Recall that stack is a category fibered in groupoids where descent holds for all covering maps.
命题 220.127.116.11. If is a stack, then for all and , we have is a sheaf on .
证明. If is a covering, then considerin , i.e. we also have an arrow by default. Then we get and hence we get a bunch of pullbacksNote here the double arrows of pullback is actually a square
命题 18.104.22.168. Letbe maps of stacks over . Let be the fibered product of category fibered in groupoids. Then is a stack.
证明. By definition, . Similarly, an easy check shows
Just like for sheaves we have sheafification, for stacks we have “stack-fication”.
定理 22.214.171.124 (Theorem 4.6.5 in Martin). Let be category fibered in groupoids over . Then there exists stack and such that for all stacks , we have .
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, .
定义 126.96.36.199. A morphism of sheaves is representable by schemes if for all with scheme, the fibered product (as cat fibered in sets) is a scheme.
So, when we say is representable by schemes, we mean for all we get the following diagram
定义 188.8.131.52. Let be a property of morphisms of schemes. If for all diagramswe have:
has implies has , then we say is stable under base change.
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 .
定义 184.108.40.206. If is representable by schemes and is a property which is stable under base change and local on the base, then we say has iff for all with schemes, has .
We remark that if and then has iff has because we can consider the identity map and pull it back using this.
引理 220.127.116.11. Let be a presheaf on . Then is representable by schemes if and only if for all is representable by schemes for all schemes .
证明. ConsiderThen we get
定义 18.104.22.168. An algebraic space over is a sheaf for the big etale topology on such that:
is representable by scheme.
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.
注 22.214.171.124. Schemes are algebraic spaces because we see is true as is a scheme, and for we choose . This is because is sheaf for fppf topology, so also it is also sheaf for etale topology.
定义 126.96.36.199. A morphism of stacks is representable if for all diagrams with algebraic spacewe get is an algebraic space.
命题 188.8.131.52. A morphism of stacks is representable iff for all diagrams with a schemewe get is algebraic space.
证明. If is algebraic space then we know is a stack, which proves the forward direction. We show the converse. Hence, suppose we are given where is algebraic space, we want to show is an algebraic space, where the stack is defined by
: 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.