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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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    I clarified this assumption in the OP.2011-05-25

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The natural bijection $$ \mathrm{Hom}_{\mathrm{LRS}}(X, \mathrm{Spec}(A)) \cong \mathrm{Hom}_{\mathrm{CRing}}(A,\Gamma(X, \mathcal{O}_X)) $$ might be the universal property of $\mathrm{Spec}(A)$ you are looking for. Here $(X,\mathcal{O}_X)$ denotes a locally ringed space which is not necessarily a scheme.

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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