Firstly, sorry for the blog silence! Christmas, New Year, birthday and exams all got in the way… but now we’re finally onto this Zariski site post I’ve been going on about!

If you remember some of the stuff from this post about sheaves on sites, and this post about the functor of points of a scheme , then you’re good to go here. We introduced the functor of points of a scheme , namely the representable presheaf , and looked at how it relates to the solutions of Diophantine equations in the affine case . We also saw here how if is a -scheme then the set of “rational points” of identifies with and with the sections of the morphism . And then I mentioned somewhere that it is possible to “identify” a scheme with its functor of points.

What this really means is that we can “embed’ the category of schemes into the larger category of presheaves on the category of schemes (this is just the Yoneda embedding), but we can also completely characterise when a presheaf on the category of schemes is the functor of points of some scheme. To do this, we had to introduce the notion of what a **sheaf on a category** was, so we needed Grothendieck (pre)topologies. It will turn out that a presheaf on the category of (affine) schemes is the functor of points of a scheme if and only if it is a sheaf on the Zariski site and a certain “covering condition” is satisfied.

To explain why I have put the word “affine” in brackets above, we’ll prove the following nice lemma:

**Lemma**: Let be a ring, and the category of schemes over . The functor

is full and faithful, where the presheaf $text{Mor}_R (-,X)$ denotes the presheaf on the category of -schemes.

This looks very like the usual Yoneda embedding except that the functor category here is slightly different. We’re sending each -scheme to its functor of points , but this time not considered as a presheaf on the category of all -schemes but now just as a presheaf on the subcategory of affine -schemes; in other words, as a covariant functor from the category of -algebras to sets. The lemma says that each functor of points is completely determined by its action on affine -schemes, which in some way just reflects the fact that all schemes are just glued together from affine schemes.

**Proof of the lemma**: Let and let denote the restriction of the presheaf to the category of affine -schemes. We want to show that every natural transformation comes from a **unique** morphism of -schemes . To do this, let be a cover of by affine schemes, and let

denote the inclusions (which are morphisms of -schemes). Then is a morphism . Then using the “glueing” of the topological spaces, continuous maps and structure sheaves, we get a unique morphism of schemes such that the restriction of to is .

You can check that the natural transformation is the image of under , so is a full functor. Now again using that schemes are glued up from affine schemes, we see that if we have two morphisms that differ then they must differ on one of the affine covers . Then by the construction above, the induced natural transformations will not be equal because they differ on their component at . Thus is a full functor. This completes the proof.

So a (relative) scheme is determined by the values its functor of points takes on the category of affine (relative) schemes, which really is just the dual of the category of rings. This gets around the awkward circular attempt to **define** a scheme as a certain type of presheaf on the category of schemes, because we already know what the category of rings is. When we eventually categorise schemes as certain sheaves on the category of affine schemes (the dual of the category of rings) we won’t need to already know what a scheme is because everything goes through purely categorically using that the category of affine schemes is dual to rings, and we can define topologies on this dual category.

**The Zariski Topology**

Let denote the category of affine schemes (i.e. the dual of the category of rings). To specify what the Zariski pretopology is on we need to say what the covering families are:

Suppose is a family of morphisms in . This family is a **covering family for the Zariski pretopology** iff

- Each ring is the localisation of at a single element ;
- The morphism is the functorial inclusion induced by the localisation map ;
- There exists a finite set such that .

The first condition means we want to think of each as a function on ; the second condition means we want the inclusion to be the subset of the space on which the function is nonzero; the third condition expresses that these functions should form a “partition of unity“, a useful geometric condition.

You can check that these satisfy the axioms for a pretopology on given in this post – clearly, the family is a covering family induced by localisation at ; the pullback condition holds by looking at the images of these localised elements under homomorphisms; the final condition holds via composition of localisations.

