Today I’m going to be talking about the structure sheaf. Suppose is a sheaf of rings on a topological space (i.e. a sheaf on with values in the category of Rings), then is a ringed space. What does a sheaf of rings mean? Wikipedia says all it means is that for any open set , is a ring. An example of a structure sheaf is the sheaf of continuous functions on an open set.
Now let us talk about -modules. What are these? You have another sheaf- a sheaf of abelian groups (we will call this sheaf ), over the same topological space, and then ensure that it is closed under scalar multiplication with elements of the structure sheaf. How does this make sense? We know that the union of all elements of over all open sets in is not a ring. Like the union of all continuous functions on all open sets in is not a ring. However, each is. Hence, as long as we take elements from and multiply them only with elements of , we are still dealing with elements of an abelian group being multiplied with elements from a ring, and then we’re fine. We have a module.
You also have a commutative diagram condition on ringed spaces. The diagram is attached below.
What does such a diagram imply? It just means that whether you multiply and then restrict, or whether you restrict (elements of both sheaves) and then multiply, the result is the same. For example, if is the sheaf of differentiable functions, and is the sheaf of continuous functions, then is an module. Moreover, the order in which we restrict and multiply is irrelevant. Hence, the commutative diagram above is satisfied.
Why is Algebraic Geometry littered with such commutative diagrams whenever new concepts are defined? What structure are we trying to preserve? We just want to say that the “same thing is going on in both places”. For instance, when we have a mapping between sheaves, we want to say that whether we map and then restrict, or whether we restrict and then map, the result will be the same. So the operations of restrictions are “the same”, modulo mapping, in both sheaves. If we identify the elements of a sheaf with their image, then they’re being restricted in exactly the same way in both sheaves. This condition is a sanity check in some way. If any element could map to any element, regardless of the way they restrict, then we wouldn’t be able to map stalks to stalks and germs to germs, and the map between pre-sheaves would never become a map between stalks. In some sense, having such a commutative diagram makes mapping between direct limits possible.