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Commutative algebra

Yoshifumi Tsuchimoto

\fbox{02. Localization}

DEFINITION 02.1   Let $ A$ be a commutative ring. Let $ S$ be its subset. We say that $ S$ is multiplicative if
  1. $ 1\in S$
  2. $ x,y \in S  \implies  x y \in S $
holds.

DEFINITION 02.2   Let $ S$ be a multiplicative subset of a commutative ring $ A$ . Then we define $ A[S^{-1}]$ as

$\displaystyle A[\{X_s ; s \in S\}]/(\{ s X_s -1; s \in S\})
$

where in the above notation $ X_s$ is a indeterminate prepared for each element $ s \in S$ .) We denote by $ \iota_S $ a canonical map $ A\to A[S^{-1}]$ .

LEMMA 02.3   Let $ S$ be a multiplicative subset of a commutative ring $ A$ . Then the ring $ B=A[S^{-1}]$ is characterized by the following property:

Let $ C$ be a ring, $ \varphi:A\to C$ be a ring homomorphism such that $ \varphi(s)$ is invertible in $ C$ for any $ s \in S$ . Then there exists a unique ring homomorphism $ \psi=\phi[S^{-1}]:B\to C$ such that

$\displaystyle \varphi=\psi \circ \iota_S
$

holds.

COROLLARY 02.4   Let $ S$ be a multiplicative subset of a commutative ring $ A$ . Let $ I$ be an ideal of $ A$ given by

$\displaystyle I=\{ x \in I; \exists s \in S$    such that $\displaystyle s x=0\}
$

Then $ I$ is an ideal of $ A$ . Let us put $ \bar{A}=A/I$ , $ \pi:A\to \bar{A}$ the canonical projection. Then:
  1. $ \bar{S}=\pi(S)$ is multiplicatively closed.

  2. We have

    $\displaystyle A[S^{-1}]\cong\bar{A}[\bar{S}^{-1}]
$

  3. $ \iota_{\bar{S}}: \bar{A}\to \bar{A}[\bar{S}^{-1}]$ is injective.

There is another description of $ A[S^{-1}]$ . Namely, We consider an equivalence relateion $ \sim_S$ on a set $ S \times A $ by

$\displaystyle (s_1,a_1)\sim_S (s_2,a_2)  \iff  t(s_1 a_2 -s_2 a_1)=0 (\exists t \in S)
$

We call the quotient space space $ S\times A/\sim_S $ as $ S^{-1}A$ . The equivalence class of $ (s,a)\in S\times A $ in $ S^{-1}A$ is denoted by $ s^{-1} a$ . Then it is easy to introduce a ring structure of $ S^{-1}A$ and see that $ S^{-1}A$ actually satisfies the universal property of $ A[S^{-1}]$ . We thus have a canonical isomorphism $ S^{-1}A\cong A[S^{-1}]$ .

EXAMPLE 02.5   $ A_f=A[S^{-1}]$ for $ S=\{1,f,f^2,f^3,f^4,\dots\}$ . The total ring of quotients $ Q(A)$ is defined as $ A[S^{-1}]$ for

$\displaystyle S=\{ x \in A; x$    is not a zero divisor of A$\displaystyle \}.
$

When $ A$ is an integral domain, then $ Q(A)$ is the field of quotients of $ A$ .

DEFINITION 02.6   Let $ A$ be a commutative ring. Let $ \mathfrak{p}$ be its prime ideal. Then we define the localization of $ A$ with respect to $ \mathfrak{p}$ by

$\displaystyle A_\mathfrak{p}=A[ (A\setminus \mathfrak{p})^{-1}]
$

DEFINITION 02.7   Let $ S$ be a multiplicative subset of a commutative ring $ A$ . Let $ M$ be an $ A$ -module we may define $ S^{-1}M$ as

$\displaystyle \{ (m/s); m\in M , s\in S\} / \sim
$

where the equivalence relation $ \sim$ is defined by

% latex2html id marker 1108
$\displaystyle (m_1/s_1)\sim (m_2/s_2)  \iff  t (m_1 s_2 -m_2 s_1)=0 \quad (\exists t \in S).
$

We may introduce a $ S^{-1}A$ -module structure on $ S^{-1}M$ in an obvious manner.

$ S^{-1}M$ thus constructed satisfies an universality condition which the reader may easily guess.

By a universality argument, we may easily see the following proposition.

PROPOSITION 02.8   Let $ A$ be a commutative ring. Let $ S$ be a multiplicative subet of $ A$ . Let $ M$ be an $ A$ -module. Then we have an isomorphism

$\displaystyle S^{-1} A \otimes_A M \cong S^{-1}M
$

of $ S^{-1}A$ -modules.

PROPOSITION 02.9   Let $ A$ be a commutative ring. Let $ S$ be a multiplicative subet of $ A$ . Then the natural homomorphism $ A\to S^{-1} A$ is flat.




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Next: local rings
2011-04-21