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?

  • 2
    $\begingroup$ 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. $\endgroup$ Nov 12, 2013 at 19:34
  • 4
    $\begingroup$ 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, 2013 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? $\endgroup$ Nov 12, 2013 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, 2013 at 19:54
  • 1
    $\begingroup$ Take $X$ to be a single point. What is it you want to say in that case? $\endgroup$
    – Marguax
    Nov 13, 2013 at 3:21

1 Answer 1


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".

  • 2
    $\begingroup$ 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, 2013 at 22:54

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.