This defines a pretopology on denoted , the **Zariski pretopology**. This induces a unique Grothendieck topology on the category, and we denote the site by , the affine Zariski site. A **sheaf** on this site is then a functor

such that for every covering family

the set is the equaliser of the two maps induced by restrictions, where the pullback in the category of affine schemes is defined as

i.e. the spectrum of the tensor product of the -algebras and .

**Open subfunctors**

We’re nearly ready to state the classification theorem for schemes as functors; we just need one final notion:

Suppose that is a subfunctor, where both are functors from rings to sets. We say that is an **open subfunctor** if whenever is an affine scheme and is a natural transformation, the morphism

is isomorphic to the natural transformation induced by the inclusion of an open subscheme.

One can show that the open subfunctors of a functor of points are precisely the functors of points of open subschemes of .

**Characterisation of schemes amongst all functors**

We’ve seen that the whole category of schemes embeds inside the category of presheaves on the category of affine schemes. Let’s now see what that embedded category looks like:

Functor** classication theorem:** Let be a presheaf on the Zariski site (i.e. a functor from rings to sets). Then is the functor of points of a scheme if and only if

- is a sheaf on the Zariski site ;
- there exists rings and open subfunctors such that for each field , is the union of the images of the sets under the -components of the natural transformations .

This is actually not hard to prove – just using that schemes are glued up from affine schemes. Although it’s abstract it’s somehow closer to home than viewing schemes as geometric spaces, because we understand algebra (rings) well in the abstract sense, and we understand categorical algebra well too – the geometry of schemes then emerges as categorical algebra on the category of rings. I’ll try to emphasise the two points of view throughout this blog, although the “ringed space” viewpoint will undoubtedly serve us better until we start doing more advanced stuff, where it is possible to change the Zariski site to “finer” sites like the étale site having better properties. It’s also necessary to use the functor of points viewpoint for anything involving stacks, and generalisation of schemes, and this particular arm of algebraic geometry makes heavy use of this “geometry-through-algebra” idea.

Next time I’m leaving all this abstract nonsense and going back to doing some real geometry with schemes, starting from the basics.

Hey, thought I’d stop by your blog (I just answered one of your questions on MSE).

A few comments.

1) Missing TeX: control f: “$text{Mor}”.

2) This is a nice result philosophically, but it’s also nice technically. As you probably know, a huge, and persistent, problem is writing down some functor and showing it’s representable. This gives a very clean way of checking this by rephrasing the general `gluing construction’ into a ‘check these conditions’ type result.

3) An interesting question is when a map of schemes can be totally encapsulated in (i.e. phrased entirely in terms of rings). In particular, one might call a map of schemes *representable* if when one takes the associated map of sheaves on it has the property that is affine for any map —so, your notion of an open embedding is a representable morphism. If one has this, then one could essentially reduce the entirely study of to these various ring maps. Of course, this is just the notion of an affine morphism of schemes!

4) One might wonder what happens if you takes your abstract classification theorem and replaces it with finer topologies (e.g.e the étale topology, the fppf topology, the fpqc topology, etc.) on $\mathsf{Aff}$ or $\mathsf{Sch}$. This is a very important concept and leads to, at least in the case of the étale topology, the notion of an *algebraic space* (with some minor details about the diagonal map).

This may not seem like such a big deal, but it actually amplifies your result. Namely, someone might say that your result is purely academic since, after all, it’s much more natural to think about schemes as locally ringed spaces. Unfortunately, many natural sheaves on the category of schemes are NOT representable (by schemes) (e.g. natural notions of the Picard scheme which essentially classifies line bundles) but the sheaves are, with the correct definition, algebraic spaces.

The skeptic says “great, let’s just find what locally ringed spaces these ‘algebraic space’ sheaves are representing and deal with those”. Well, as I recently learned, algebraic spaces aren’t just defined purely as sheaves because “screw it, we’ll identify it with its sheaf anyways” but because they CAN’T be faithfully embedded into the category of locally ringed spaces. They can only be thought about sheaves.

