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ring homomorphism and spectrum

LEMMA 1.5   Let $ A$ ,$ B$ be two ring homomorphisms. Let

$\displaystyle \alpha: A\to B
$

be a ring homomorphism (which we always assume to be unital).

Then we have a associate map

$\displaystyle \operatorname{Spec}(\alpha): \operatorname{Spec}(B)\to \operatorname{Spec}(A)
$

defined by

$\displaystyle \operatorname{Spec}(\alpha)(\mathfrak{p})=\alpha^{-1}(\mathfrak{p}) \qquad(\forall \mathfrak{p}\in \operatorname{Spec}(B)).
$

The map $ \operatorname{Spec}(\alpha)$ has the following properties.
  1. $\displaystyle \operatorname{Spec}(\alpha)(\mathfrak{p})=\{ f \in A; \rho_\mathfrak{p}(\alpha(f))=0 \}
$

  2. $\displaystyle \operatorname{Spec}(\alpha)^{-1}(O_f)=O_{\alpha(f)}
$

    for any $ f\in A$ .
  3. $ \operatorname{Spec}(\alpha)$ is continuous.



2007-12-11