Stacks 2: An example


This will hopefully be a short, yet enlightening post in which the concept of a stack starts to make more sense than the abstract nonsense of the last few posts. Recall that we formed a category of line bundles on a manifold {L(X)} and had a natural forgetful functor: {L(X)\rightarrow \text{Top}(X)}.

If one is not writing a blog and wants to be much more careful, one should probably check that {L(X)} is indeed a category and the given functor is actually a functor. Most readers that have made it this far probably aren’t concerned with this part, though.

Is this fibered in groupoids? Well for checking what sorts of things lie over certain objects {U\in \text{Top}(X)}, the situation has been rigged so that the objects over {U} are precisely the line bundles on {U} as a manifold. The first type of “square” we have to be able to complete is as follows {\begin{matrix} & & (V, L_V) \\ \\ U & \hookrightarrow & V \end{matrix}}. Well, all we need to be able to do is find some {(U, L_U) \rightarrow (V, L_V)}. But by definition of our category this would consist of an iso {L_V|_U \rightarrow L_U}. These are line bundles, so we can always restrict to get another one, so just take {(U, L_V|_U)} lying over {U} to complete it.

What about the second diagram of being fibered in groupoids? The base is just {U\hookrightarrow V \hookrightarrow W}. Now suppose we have line bundles over these {L_U, L_V,} and {L_W} and isomorphisms {L_W|_V\rightarrow L_V} and {L_W|_U\rightarrow L_U}. This certainly tells us that there is an iso {L_V|_U\rightarrow L_U}, and uniqueness is just from the fact that it has to be the one that makes the composition what we said it had to be. You could think of this coming from the fact that {L_V|_U\rightarrow L_U} is unique up to automorphism of {L_U}, and we know which automorphism it from the other condition.

Now we check that Isom forms a sheaf. Let {U} be some open set. Let {L} and {S} be two line bundles over {U} (in this case, this literally just means line bundles on {U} as a topological space). Now we want to check that the presheaf (of sets) {\mathcal{F}(V)=\{L_V\stackrel{\sim}{\rightarrow} S_V\}} is actually a sheaf. This is a presheaf just because isomorphisms restrict. It is a sheaf because all the information is local. If you have isomorphisms defined on open subsets that agree on overlaps, then you can glue them to make an isomorphism on the union. These are just two basic properties of line bundles that most people have already seen. So Isom is a sheaf.

Lastly we need to check the stack condition. Maybe I should remark on the terminology here. A collection of objects and isos over a covering of an open set that satisfies the cocycle condition is called a descent datum. If the objects glue in the way of the stack condition, then that descent datum is said to be effective, so the stack condition is sometimes stated that every descent datum is effective.

Given a descent datum, the fact that you can glue to get an object over the whole open set is just a standard exercise or proven proposition in basically any text on manifolds. In fact all of the above things are true for any rank {r} vector bundle. So we actually get the stack of rank {r} vector bundles on {X} for any {r}. Since I’m not sure we’ll return to this example, we’ll just temporarily notate it {\text{Vect}^r(X)}, and hence {\text{Vect}^1(X)=L(X)}.

If you’ve been following along, it should be pretty clear how to translate all of this over to a stack on the Zariski site rather than on {\text{Top}(X)}, but we’ll make that more explicit next time and get some more examples.

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