Of course, if one also relaxes the idea that you’re dealing with a sheaf of sets and, instead, passes to a ‘sheaf of groupoids’ (which is really a -sheaf) one recovers the incredibly useful, incredibly powerful notion of a stack, which is really an ‘algebraic stack’ if one imposes axioms on the -sheaf similar to those in your classification theorem.

3) Someone might ask *why* would someone think to look for a larger geometric category, containing the category , in its category of presheaves? Well, here’s a really nice result. One thinks about a colimit of objects as being a ‘formal gluing’ of those objects. So, the natural place to look for a category of geometric objects that come from ‘gluing together affine schemes’ would be the minimal category where all such formal gluings exist.

As it turns out, the category of presehaves is precisely this category. Namely, the presheaf category on is the ‘free cocompletion’ of , and so the natural place to look for these glued objects!

Thanks for writing this nice post!

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Hey Alex, thanks for your comments and sorry it’s taken me a while to get back to you! I really appreciated this comment – I found this result about schemes as certain sheaves to be really beautiful and it’s great to see a few extensions and some different ways of thinking about the idea!

I’ve been aware of the concept of “representability” for a few years now, but I think it’s only really quite recently that I had a eureka moment and started to understand what it truly means – snapshots of an object “from all possible viewpoints” determine that object uniquely, but when does a collection of different snapshots fit together into a picture of a uniquely defined object? Coupling this with the result you mentioned about being the free cocompletion of , I suddenly realised how useful it was to think of schemes functorially. In some ways although it’s more abstract it helps to remove all the horrendously weird topological stuff with nonclosed points etc that makes schemes as ringed spaces also difficult to understand at first.

So: firstly, you mentioned algebraic spaces are what you get when you look at sheaves on the site of affine schemes with the étale topology (modulo some diagonal condition). Algebraic spaces seem to be a pretty big thing in the world of algebraic geometry “beyond schemes”, and it’s really cool how they can’t be faithfully embedded in the category of locally ringed spaces! Although I haven’t looked at any of the other Grothendieck topologies out there (e.g. fppf, fpqc), are there other notions of space that you’d get if you changed the étale site to one of these sites, for instance? And if so, why aren’t they as popular as algebraic spaces??

Secondly, I have heard that stacks are generalisations of algebraic spaces. Are stacks then certain “2-sheaves” on the étale site specifically, or can they be defined on other sites? What about the Zariski site?

Finally, stacks seem to be popular because they remove “automorphism problems” when trying to construct moduli spaces. This is something I’ve been trying to wrap my head around from a philosophical “representability” viewpoint, but I’ve been struggling with this. My limited attempt at making sense of the idea is this: suppose we wanted to know if a moduli space existed for a functor assigning isomorphism classes of some interesting geometric objects (e.g. line bundles, not that I know anything about these). So a morphism into this moduli space should correspond to isomorphism classes of these geometric objects on , if this moduli space existed. The problem is then that if we precompose with an automorphism of then we get a different element of but (morally) the same isomorphism class of objects in , so a fine moduli space might not exist. Stacks then come into the picture because somehow keeping track of the actual isomorphisms of these geometric objects (e.g. line bundles) rather than just isomorphism classes removes this defect. The downside being having to use a sheaf that takes values in groupoids rather than sets.

Is this the *right* explanation for the problems of automorphisms and the need for stacks? Can you think of a better way to explain what goes wrong with automorphisms? Stacks are, ultimately, something I’m really interested in learning more about, but this confusion about automorphisms has been slowly taking shape my mind for a while.

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I agree with your sentiment—seeing it ‘from all perspectives’ was a changing point in my idea of presheaves. I used to fancifully think of analyzing the shadow cast by position a light source from all possible angles, and how these composite shadows give you a full picture of the object involved. That said, while I agree that the functor of points perspective is nice, it certainly is missing *something*. The reason why algebraic geometry is so great is that we *can* see geometry in situations which have no right to possess it (e.g. varieties over finite fields). For example, in those Hida notes I sent you, he defines as being the functor which assigns to the set of surjections for a projective -module . This, while of course correct (being an amelioration of a line bundle with globally generating sections) sort of misses the point of —I can’t ‘see it’ and that makes me unhappy. To boot, there was a question in those notes which seemed non-obvious to me until I thought about the topological picture!

