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general localization of a commutative ring

We define a localization of a commutative ring in a more general situation than in subsection 1.3.

DEFINITION 1.37   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  xy \in S $
holds.

DEFINITION 1.38   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 1.39   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 1.40   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 (1) $ I$ is an ideal of $ A$ . Let us put $ \bar{A}=A/I$ , $ \pi:A\to \bar{A}$ the canonical projection. Then:

(2) $ \bar{S}=\pi(S)$ is multiplicatively closed.

(3) We have

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

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

EXAMPLE 1.41   $ 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 1.42   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}]
$


next up previous
Next: general localization of modules Up: (Usual) affine schemes Previous: kernels, cokernels, etc. on
2007-12-11