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let $\phi: R \to A$ a ring homomorphism and $f:Spec(A) \to Spec(R)$ the induced morphism on Specs. Let $p \in Spec(R)$. So $ R_p$ is local. This induces following commutative diagramm:

$$ \require{AMScd} \begin{CD} Spec(A_p) @>{a} >> Spec(A) \\ @VVgV @VVfV \\ Spec(R_p) @>{b}>> Spec(R); \end{CD} $$

a, b are inclusions (see property of localisations) and g is induced by f canonically. My question is how to see that $Spec(A_p) \cong Spec(A) \times _{Spec(R)} Spec(R_p)$ (therefore a pullback)?

user267839
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  • By definition $A_p=R_p\otimes_R A$ (note that $p$ is not necessarily a prime ideal in $A$). Tensor products are fibered coproducts and $\operatorname{Spec}$ sends tensor products to fibered products. – Roland Jul 10 '17 at 16:08
  • Why does Spec sends tensor producs to fibered products? – user267839 Jul 10 '17 at 16:20
  • Do you know that there is a natural isomorphism $\operatorname{Hom}{Alg}(A,\Gamma(X,\mathcal{O}_X))=\operatorname{Hom}{Sch}(X,\operatorname{Spec}A)$ ? From this, it follows that $\operatorname{Spec}$ sends colimits to limits. – Roland Jul 10 '17 at 16:27

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