As for your second question that depends on what we mean by changing the topology. Let us say that a map of sheaves on a site (with underlying category ) is *representable* if for all schemes and maps (conflating with ) the fiber product is a scheme. An algebraic space is then just a sheaf on the étale site over such that there exists an étale surjection for some scheme —what it means to be an étale surjection is that the map is representable and for all schemes and maps the map of schemes is an étale surjection (again, ignoring diagonal issues which are actually important). Then, you might try and modify this in two ways. First, you might try and demand that is a sheaf on a finer topology. Thanks to Gabber though, we know that this is not actually going to give you anything new (cf. http://stacks.math.columbia.edu/tag/0APL). You might then try and get some sort of fpqc covering to replace the étale covering by schemes. This is much more finnicky. See this post of de Jong: http://math.columbia.edu/~dejong/wordpress/?p=1927. So, I think morally, finer topologies don’t really give you anything new—might as well deal with the simplest one!

Stacks can be defined on any site, and the notion of -sheaf is a correct one. But, it’s a lot more scary than it sounds (but to understand why it’s correct takes some introspection). Naively, you want a stack to just be a functor from site to the category of categories (or, in practice, almost always lying in the full subcategory of groupoids). Unfortunately, for technical reasons, this is never what you actually have in your possession. Let me give you a silly example. Take the big Zariski site on (i.e. underlying category is and coverings are Zariski coverings). Then, there is a natural ‘presheaf’ given by is just, well, the category of quasi-coherent sheaves on . The mappings take a map of schemes to the pullback map .

This is a nice functor, right? Well, not really. Namely, even though it’s silly it’s not true that this map satisfies the axioms for a functor. For example, if you have consider the map then . So, is EQUAL to the identity functor on ? Well, no. But, it’s naturally isomorphic to. Much more cumbersome is the fact that in general is NOT EQUAL to —just naturally isomorphic to. The issue is that functors are never really equal (in practice) they are just isomorphic, but the idea of a literal functor would require that on the nose—not just that they’re isomorphic. So, the -part is literally just accounting for this difficulty—it’s not a functor, but it’s a functor up to natural equivalences (which makes sense since the -morphisms in are natural equivalences).

An algebraic stack, the real correct generalization of an algebraic spaces, is basically a stack with a smooth cover by a scheme (in the similar sense to a smooth cover talked about above).

But, yes, you can have a stack on any site. A good example of one that is non algebro-geometric is the following. Consider the category of topological spaces with the ‘Zariski topology’ (coverings are actually open coverings). Then, there is, for a topological group , the stack which associates to any space the groupoid of -torsors on (i.e. principal homogenous spaces under over ). So, for example, is just real line bundles on . Then, this is a stack on (with the Zariski topology). In fact, (up to homotopy considerations) these ‘classifying stacks’ are actually representable by classifying spaces (or ‘s).

What you said about automorphisms is good intuition. Let me provide another perspective which, to me, is very enlightening. So, the most rigorous way of phrasing what I’m about to say is in terms of stacks, but since you’re trying to learn stacks, it’d be a little circular. So, let me instead just be a little imprecise. Let’s think about a very nice example of a non-representable functor and how automorphisms are getting in the way. Namely, let’s define to be the presheaf on which takes a scheme to the isomorphism classes of elliptic schemes oveR —if you’re not comfortable with elliptic schemes, just think that they’re relative elliptic curves and so, in particular, if is a field they’re just elliptic curves—and which on morphisms is just pullback(=fiber product). I claim that this is NOT representable.

