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Speculation and background

Let $\mathcal{C}:=CRing^{op}_{Zariski}$, the affine Zariski site. Consider the category of sheaves, $Sh(\mathcal{C})$.

According to nLab, schemes are those sheaves that "have a cover by Zariski-open immersions of affine schemes in the category of presheaves over Aff."

In SGA 4.1.ii.5 Grothendieck defines a further topology on $Sh(\mathcal{C})$ using a "familles couvrantes", which are families of morphisms $\{U_i \to X\}$ such that the induced map $\coprod U_i \to X$ is an epimorphism. Further, he gives another definition. A family of morphisms $\{U_i \to X\}$ is called "bicouvrante" if it is a "famille couvrante" and the map $\coprod U_i \to \coprod U_i \times_X \coprod U_i$ is an epimorphism. [Note: This is given for a general category of sheaves on a site, not sheaves on our affine Zariski site.]

Speculation: I assume that the nLab definition means that we have a (bi)covering family of open immersions of representables, but as it stands, we do not have a sufficiently good definition of an open immersion, or equivalently, open subfunctor.

It seems like the notion of a bicovering family is very important, because this is precisely the condition we require on algebraic spaces (if we replace our covering morphisms with etale surjective morphisms in a smart way and require that our cover be comprised of representables).

Questions

What does "open immersion" mean precisely in categorical langauge? How do we define a scheme precisely in our language of sheaves and grothendieck topologies? Preferably, this answer should not depend on our base site. The notion of an open immersion should be a notion that we have in any category of sheaves on any site.

Eisenbud and Harris fail to answer this question for the following reason: they rely on classical scheme theory for their definition of an open subfunctor (same thing as an open immersion). If we wish to construct our theory of schemes with no logical prerequisites, this is circular.

Once we have this definition, do we require our covering family of open immersions to be a "covering family" or a "bicovering family"?

Further, how can we exhibit, in precise functor of points language, the definition of an algebraic space?

This last question should be a natural consequence of the previous questions provided they are answered in sufficient generality.

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I have found a definition of an open subfunctor, but it is still not sufficient. In Gabriel-Demazure, an open subfunctor U of X is defined as a subfunctor of U such that every map from a representable functor f:specA -> X, f^-1(U) can be defined by an ideal in A. –  Harry Gindi Feb 7 '10 at 22:09
    
Meanwhile, the topology I cited from SGA is still useful here, as it restricts to the canonical topology of "universally strict epimorphisms" on the category of sheaves that Grothendieck defines in 4.1.ii.2.5. This is useful because it gives us an explicit characterization of coverings. The only thing left to tie up all of the loose ends is decide whether our covers need be covers or bicovers. –  Harry Gindi Feb 8 '10 at 9:26
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5 Answers

up vote 7 down vote accepted

Check out the paper of Kontsevich-Rosenberg noncommutative space., they defined formally open immersion and open immersion completely functorial way. This definition is nothing to do with "noncommutative"

Definition: Formally open immersion is formally smooth monomorphism.

But one thing need to point out, they are working on Q-category which is a generalization of Grothendieck topologies(destination is dealing with topology without base change property).But just disregard this notion and work in usual grothendieck topology as you want

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I'm not sure if this will satisfy you, but a map of schemes is an open immersion if and only if it is an etale monomorphism. Etale means, by definition, formally etale and locally of finite presentation, both of which conditions have simple formulations in terms of functors of points, from Rings to Sets. Likewise, a map of schemes is a monomorphism if and only if the map of underlying functors is a monomorphism.

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Yes! This is the answer I was looking for. I thought that etale (at least for rings) means finitely presented (as an algebra) + formally etale. That's why my lecture notes from commutative algebra say, anyway. –  Harry Gindi Feb 8 '10 at 1:58
    
etale monomorphism is equivalent to smooth monomorphism –  Shizhuo Zhang Feb 8 '10 at 2:01
    
Actually, a morphism of sheaves can be formally étale and locally of finite presentation without being representable by algebraic spaces. One needs some kind of algebraizability hypothesis. –  Jonathan Wise Feb 8 '10 at 2:21
    
@Jonathan Wise: I think he's talking about representability by affines. That is, we can define a "covering family" as a family of morphisms $\{U_i\to X\}$ where all of the U_i are affine, and all of the morphisms are etale monic. Then if we also require that this family is a "famille (bi?)couvrante", we should meet the requirements for the sheaf to be a scheme. Correct me if I'm wrong. –  Harry Gindi Feb 8 '10 at 2:45
    
@Jonathan: I hope I didn't suggest otherwise. Does that matter? But now that you mention it, it would be nice to see an example! –  JBorger Feb 8 '10 at 3:02
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I'm having a little trouble teasing out exactly what your question is, so I'll just write some things about sheaves that seem related and hope they are helpful.

Suppose $C$ is a site. Let $\hat{C}$ be its category of presheaves and $\tilde{C}$ its category of sheaves. The topology defined in SGA 4, II.5 is on $\hat{C}$, not on $\tilde{C}$ as you suggest in your question. Its purpose is to give a topology on $\hat{C}$ such that the category of sheaves on $\hat{C}$ (that is, contravariant functors satisfying descent on the category of contravariant functors on $C$) should coincide with the category of sheaves on $C$ (i.e., $\tilde{\hat{C}} = \tilde{C}$).

