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I've heard it been said that the construction of Spec$R$ is a canonical way of taking the ring $A$ and producing a locally ringed space with $A$ as the ring of global sections. This is certainly informal; but is it correct in some technical sense? If it was, we might expect to find $\text{Spec}(-):\text{Ring}^{op}\to\text{LRSpace}$ (or indeed $\text{Spec}(A)$) characterized by some universal property. So I wonder: is this so?

Sincerely, Eivind

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    Isn't the thing you're looking for simply the [anti-equivalence between affine schemes and commutative rings](http://en.wikipedia.org/wiki/Scheme_(mathematics)#The_category_of_schemes), so it's hardly possible to get "more canonical"? If you don't want to restrict to affine schemes you still get an adjoint pair, so you have a universal property "for free".2011-05-25
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    I knew about that, but I'm thinking about $\text{Spec}$ as a functor $\text{Ring}^{op}\to\text{LRSpaces}$. This question may still be silly though :)2011-05-25
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    I clarified this assumption in the OP.2011-05-25

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The functorial bijection (where ($X,\mathcal O_X$) denotes a locally ringed space which is not necessarily a scheme) $$Hom_{LRS}(X, Spec(A))=Hom_{Ring} (A,\Gamma(X,\mathcal O_X))$$

might be the universal property of $Spec(A)$ you are looking for .

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    In other words, there is a contravariant right adjunction between $\mathrm{Spec}\colon \mathcal{R}ing^{op}\to LRS$ and $\Gamma\colon LRS^{op}\to\mathcal{R}ing$ (global sections)2011-05-25