Recall that we are assuming that is a smooth projective variety. Let’s also say it is of dimension . We’re going to be lazy (i.e. sane) and all functors will be derived when talking about the derived category even though the and will be omitted. Our derived category always has some (auto-) functors. For example, we definitely have the shift functor that comes with any triangulated category just by shifting where the sheaves occur in the complex.
Also, given any coherent sheaf we have the functor . In particular, we could tensor with the canonical bundle and shift by . This functor is so useful it has a name and notation (again, is any object of and hence a complex even though I didn’t write ). We call this the Serre functor.
This name just comes from the fact that the generalized form of Serre duality for the derived category says that there is a functorial isomorphism
Notice that if and are honest sheaves sitting in degree we can use that to derive the special case which is the standard form of Serre duality given in classic texts like Hartshorne.
For the rest of today let’s look at a very important concept from triangulated categories. One might wonder how much we can know about certain triangulated categories just from knowing certain special classes of objects. A collection of objects is called a spanning class for a triangulated category if the following hold:
If for all and , then .
If for all and , then .
It is not in general true that these two conditions are equivalent, but it is easy to check that Serre duality for will allow us to only have to check one of the conditions. The idea of spanning classes (which may not come up for awhile) is that you can check certain properties just on these objects to get properties on the whole category. For example, one can use this idea to prove necessary and sufficient conditions for a Fourier-Mukai transform to be fully-faithful.
Since our triangulated category is somehow built out of , to any (closed) point of we have a natural object associated to it that we’ll call . This is just the skyscraper sheaf at the point . One hope would be that the set of objects of this form is a spanning class. This intuitively makes sense, because checking a property on this class in the derived category is sort of like checking a property on “points” of variety. It is indeed the case that this forms a spanning class.
Suppose is a non-trivial object of . We’ll check the second condition. This says that we must produce some closed point and some integer so that (well, almost, we used Serre duality again to flip the i over to the other side). We will use the standard local-to-global spectral sequence
If we plug in and we get
Let be the maximal such that . The sheaf itself is assumed non-trivial, so there exists with that sheaf non-zero, but is regular so there are only finitely many non-zero and hence such an exists. We will now argue that by showing that all differentials with source and target for any must be trivial.
On the one hand, is the -th Ext group between coherent sheaves, so when it always vanishes. This means that any differential with target must be trivial. On the other hand, our choice of maximal implies that any differential with source is trivial.
Now is non-trivial, so in particular it has non-trivial support which is a closed set and hence contains some closed point . This tells us that . But this says that which is what we set out to prove and hence the collection of skyscraper sheaves of closed points do form a spanning set.