be a local ring and f : M → N a homomorphism.
Assume N is finite. The following conditions are equivalent.
(1) f is surjective.
(2) f(P ) is surjective.
Proof. f is surjective if and only if Cok f = 0. Cok f is finite, so it is zero if and
only if Cok f ⊗R k(P ) = Cok(f(P )) = 0, 6.4.1.
6.4.6. Corollary. Let R be a ring and f : M → N a homomorphism. Assume N
is finite. The following conditions are equivalent.
(1) f is surjective.
(2) f(P ) is surjective for all prime ideals P .
(3) f(P ) is surjective for all maximal ideals P .
6.4.7. Corollary. Let (R, P ) be a local ring and M a finite module. Let x1, . . . , xn ∈
M. The following conditions are equivalent.
(1) x1, . . . , xn generate M.
(2) x1, . . . , xn generate M ⊗R k(P ).
6.4.8. Corollary. Let R be a ring and M a finite module. Let x1, . . . , xn ∈ M.
The following conditions are equivalent.
(1) x1, . . . , xn generate M.
(2) x1, . . . , xn generate M ⊗R k(P ) for all prime ideals P .
(3) x1, . . . , xn generate M ⊗R k(P ) for all maximal ideals P .
6.4.9. Proposition. Let f : R n → R m be a homomorphism represented by an
m × n-matrix A. The following are equivalent.
(1) f is surjective.
(2) n ≥ m and the ideal (m − minors) = R.
Proof. (1) ⇒ (2): If f is surjective, then for any maximal ideal f(P ) is surjective
linear map. So n ≥ m and some m-minor is nonzero in k(P ). Therefore (m −
minors) is not contained in P , so (m − minors) = R. (2) ⇒ (1): f(P ) is
surjective for all maximal ideals, so f surjective by 6.4.6.
6.4.10. Proposition. Let f : R n → R m be a homomorphism represented by an
m × n-matrix A. The following are equivalent.
(1) f is injective.
(2) n ≤ m and the ideal Ann(n − minors) = 0.
Proof. See the proof 6.2.5 (2).
6.4.11. Exercise. (1) Let R be a domain and f : R n → R m a homomorphism represented
by an m × n-matrix. Show that f is injective if and only if n ≤ m and some
n − minor = 0.