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Commutative algebra

Yoshifumi Tsuchimoto

\fbox{07. Regular local ring is UFD(0) strategy.}

Supplement:

PROPOSITION 07.1   Let $ A$ be a ring. Then $ \dim A=\sup_{\frak m \in \operatorname{Spm}(A)}A_{\frak m} =
\sup_{\mathfrak{p}\in \operatorname{Spec}(A)}A_\mathfrak{p}$ .

DEFINITION 07.2   Let $ \mathfrak{p}$ be a prime ideal of a ring $ A$ . Then we define its height $ \operatorname{ht}\mathfrak{p}$ to e the supremum of the lengths of prime chaines

% latex2html id marker 685
$\displaystyle \mathfrak{p}=\mathfrak{p}_0 \supsetne...
...rak{p}_1 \supsetneq \mathfrak{p}_2 \supsetneq \dots \supsetneq \mathfrak{p}_r.
$

We have $ \operatorname{ht} \mathfrak{p}= \dim A_\mathfrak{p}$ .

DEFINITION 07.3   For an ideal $ I$ of a ring $ A$ , we define its height $ \operatorname{ht}I$ to be

$\displaystyle \operatorname{ht}I=\inf \{\operatorname{ht}\mathfrak{p}\vert I \subset \mathfrak{p}\in \operatorname{Spec}(A)\}
$

DEFINITION 07.4   Let $ A$ be a ring and $ M$ be an $ A$ -module. Then a prime ideal $ \mathfrak{p}$ of $ A$ is called an associated prime ideal of $ M$ if It is the annihilator ann(Mx of some $ x\in M$ . The associated primes of the $ A$ -module $ A/I$ are refereedd to as the prime divisorsof $ I$ .

THEOREM 07.5   Let $ A$ be a Noetherian ring, and $ I=(a_1,\dots,a_r)$ an ideal generated by $ r$ elements; then if $ \mathfrak{p}$ is a minimal prime divisor of $ I$ we have % latex2html id marker 742
$ \operatorname{ht}\mathfrak{p}\leq r$ .

&dotfill#dotfill;

THEOREM 07.6   Regular local ring $ (A,\mathfrak{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 \in \mathfrak{m} \setminus \mathfrak{m}^2$ . It suffices to prove:

  1. If $ A[x^{-1}]$ is UFD, then $ A$ is UFD.
  2. $ A[x^{-1}]$ is UFD.


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2012-06-15