Serre Duality is the statement, for a smooth projective (integral) variety and a locally-free sheaf on ,
where is the canonical bundle and . This isomorphism is almost canonical: it depends on the choice of an isomorphism
called a trace map. I’m going to sketch out my understanding of what’s going on with this duality statement and how it comes up (non-rigorously).
Derived Functors of
The starting point for sheaf cohomology on is that the global sections functor is only left-exact: for a short exact sequence of sheaves
taking global sections gives
not, in general, exact on the right. So, since the category of sheaves on has enough injectives, we can define derived functors , which continue the above sequence to the right. If has dimension , the sequence necessarily terminates at the -th level:
since for any sheaf , for . This is the Grothendieck Vanishing Theorem.
Consider dualizing this sequence (of vector spaces):
This is kind of neat: we see that is again a left-exact (contravariant) functor. And, evidently, its higher derived functors are the duals of sheaf cohomology, in reverse order, .
So we can think of the two ends of the familiar long exact sequence in sheaf cohomology as being interchangeable — either one could be the “starting point”. (I’m glossing over the fact that our category doesn’t have enough projectives, so technically these are “universal -functors,” not strictly speaking derived functors.)
Another standard fact about sheaves on is that a map of sheaves is the same as a global section of :
In other words, the sheaf represents the global-sections functor for sheaves on . This ties in nicely with higher cohomology as well: taking derived functors “of both sides” of the equation above, we get
(To establish this, we observe/show that both sides are universal -functors that agree in degree 0. Or we observe, informally, that the functors are the same (naturally isomorphic) for , hence must have naturally isomorphic derived functors. This is the main tool for justifying this kind of manipulation, and I’ll gloss over it from now on.)
Now let’s make the following observation. Let be any sheaf whose top cohomology is -dimensional, and fix a trace map . Then, for any map , we have an induced map
that is, we get an element of . If every element arises naturally and uniquely in this way — this is a big if — then we can say
that is, , combined with the choice of trace map , (co)represents the dual-of-top-cohomology functor . We call such a pair a dualizing sheaf.
And, thinking optimistically, we might hope for the derived functors to agree as well — namely,
Note that this is Ext in the second argument, not the first, so there are some technical hiccups here. In any case, plugging in a locally free sheaf , we can get rid of the Ext part and get a statement entirely in terms of sheaf cohomology. Specifically, we know that
holds by hom-tensor adjunction, so “taking derived functors of both sides” gives
Note that here, we’re letting be fixed in the equation, taking derived functors in the other argument, and then plugging back in . And to even talk about , we need to be locally free. Luckily, that’s enough to make this justifiable and correct (you can prove this, for example, with the spectral sequence relating “Ext in the first argument” to “Ext in the second argument”.)
Anyway, combining the two equations gives the usual statement of Serre duality, in terms of the dualizing sheaf :
Finding a Good Dualizing Sheaf
The construction above relied, crucially, on the existence of a dualizing sheaf to (co)represent the functor. So a few remarks are in order to say where this comes from. Personally, I find the exposition in Hartshorne to be a bit frustrating, since the method is as follows:
- (Projective Space) Observe that it just so happens to work for , with , by inspection.
- (Duality for Closed Embeddings) Use sheaf Ext to construct explicitly a dualizing sheaf on , where is an embedding, are smooth (or, more generally, Cohen-Macaulay), and has a dualizing sheaf . Explicitly, the dualizing sheaf on is . This gains the desired functor-representation property from the Cohen-Macaulay formalism, using regular sequences and/or Koszul complexes and/or vanishing of certain Ext functors.
- (Canonical Bundle). Observe that the adjunction formula for the canonical bundle, , where is the ideal sheaf of , is compatible with the transition functions when building the dualizing sheaf out of Koszul complexes. In other words, the canonical bundle on is, itself, a dualizing sheaf. This allows us to replace the more abstract definition, using sheaf Ext, with more familiar (not to mention geometric) differential forms.
An alternate approach is to replace “closed embedding” with “finite cover”, but I’m not familiar with it. Ravi Vakil’s notes do it this way, so refer to those for the exposition.
While the approach of “pulling back” Serre Duality across embeddings or finite maps is elegant and (I think) fairly theoretically satisfying, it still leaves that magical step 1 above, where Serre Duality is practically coincidental, and unenlightening, when .
I remember thinking this when reading Serre’s GAGA paper as well: the approach for general smooth projective varieties is to reduce to the case of projective space, then to observe that all the statements of GAGA (particularly the Betti numbers) just happen to be true for projective space, whether considered algebraically or analytically. It’s a little weird that there’s no “deeper” reason, or that the proof isn’t more intrinsic for arbitrary smooth varieties. If anyone reading this has any thoughts, I’d be interested to hear them.
* I actually have no idea how one actually does this with derived categories. Someone should explain that to me…