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If X is a scheme, we know there is a one-one correspondence between quasi-coherent sheaves of $\mathcal O_X$-algebras on X and affine morphisms $Y \longrightarrow X$

But what about arbitrary (not necessarily quasi-coherent) sheaves of $\mathcal O_X$-algebras? Do they correspond to schemes $Y\longrightarrow X$?

It seems to me that given a morphism $f:Y\longrightarrow X$ of schemes, for any $U\subseteq X$, the association $U\mapsto \mathcal O_Y(f^{-1}(U))$ defines a sheaf of $\mathcal O_X$-algebras. That's one direction. Does it not work in the other direction for some reason?

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    $\begingroup$ IIRC Hartshorne discusses the $\mathbf{Spec}$ (notice the boldness) of such a sheaf of algebras (in some exercise?); surely, he discusses the $\mathbf{Proj}$ of a sheaf of graded algebras. $\endgroup$ – Mariano Suárez-Álvarez Nov 12 '13 at 19:34
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    $\begingroup$ At least all I could find in Hartshorne is Spec of sheaves of quasi-coherent algebras. I tried several obvious sources: Hartshorne, Vakil's notes and some google searches, but both Proj and Spec appear to be for quasi-coherent sheaves of algebras. $\endgroup$ – user42697 Nov 12 '13 at 19:38
  • $\begingroup$ Ah. It may be the case that I did not recalled correctly :-) But is quasi-coherence really used for what you want? $\endgroup$ – Mariano Suárez-Álvarez Nov 12 '13 at 19:49
  • $\begingroup$ Quasi-coherence definitely matters in the proof. I had forgotten about the quasi-coherence, tried to prove it generally, got stuck...then pulled out Hartshorne and said: Ah! They assumed quasi-coherent! :P But I really want to know if the general result is true. $\endgroup$ – user42697 Nov 12 '13 at 19:54
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    $\begingroup$ Take $X$ to be a single point. What is it you want to say in that case? $\endgroup$ – Marguax Nov 13 '13 at 3:21
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You could ask about this in the category of locally ringed spaces, rather than just schemes. For every locally ringed space $(X,\mathcal{O}_X)$, and for every sheaf $\mathcal{A}$ of $\mathcal{O}_X$-algebras, there is a locally ringed space, $(S,\mathcal{O}_S)$, a morphism of locally ringed spaces, $$(\pi,\pi^\#):(S,\mathcal{O}_S)\to (X,\mathcal{O}_X),$$ and a morphism of $\mathcal{O}_S$-algebras, $\phi:\pi^*\mathcal{A} \to \mathcal{O}_S$, that represents the contravariant functor from the category of locally ringed spaces over $X$ to the category of sets that associates to each morphism of locally ringed spaces, $(f,f^\#):(Y,\mathcal{O}_Y)\to (X,\mathcal{O}_X)$, the set of morphisms of $\mathcal{O}_Y$-algebras, $\psi:f^*\mathcal{A}\to \mathcal{O}_X$. Of course when $(X,\mathcal{O}_X)$ is a scheme and $\mathcal{A}$ is quasi-coherent, $(S,\mathcal{O}_S)$ is isomorphic (over $X$) to $\textbf{Spec}_X(\mathcal{A})$ as constructed in Hartshorne's book. In this sense, the locally ringed space $(S,\mathcal{O}_S)$ deserves to be called "Spec".

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  • $\begingroup$ In their paper « Factorisation de Stein topologique. Découpe », Malgoire and Voisin use this construction in a topological setting. So let $X$ be a topological space, $F$ a closed subset, $j\colon U\to X$ the immersion of the complementary open subset, and set $\mathcal A=j_*(\mathbf Z/2)_U$. The spectrum of this algebra is a topological space $Y$ equipped with a morphism to $X$, whose fibers are compact and totally disconnected. This space $Y$ is the space obtained by cutting $X$ with scissors along the closed subset $F$. $\endgroup$ – ACL Nov 15 '13 at 22:54

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