COMMUTATIVE ALGEBRA

YOSHIFUMI TSUCHIMOTO

07. Regular local ring is UFD(0) strategy. Supplement:

Proposition 7.1. Let A be a ring. Then dim A = sup_{𝔪∈Spm(A)} A_{𝔪} = sup_{𝔭∈Spec(A)} A_{𝔭}.

Definition 7.2. Let 𝔭 be a prime ideal of a ring A. Then we define its height ht 𝔭 to e the supremum of the lengths of prime chaines

𝔭 = 𝔭_{0} ⊋ 𝔭_{1} ⊋ 𝔭_{2} ⊋ ⋯ ⊋ 𝔭_{r}.

We have ht 𝔭 = dim A_{𝔭}.

Definition 7.3. For an ideal I of a ring A, we define its height ht I to be

ht I = inf{ht 𝔭|I ⊂ 𝔭 ∈ Spec(A)}

Definition 7.4. Let A be a ring and M be an A-module. Then a prime ideal 𝔭 of A is called an associated prime ideal of M if It is the annihilator ann(Mx of some x ∈ M. The associated primes of the A-module A/I are refereedd to as the prime divisorsof I.

Theorem 7.5. Let A be a Noetherian ring, and I = (a_{1}, …, a_{r}) an ideal generated by r elements; then if 𝔭 is a minimal prime divisor of I we have ht 𝔭 ≤ r.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Theorem 7.6. Regular local ring (A, 𝔪) is UFD.

Step 1. Induction on dim(A). If dim(A) = 0, Then A is a field. Thus it is UFD. Assume dim(A) > 0. Take x ∈ 𝔪 \ 𝔪^{2}. It suffices to prove:

(1) If A[x^{−1}] is UFD, then A is UFD.

(2) A[x^{−1}] is UFD.