# 4.3. 代数空间

Recall that stack is a category fibered in groupoids where descent holds for all covering maps.

Suppose . We want to descent to . By definition, defines an isomorphism in . However, we know as is a stack, thus we are done as desired.

Each and hence is a stack.

Just like for sheaves we have sheafification, for stacks we have “stack-fication”.

Next, we are going to define algebraic spaces, but first, we give some ideas.

Idea: what is a scheme? A scheme is affine schemes glued in Zariski topology. Then algebraic space is affine schemes glued in etale topology.

Of course, now we are just begging for the question of what if fppf topology. It turns out, it is not so easy to answer this question. The answer is that a theorem of Artin, where he showed they are the same as algebraic spaces.

Let be a scheme, .

So, when we say is representable by schemes, we mean for all we get the following diagram

 1 has implies has , then we say is stable under base change. 2 if is a Zar (et, sm, fppf) covering, then has iff has , then we say is local on the base for the Zar (et, sm, fppf) topology.

Finally, we say is local on the source for the Zar (et, sm, fppf) topology, if for all diagramswith a Zar (et, sm, fppf) covering, has iff has .

We remark that if and then has iff has because we can consider the identity map and pull it back using this.

Here is with . So, . representable by schemes, so is scheme, so is scheme, as desired.

 1 is representable by scheme. 2 there exists scheme and etale covering

We note makes sense because implies is representable by schemes and etale surjections are property that are stable under base change and local on the base.

: we want to show is a sheaf. Since is a stack, so we just need to show is fibered in sets. Now consider this diagramwhere we know is a scheme, an algebraic space, and a scheme. We have in and we want to be the identity map. However, is algebraic space by hypothesis, is a scheme by definition of being algebraic space. We get and is algebraic space, so . is etale covering and is a stack, and , hence .

: we want to show has etale covering by scheme. We get where is algebraic space and all arrows are etale, hence we are done.

: It remains to show is repable by schemes. We do a diagram again:where is a scheme. This giveswhere is a scheme since is an algebraic space. This concludes the proof.