# 4.4. 代数叠

Today we are going to define algebraic stacks.

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

 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.

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.

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.

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 !

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 .

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

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.

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.

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