4.4. 代数叠

Today we are going to define algebraic stacks.

定义 4.4.0.1. Let be property of morphisms which is local on the source for the etale topology, local on the target for the sm topology, and stable under base change. Let be a representable map of stacks over . We say has if for all schemes and diagramwe have has property , where we know is algebraic space.

Recall that, if where is scheme and is algebraic space, then we say has if there exists etale covering withsuch that has .

例 4.4.0.2. could be etale, smooth, relative dim , affine, finite, closed, immersion, open immersion, surjection, and so on. We note this list is smaller than the list for fppf descent as we require local on the source.

定义 4.4.0.3. A stack is an algebraic/Artin stack if:

1.

is representable,

2.

There exists scheme so that

We note is representable implies for all algebraic spaces , the maps are representable. So, being smooth surjection is well-defined.

定义 4.4.0.4. A morphism of stacks over , say , is defined as an element of , i.e. they are morphisms of fibered categories over .

引理 4.4.0.5. Let be stack over . Then is representable iff for all -schemes , and withwe have is an algebraic space. This is also equivalent to: for all , is an algebraic space.

证明. Equivalence of last conditions: for all ,so we see is algebraic space imply is algebraic space. Conversely, is the special case with , , .

Next, consider diagramJust like the proof for algebraic space, we see we getSo, giveswhere . Then implies and hence . Thus we see which concludes the proof.

The next goal is to define one of the most important example of stacks, namely quotient stacks.

例 4.4.0.6. Consider acts on , i.e. via . In particular, we see where the ring is the ring of invariant of the action. We also ntoe which is given by . Thus we get is sort of like the -plane map to the cone defined by . But this is bad, because is singular.

That’s why we want stakcs, where we replace the origin by the point , i.e. we get a “quotient stack” .

In particular, we get a diagramwhere is the minimal resolution of , and is the canonical stack we discussed above, and is any surface with mild singularity.

In physics, we get McKay correspondence that comparing and . We see a lot of interesting math about comparing the two, and it also relates to deriving categories.

Before we jump to definition, we give one or two words about the idea. Say we have group scheme over . Then we get and what we want is to have the arrow to be a -torsor.

We don’t realy know what is -torsor means, thus we want to pullback and getThis is going to be our definition.

定义 4.4.0.7. Continue the above set-up, for any scheme the category is defined as follows. The objects areThe morphisms are is given bywhere

We let . Why is a stack? We know -torsors are sheaves, so we have descent for sheaves. Then descent as sheaf with -action we see -action is map of sheaves, so those descent as well. We see is torsor if given by and we can check this isomorphism locally.

Why is representable? Let , be over . LetWe want to show is algebraic space.

First, we claim (its an exercise!) that if where is scheme and a sheaf, then we can check is algebraic space etale locally on .

Thus, to check is algebraic space, we can make etale base change on so that are trivial torsors. Thus now we havewhere . However, note if we have , then for any we get and hence is right multiplication by . Thus we see . Thus we see has a very simple description:In particular, since are all schemes, hence is a scheme, as desired.

Why is has smooth cover by scheme?

We see we getwhere is defined asWhy is a smooth surjection? Well, we getwhere is given by . So, we get map given by .

The point is that we getand since is smooth, is smooth surjection as desired. Thus, we showed, every torsor is the pullback of the torsor !

例 4.4.0.8. Let and be the trivial action. Then we let and hence we getHowever, note the action is trivial, thus -equivariant is just any arrow . Hence we see is just -torsors on .

例 4.4.0.9. Let , then is just line bundles, and is vector bundles.

定义 4.4.0.10. We say is quotient stack if for some .

Last time we talked about quotient stacks where is a group scheme over and with a -scheme.

We also showed that is universal, in the sense that if we have then we get the following diagram

We also talked about examples. In particular, and more generally, if is a free action, i.e. all stablizers are trivial, then is an algebraic space that is exactly . For example, if then .

命题 4.4.0.11. If are Artin stacks withthen is an Artin stack.

Next, we are going to define a very useful stack, called the inertia stack.

定义 4.4.0.12. For , we define the inertia stack to be the pullback

The point of this is that, suppose we havethen a lift of the arrow is equivalent to giving and , i.e. . In particular, we get that for , we have .

A different point of is . This new point is isomorphic to the old point: So, we seeIn other word, we get and is a relative group algebraic space, but it is normally not flat!

The next topic is properties for stacks and morphisms.

定义 4.4.0.13. Let be a property that is local for smooth topology. Then we say has if there exists smooth cover with scheme, such that has .

例 4.4.0.14. could be local Noetherian, regular, of finite type over , of finite presentation over .

引理 4.4.0.15. If is local for smooth topology, and has , and for a scheme we have then has .

证明. Considerwhere we assume has P. But has implies has and hence has as desired.

注 4.4.0.16. The proof shows that if is a morphism, then smooth locally on factors through .

定义 4.4.0.17. If is morphism of Artin stacks, then a chart for is a diagramIf is property of morphisms stable under base change, local on source and target for smooth topology, then we say has if has . In this case we also say that this chart has .

例 4.4.0.18. could be smooth, locally of finite presentation, surjective, etc.

例 4.4.0.19. If we are given quotient stack over , andThen is smooth iff is smooth. For example, we see is smooth because is smooth. On the other hand, is singular as its equal .

命题 4.4.0.20. The morphism has iff every chart has .

证明. We start with a chartThen we get another chartNow we want that: has iff has .

Now take pullbacks of the squares of the two sides, we getand we also get natural arrows from . Viz we getThus it is good enough to show has iff has , i.e. it is good enough to handle the case when factors as .

First, consider the case and we get diagramwhere is pullback of under . We see is smooth local on the target so has iff has .

Now, we just need to compare two charts with the same . To see this, we note we have the following diagramand we see has iff has iff has . This concludes the proof.

Next, we consider separatedness.

命题 4.4.0.21. Consider the diagramof stacks with representable. Then is representable iff is representable.

证明. If is representable, is representable by definition. If is representable, given we want is an algebraic space. Thus we get the following diagramwhere the bottom square is also Cartesian. We note are algebraic spaces because , are representable, respectively. Then, note we get a section defined by and hence we obtain the diagramNow we see is an algebraic space as are all algebraic spaces.

命题 4.4.0.22. Let be Artin stacks over and , then is representable.

证明. We have the following diagramTHen repable implies is repable. Hence repable implies, by the proposition above, that is representable as desired.

定义 4.4.0.23. We say is separated if is proper.

注 4.4.0.24. For stacks, keeps track of or which is a group so that is is rarely a closed immersion.

注 4.4.0.25. FOr schemes, is always an immersion, so they are separated and of finite type. Thus is proper iff universally closed but base changes of immersion is immersion and hence is universally closed iff its closed.

Thus, separated in the usual definition ( closed) iff its separated as stacks.

例 4.4.0.26. Let , then we see we getwhere is the of universal torsor. Hence we seewhere because we has a section . Thus, we see the above diagram’s arrows are given bySo, we see