Why? Well, if were representable, then the map would have to be injective. Why? Well, there is a super elementary proof just thinking about what this map does, but there is also a highfalutin way of thinking about it. Namely, is an fpqc cover (fpqc=fidelment plat, quasi-compact=faithfully flat+quasi-compact=flat+surjective+quasi-compact, all of which the map are). Grothendieck proved that every representable presheaf is a sheaf for the fpqc topology and so this covering should give an injection on the level of points. But, this map is NOT injective. Why? Well, there are elliptic curves non-isomorphic over which become isomorphic over (e.g. and for any non-square ). So, is NOT a sheaf for the fpqc topology, and so not representable.

But, how can we measure what went wrong. Namely, how can we, in the example above, think about the non-injectivity of the map ? In a more pointed format, given what is ? Well, it turns out that, using the theory of twists, it’s precisely . Why does this make sense?

To make sense of this, think about the isomorphism ? Why does this make sense? Because, to give a line bundle on one can give a cover of , consider the trivial line bundles on each , and then glue along intersections which amounts to choosing identifications/isomorphisms (where ). But, not any set of isomorphisms will work, they must satisfy the cocycle condition that . Moreover, two sets of gluing data can give isomorphic line bundles, so we must not consider all gluing data, but mod out by ‘trivial gluing data’. So, with this observation, and the realization that the automorphism sheaf of is one can see that this gluing data modulo trivial gluing data is precisely an element of the first Cech cohomology group .

Thus, to summarize: we see that the the cohomology of the automorphism group latex x$ and glue them together to a global object possibly NOT isomorphic to . Thus, in the elliptic curves case, thinking about as being a cover, the group is just a group describing how to take the local piece and ‘glue it’ down to an object over the global space . In other words, it’s completely describing (remember .

Thus, to summarize, the first cohomology group of the automorphism sheaf of an object (NB: one should really say that the object is in a stack!) is describing exactly objects locally isomorphic to , which is exactly describing the non-separatedness of the presheaf of isomorphism classes of objects (or at least, how multiple things can map to the same point as which breaks separatedness).

One then tries to fix this problem by completely separating objects, no longer identifying them if they are isomorphic, not equal. This, in some sense, totally eliminates the issue with isomorphisms.

There are more nice things to say, but, at this point, maybe I should just write a blog post about it. 🙂

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(Sorry about the bad LaTeX ;;, there is no preview button!)

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Hey Alex, sorry for cluttering your comment section. 🙂

I found this nice result today that, I think, is much nicer than what I claimed above.

The following four definitions are equivalent:

1) is an étale sheaf with an étale surjection for some scheme .

2) is an étale sheaf with a smooth surjection for some scheme (i.e. roughly, Artin stacks which are sheaves are algebraic spaces).

3) As in 1), but it’s an fppf sheaf.

4) As in 2), but it’s an fppf sheaf.

These are all equivalent definitions of an algebraic space. Some of these are easy to show equivalent (e.g. 1) and 2)), some are hard (e.g. 4), which is proved in Chapter 10 of Moret-Bailey’s Champs Algebrique).

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Alex – you’ve written some fantastic comments here and given me loads to think about! Thank you for taking so much time to write these 🙂

Firstly, it’s nice to see these four equivalences about different types of sheaves and it gives good evidence for why I only see the term “algebraic spaces” all over the place rather than sheaves built on other more exotic topologies.

Secondly, what you wrote about stacks was great! There’s lots there that I’m still technically out of my depth with, but I got some real intuition and motivation for why they’re important. The example involving elliptic schemes was especially clear – it’s cool to see such abstract stuff such as representability coming into close contact with the fact that the two elliptic curves you described are nonisomorphic over .

So this functor you described isn’t representable by any scheme. I’m guessing with your remark at the end about completely separating objects that if we looked instead at categories of elliptic schemes/curves over each scheme in then this would become (after appropriate technical meddling) an example of an algebraic stack, right? Is this the “moduli stack of elliptic curves” that I’ve heard about, or is that something else?

