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

9.2. ASS OF MODULES 105
9.2.12. Definition. A non minimal prime ideal in Ass(M) is an embedded prime
of M.
9.2.13. Proposition. Let R be a noetherian ring and M a finite module. Then
Ass(M) is a finite set.
Proof. Follows immediately from 9.2.4 and 8.1.5.
9.2.14. Corollary. Let R be a noetherian ring and M a finite module. The following
are equivalent
(1) M has finite length.
(2) Supp(M) consists of maximal ideals.
(3) Ass(M) consists of maximal ideals.
Proof. (3) ⇒ (2): The minimal ideals in the support are maximal.
9.2.15. Lemma. Let (R, P ) be a local ring M a module. Then P ∈ Ass(M) if
and only if HomR(k(P ), M) = 0.
9.2.16. Proposition. Let R be a noetherian ring and M a finite module. For any
module N
Ass(HomR(M, N)) = Supp(M) ∩ Ass(N)
Proof. By 8.2.9 HomR(M, N)P HomRP (MP , NP ). So reduce to the case
where (R,P) is local. Now
HomR(k(P ), HomR(M, N)) = HomR(M, HomR(k(P ), N))
Conclusion by Nakayama’s lemma 6.4.1 and 9.2.15.
= Hom k(P )(M ⊗R k(P ), HomR(k(P ), N))
9.2.17. Proposition. Let R be a noetherian ring and F a finite module. Assume
rank F ⊗R k(P ) = n for all primes P . Then F is locally free (projective) if and
only if FP is free for all P ∈ Ass(R).
Proof. Let Q be a maximal ideal and 0 → K → R n Q → FQ → 0 exact. Ass(K) ⊂
Ass(R), so KP = 0 for all P ∈ Ass(K). By 9.2.7 K = 0.
9.2.18. Proposition. Let (R, P ) be a noetherian local ring. If there is a nonzero
finite injective module E, then R is artinian.
Proof. Let Q be a prime and f : R/Q → E. If a ∈ P \Q then a R/Q is injective,
so there is f ′ : R/Q → E such that f = f ′ ◦ a R/Q. That is P HomR(R/Q, E) =
HomR(R/Q, E), so by Nakayama’s lemma 6.4.1 HomR(R/Q, E) = 0 if Q = P .
By 9.2.7 0 = HomR(R/P, E) ⊂ HomR(R/Q, E), so Q = P . R is artinian by
8.6.6.
9.2.19. Exercise. (1) Show that
(2) Show that
Ass(Z/(n)) = {(p)|p prime dividing n}
Ass(K[X, Y ]/(X) ∩ (X 2 , Y 2 )) = {(X), (X, Y )}
and point out an embedded prime.
(3) Let I ⊂ R be an ideal such that √ I = I. Show that R/I has no embedded prime
ideals.