定义 184.108.40.206. A stack on a site is a category fibered in groupoids , such that descent data is effective for covering maps, i.e. if , then is an equivalence between categories.
In short, stacks are just category fibered in groupoids where descent holds.
注 220.127.116.11. This is what we really call categorical stack, as we haven’t done any actual geometry yet. Fibered categories are analogous to presheaves (they are presheaves when fibered in sets). So stacks are presheaves where sheaf axiom 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.
注 18.104.22.168 (Spoiler Alert). The idea for algebraic stacks is that, if is a stack over schemes. To import/involve geometry, we require that there exists and arrow , so that is a “smooth cover”. This makes no sense, as we don’t know what smooth covers 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.
命题 22.214.171.124. If admits a section then descent data is effective for .
证明. We have by . Thus we define bywhere we recall and . Let’s check this is what we wanted. Indeed, we see .
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.
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.
定理 126.96.36.199. If is a covering in , then is effective descent morphism for .
证明. Consider . This has objects where and in that satisfies cocycle condition. Forwe want to construct an inverse functor .
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).
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
命题 188.8.131.52. Let be schemes with diagram:where , and similarly for . THe over is such that . Then there exists unique over so that .
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 .
定理 184.108.40.206. For all covers 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.
引理 220.127.116.11. is an fppf sheaf.
证明. Recall from awhile ago, to prove is an fppf sheaf, we just need to check:
, is a sheaf, and
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.
So, yields . Conversely, given sheaf of -mods, and , we get defined byBy construction, is given by .
命题 18.104.22.168. Let , a fppf sheaf of -mods. Thengiven by .
证明. Exercise: it is enough to check Zariski local on , so we can assume is affine. Then we getThen is right exact, so . As a result, we getSo, we can assume . Therefore, we may assume , i.e. . Now, observe elements in means that we have compatible maps . In particular, this means for any with diagramwe get . Thus we get diagramso is determined by . Thus the map is isomorphism and hence iff then .
定义 22.214.171.124. A big quasi-coherent sheaf on of -mods is on such that:
, is an isomorphism.
命题 126.96.36.199. THere is an equivalent of categories
We get fibered category .
定理 188.8.131.52. If we have fppf then .
证明. We only show the local case and the full proof can be found in the book.
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. .
Now we consider something that we already know, but in the new language.
We consider descent for closed subschemes:
命题 184.108.40.206. Suppose we have fppf coverThen the set of closed is equivalent to the set of closed such that given by the map .
In a very similar manner, we get descent for affine maps.
Let be the fibered category with objects with affine map.
命题 220.127.116.11. If we have fppf cover , then .
证明. Say is affine, this is the same as where is qcoh sheaf of -algebra. Then we have descent for quasi-coherent sheaves, and we want to make sure it is still -algebra.
That is, how do we know if and a -algebra, then is -algebra?