And with regards to what you said about 2-sheaves: it’s interesting to note that they’re not even functors! Having idly thought about what a 2-sheaf meant I always just assumed it was the sheaf condition that needed to be modified with some natural isomorphisms instead of actual equalities – I didn’t realise things could be so “2-categorical” at the most basic level. Would you say one of the hardest parts about initially understanding stacks is getting your head around things only being equal up to natural isomorphisms?

Anyway, if you do decide to write a blog post extending any of this stuff I look forward to reading it!

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Dear Alex (again),

Essentially the answer to your first question is yes. To any stack valued in groupoids, there is an associated presheaf (I used to write —but people got angry at me :P) defined by declaring that are the isomorphism classes of objects in the groupoid (the notation then seems apropos since in a groupoid two objects are isomorphic if and only if they’re connected [there is an arrow between them]). So, one might then phrase your question as “Is there a stack with ?”

Of course, there is! Namely, let be the -presheaf (bad terminology for those who are used to classical definitions) that sends a scheme to the category with objects elliptic schemes and morphisms isomorphisms. Then, one can show that is, in fact, a stack (even a smooth Deligne-Mumford stack!) and, moreover, evidently.

It should be noted here that the ‘elliptic curve’ quality of our objects is serving a little different purpose than we usually think about. Namely, we usually think about the difference between an elliptic curve and a genus curve as the former having a group structure, and the latter not. But, of course, this is equivalent to choosing a point (a point determines the group structure uniquely, and vice versa). Here though, it’s *not* the group structure that’ important—it’s the point.

Namely, it’s *not* true in general that schemes satisfy effective descent—it’s not true that the -presheaf (with isomorphisms as morphisms) is a stack. We can’t, in general, glue étale local schemes together to get a scheme (this is, in fact, essentially what an algebraic space is, and there are algebraic spaces that are not schemes!). In fact, it’s not even true that restricting this -presheaf to something like maps to the category of projective -schemes (with morphisms as isomorphisms) is a stack.

There are two smaller subcategories where this is fixed. First, it’s true that *affine* schemes form a stack. It’s also true that quasiprojective schemes with a *fixed* relatively ample bundle are a stack. One can then think about the point (really, a section) of an elliptic scheme as specifying this relatively ample bundle, thus allowing gluing to occur.

In particular, the -presheaf sending to just ‘genus curves over ‘ (with morphisms isomorphisms) is *not* a stack, the point—which is a relatively ample divisor—on an elliptic curve is needed to make it a stack.

Of course, one can also define —the -presheaf of relative genus -curves with -specified points. This is stack as long as , and, in this case, it’s even a DM stack. One can then try and compactify these objects (which are not proper) by consider ‘stable curves’, which give you a proper stack .

As to your question about whether the initial difficulty is wrapping one’s head around things only being up to equivalence—sort of. It is the initial headache but, honestly, while it’s important, it’s good to just know that it is a thing and, by and large, ignore it. Of course, one needs to be careful when one is doing something serious, but one is fairly well-served in general ignoring these technical details (always, as I said, being aware that they exist) and focusing on the actual content of these objects.

Amusingly enough, I am actually in the process of writing a blog post. It’s not about algebraic stacks per se but, more generally, stacks. The goal is to understand intuitively what a ‘torsor’ and ‘gerbe’ are, and how to use them.

Hope you enjoy it. 😛

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OK, this makes sense – so the presheaf is sort of a “de-(2-)categorification” of the stack . To try and work out my own intuition for this, the cases you mentioned where this does actually give a stack – e.g. for affines or for quasiprojective schemes with a distinguished point – seem to be “easy to glue”. I suppose the presheaf assigning the category of affine schemes is actually a stack because all schemes are glued up from affines, and in the second case the additional data of the fixed bundle rigidifies things to make glueing easier.

Anyway, I don’t want to ask you too many more questions as you’ve already written so much on here for me. Thank you so much for this – and I really look forward to reading your own post about stacks!

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