# 4.1. 范畴叠

In short, stacks are just category fibered in groupoids where descent holds.

What this means is that, for example, take fppf topology on schemes and choose any sheaf . Then we say is “geometric” if for some scheme . Those are of course example of stacks.

Hence, in general, we want “representable stacks” (the technical term is algebraic stacks, or Artin stack) instead of arbitrary stacks.

Our next goal is to get a feeling about cat stacks, and then we try to find what would be a nice notion for algebraic stacks.

Well, I lied. Here is the punch line for what algebraic stacks are.

Thus, what should smooth cover be? Well, it should have the property that, for any scheme , if we take fibered productthen the arrow is a smooth cover. Well, this helps a little bit, as now over base becomes a scheme . But, what is ? We don’t know, hence we just insists that it should be nice, i.e. it should be a scheme by definition (this is actually not the full definition, i.e. the actual def is should be algebraic space).

In other word, algebraic stacks are stacks over scheme that we get a smooth cover , where smooth cover means when we pullback along scheme we always get be a scheme and is a smooth cover of schemes.

Next, we consider an example of effective descent morphism.

Note this holds for any category.

Now we need to go the other way, i.e. we start with . We get the diagramThen, we getThis commutes because and . Hence the dotted arrow in the above exists.

Then we get and hence we get diagramHere hence and similarly . This gives an arrow . Next, by pullback the cocycle diagram along the morphism , we get this map is compatible with . Hence this yields an isomorphism .

This is a silly example, but it is actually very useful, as we can frequently reduce to the case where we have sections.

Now we consider descent for sheaves. Let be a site where finite limits exist. Then take in , we get maps between their topoi (here we use to denote topos). Here we get . So, are equivalent.

Let be the following category. The objects are where and . The morphisms from to are given by plus . The projection is . We note this is not fibered in groupoids.

In this case, we get diagramThis is not a commutative diagram, thus we want to take equalizer. That is, we want to take as our definition. Here we abused notations. In particular, we write to mean the arrow given by apply to the adjunction map then take the reverse arrow of the arrow . Similarly is the arrow obtained from the above diagram. Also, that is also abuse of notations, as what we really meant is as in the above diagram.

Then, we claim . Indeed, if , then . Then . Now note and hence we just want to show that is an equalizer of the arrow , then it will conclude , where we note there is a natural map , i.e. we want to show is an equalizer diagram.

To do this, we prove it on -valued points, i.e. we take arbitrary , and we show it holds when we apply to . LetThen, we see we getThat is exactly by def of sheaf an equalizer. Hence we see by Yoneda.

Next, we let be given and set . Then by construction and so we get , i.e. we get canonical map . We want to show is isomorphism.

To show is isomorphism, it is okay to do that locally on (left as exercise). To say do this locally, we mean that ifthen on topoi is exact, so . In other word, we get commutative diagramThen . In particular, we see (this is the image of in in the above diagram) is isomorphism implies is isomorphism (this claim is left as exercise).

Now here is the trick: since we can take any cover , we choose . Now we getand is a section via the diagonal map, i.e. . Hence by the last theorem, we are done.

Last time, we defined fibered category of sheaves over . We showed is a “stack not fibered in groupoids”, i.e. we have descent for covering in .

As a corollary of the theorem we proved above, we get

Next, we talk about variant of descent for sheaves. Let be a sheaf of rings on a site . For all , let be defined by . Then for all , we get a map map of “ringed topoi”, i.e. map of topoi plus .

We will show this is almost a stack.

For , let be the category of -modules in . Then for all , we get by .

Now we define fibered category as follows: it has object where , . The morphisms are is in and in .

Now we have defined modules, the next topic is of course quasi-coherent sheaves.

Let be a scheme, be the fppf site associated with .

Now be the presheaf of rings on defined as: . This is just the global sections, i.e. it is the sheaf represented by . In other word, and hence we see this is a fppf sheaf as is fppf sheaf.

Next, we want to figure out what’s a reasonable notion of quasi-coherent for fppf topology.

For any scheme , let be the category of quasi-coherent -modules in Zariski topology, i.e. for .

Given we get , a presheaf of -modules on , defined as followsNote this depends on choice of pullback.

 1 , is a sheaf, and 2 sheaf condition on fppf arrow .

To see , we see it is enough to show is sheaf for small Zariski site. However, we see this is clearly a sheaf, becauseby definition. So it is indeed a sheaf.

For , let and be an -module. We need to checkWe showed this when showing schemes are fppf sheaves.

So, yields . Conversely, given sheaf of -mods, and , we get defined byBy construction, is given by .

 1 , 2 , is an isomorphism.

We get fibered category .

Martin reduces to the case where is qcqs (quasi-compact, quasi-separated). In this case, pushforward of quasi-coherent sheaf is quasi-coherent, i.e. .

Then, we noteSince , and equalizer of qcoh is qcoh, we see sends to .

Now we consider something that we already know, but in the new language.

We consider descent for closed subschemes:

In a very similar manner, we get descent for affine maps.

Let be the fibered category with objects with affine map.

That is, how do we know if and a -algebra, then is -algebra?

Being an algebra means we have with commutativity diagrams. But this is a map and hence descents to get and diagrams can be checked locally.