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What is the correct definition of an ind-scheme?

I ask this because there are (at least) two definitions in the literature, and they really differ.

Definition 1. An ind-scheme is a directed colimit of schemes inside the category of presheaves on the Zariski site of affines schemes, where the transition maps are closed immersions.

Definition 2. An ind-scheme is a directed colimit of schemes inside the category of sheaves on the Zariski site of affines schemes, where the transition maps are closed immersions.

This makes a difference, since the forgetful functor from sheaves to presheaves does not preserve colimits. For an ind-scheme $\varinjlim_n X_n$ according to Definition 1, we have $$\hom(Y,\varinjlim_n X_n) = \varinjlim_n \hom(Y,X_n)$$ for affine schemes $Y$. Does this also hold for non-affine schemes? Probably not. But one certainly wants the category of schemes to be a full subcategory of the category of ind-schemes. For this, Definition 2 seems to be more adequate.

Any detailed references are appreciated.

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As you observe, the problem with (1) is that it depends on the choice of site. Perhaps what is intended there is for the site to only include affine schemes. In that case the forgetful functor from sheaves to presheaves preserves filtered colimits. – Zhen Lin Feb 1 at 10:13
    
Sorry, I meant affine schemes in each case (this is what I have found in the literature). Edited. – Martin Brandenburg Feb 1 at 10:35
    
The equality $\text{Hom}(Y,\varinjlim X_n)=\varinjlim \text{Hom}(Y,X_n)$ extends quite easily from affine schemes $Y$ to quasi-compact ones. Away from qc schemes, I do not think it works. – Matthieu Romagny Feb 1 at 12:44
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@MatthieuRomagny: Don't we also need that $Y$ is quasi-separated? – Martin Brandenburg Feb 1 at 15:18
    
Dear @Martin, I appreciate you asking this question! It's something I've wondered about many times before. +1! – Keenan Kidwell Feb 1 at 15:28
up vote 9 down vote accepted

There is in fact no difference between the two definitions if you take your site to be the category of affine schemes – while it is true that the forgetful functor from sheaves to presheaves does not preserve colimits in general, for the purposes of this definition, only filtered colimits matter, and those are preserved.

Indeed, recall that the Zariski topology is a Grothendieck topology "of finite type", i.e. generated by a pretopology consisting only of finite covering families. For such Grothendieck topologies, it is straightforward to verify that the category of sheaves is closed under filtered colimits in the category of presheaves – after all, in this case, the sheaf condition amounts to saying that certain cocones under finite diagrams of affine schemes are mapped to limiting cones in the category of sets.

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Thank you. So we use the property of $\mathsf{Set}$ that finite limits commute with filtered colimits. Does $\hom(Y,\varinjlim_n X_n) = \varinjlim_n \hom(Y,X_n)$ hold for arbitrary schemes $Y$ then? I think it holds when $Y$ is quasi-compact and quasi-separated. – Martin Brandenburg Feb 1 at 15:15
    
Surely it's not true in general. For example, imagine if $Y$ is a disjoint union of infinitely many affine schemes. – Zhen Lin Feb 1 at 16:43

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