You've got the condition for being bicovering backwards: a map of presheaves $H \rightarrow G$ is called bicovering if it is covering (with respect to the topology on $C$) and its diagonal $H \rightarrow H \times_G H$ is also covering. (What it means for a map of sheaves to be covering is that for any map $X \rightarrow G$ with $X$ representable, the sieve of $X$ induced by $H \times_G X$ should be covering.)

A Grothendieck topology on $C$ is described by asking certain subfunctors (sieves) of objects of $C$ to be covering. If $H$ is a subfunctor of $G$ then the relative diagonal map is automatically an epimorphism since it is an isomorphism (by definition). The covering sieves of $X$ are the subfunctors of $X$ that become isomorphic to $X$ upon passing to associated sheaves.

The bicovering business arises when one wants to study which arbitrary morphisms of presheaves (not just inclusions) become isomorphisms upon passing to associated sheaves. The notion of a covering morphism of presheaves explains which morphisms become surjections of sheaves. The question then remains: which morphisms become injections? A map of sheaves is an injection if and only if its relative diagonal is a surjection, so the condition is that the relative diagonal be a covering map.

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Aha! That is an interesting question. However, I think you also misunderstood me. What I meant to show was that a bicovering map of sheaves is the same thing as an isomorphism of sheaves. So I don't think that worrying about bicoverings will help with what you want to do. –  Jonathan Wise Feb 7 '10 at 20:09
    
Then what more do we need, given a sheaf in the Zariski topology (satisfies all descent stuff) for it to be a scheme? I was under the impression that we need a "covering", that is, an epimorphism, that is constructed as the coproduct of a family of monomorphisms (an open covering map). –  Harry Gindi Feb 7 '10 at 20:09
    
Eisenbud and Harris describe this indirectly. I'm curious about how to represent it using topologies etc. –  Harry Gindi Feb 7 '10 at 20:11
    
The problem in E&H is that they characterize schemes among sheaves by using this nasty concept of an open immersion, which relies on a classical notion of an open subscheme to define an open subfunctor. –  Harry Gindi Feb 7 '10 at 20:56
    
So, and you are looking for a purely categorical description of "open subfunctor"? I don't really understand what is going on there but perhaps you should take a look at "Categories of commutative algebras" by Diers. I chapter 4 he shows that open immersions of affine schemes correspond to so-called "singular epimorphisms". But this might just work in special cases, I don't know, too much stuff in there... –  user717 Feb 7 '10 at 22:06
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A morphism of sheaves $f: X \to Y$ in the fpqc topology on $Aff$ [covers are finite universally epimorphic families $(Spec(R_i)\to Spec(R))_i$ in $Aff$ with each morphism $Spec(R_i)\to Spec(R)$ flat] is representable by open immersions of schemes if and only if:

1) for all local schemes $Spec(R)$ with closed point $Spec(k)$ (in the category $Sh/Y$) the natural map $Hom_Y(Spec(R), X) \to Hom_Y(Spec(k), X)$ is bijection [or with condition 3) below, just surjective].

2) it is locally finitely presented (in the presheaf theoretical sense).

3) it is a monomorphism.

Notes: a) Conditions 1) and 2) only are equivalent to the map being representable by "local isomorphisms of schemes" for example the map $X \to \mathbf{A}^1$ where $X$ is the affine line with the original double and the map just folds in the double point. However, these maps are 'no good' (i.e. they do not satisfy fpqc descent).

b) A scheme is a sheaf $X$ in the fpqc topology on $Aff$ such that there exists a cover (in the canonical topology on $Sh$) by affine schemes $(Spec(R_i)\to X)_i$ with each map $Spec(R_i)\to X$ satisfing the conditions above.

c) I haven't checked but I'm pretty sure this will work with the other natural topologies (fppf, etale, Zariski).

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I think this is just equivalent to say $f:X\rightarrow Y$ is etale monomorphism –  Shizhuo Zhang Feb 17 '10 at 0:09
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I'm not sure if this is what you are after, but when I started to look at Grothendieck-topologies I thought of being an open immersion as a topological property; somehow it should be possible to recover all open immersions from the topology, and if we changed the topology to the etalé site, the same method should give us the etalé maps as "open immersions".

Unfortunately, I doubt this is possible (it would be kind of nice if it were, so please correct me if I'm wrong). The reason why I was fooled to try, was probably due to the topological flavour of the term open immersion. But passing to the Grothendieck topology loses information about our model covering maps we started with. For instance, if {U_i -> U} is a covering, then a sheaf F would satisfy the sheaf condition also for the set {U_i -> U} U {A -> U}, the latter being an arbitrary morphism.

Instead I think it is more correct to think of the property of being an open immersion as something that we know what it is in our base category (affine schemes) and want to generalise to our new, larger category.

That said, we must look for the correct definition of open immersion in our base category. We want it to be something that is the "complement" of a closed immersion. Let F be a subfunctor of spec A. Pulling it back along spec A/I -> spec A should give us the zero scheme. Now define the complement of spec A/I as the most general subfunctor F of spec A satisfying this (i.e take the categorical limit). I've not done the details, but I suspect this gives the right concept and that a concrete description of what the subfunctor looks like would be as in exercise VI.6 in E-H. Extending it to representable morphisms of sheaves (in any reasonable topology) is now straight forward using pull-backs.

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