In "introduction to algebraic geometry and algebraic groups" Gabriel and Demazure proved the following theorem(section 4 theorem 4.1.):

Let $\mathcal{LRS}\rightarrow{Sh}_{Zar}$ be the functor that acts on objects as follows:$$X\mapsto{\mathcal{LRS}(Spec(-),X)}$$ Then it has a left adjoint.

$\mathcal{LRS}$ denotes the category of locally ringed spaces and $Sh_{Zar}$ denotes the category of sheaves on commutative rings with respect to zariski topology.

Is this theorem true? Authors are using in proof some colimits formulas over large index category. The problem is that $\mathcal{LRS}$ is small cocomplete, so there are no reasons to believe that such colimit would be contained in chosen universe. Moreover, if this theorem holds, then one will always have coarse moduli space for any zariski sheaf at least as locally ringed space. So I really have doubts about it.

  • $\begingroup$ Can you clarify what book you are referring to? Demazure and Gabriel wrote a book (in French), but with a much shorter name. $\endgroup$ – Jim Humphreys Feb 8 '13 at 11:55
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    $\begingroup$ @Jim: The first two chapters were translated into English. The precise reference is: P. Gabriel, M. Demazure, Introduction to algebraic geometry and algebraic groups, North Holland 1980. But the Theorem is also included in the original French version. $\endgroup$ – Martin Brandenburg Feb 8 '13 at 16:45

You probably mean Theorem 4.1 in Chapter I, Paragraph 1. With your formulation, it is in fact wrong. But note that Demazure-Gabriel have been careful as for the set-theoretic foundations: See page xiii for the general conventions. Instead of the full category of rings, they consider a full subcategory $\mathsf{M}$ of "very small" rings (which belong to the universe $U$, which is contained in the universe $V$ where the other rings live), also called models. The category of $Z$-functors is defined as the category of functors $\mathsf{M} \to \mathsf{Set}$. The theorem states that the functor from locally ringed spaces to $Z$-functors has a left adjoint. But this left adjoint is just the left Kan extension of $\mathrm{Spec} : \mathsf{M} \to \mathsf{LRS}$ which exists since $\mathsf{LRS}$ is $V$-cocomplete and $\mathsf{M}$ is $V$-small.

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  • $\begingroup$ Thank you. I wasn't so carefull while reading first pages of the book. $\endgroup$ – crsr Feb 7 '13 at 22:19

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