## Serre Duality

Sorry that this one is late, things are going to continue to be hit-or-miss through June, though hopefully hit more than miss. Anyway, we’ve got a cohomology theory for sheaves. Anyone who has worked with cohomology of manifolds will know just how useful Poincare Duality is. So what we want is some sort of duality theory for the cohomology of sheaves that will help us to reduce the amount of computation necessary. Here, we must appeal to the wisdom of Serre (and of course, it was generalized by Grothendieck, but I don’t quite get Grothendieck Duality yet).

Now, for this all to work, we need to start by working over a field $k$. Not an algebraically closed field, but just some field. Our construction of the cotangent sheaf and of the canonical sheaf carries over to schemes, and we’ll just recall that definition in terms of differentials for these purposes (though there is a second definition, I don’t like it as much because it obscures my main way of thinking about it). Anyway, here’s Serre Duality for projective space, and we’ll use it to make it work for coherent sheaves on projective schemes:

Serre Duality for $\mathbb{P}^n_k$: Denote $X=\mathbb{P}^n_k$. First, $H^n(X,\omega_X)=k$. Choosing an isomorphism, for any coherent sheaf $\mathscr{F}$, we have a pairing $\hom(\mathscr{F},\omega)\times H^n(X,\mathscr{F})\to H^n(X,\omega)=k$. That is, these two modules are dual to each other. And finally, for each $i\geq 0$, there is a natural isomorphism $\mathrm{Ext}^i(\mathscr{F},\omega)\to H^{n-i}(X,\mathscr{F})'$, where the apostrophe denotes dual.

Now, those Exts look bad, but in reality, we have $\mathrm{Ext}^i(\mathscr{F},\mathscr{G})\cong H^i(X,\mathscr{H}om(\mathscr{F},\mathscr{G}))$, that is, the cohomology of sheaf hom, the sheaf of local homomorphisms between sheaves.

Now, to generalize, we need to find something that acts like $\omega_{\mathbb{P}^n_k}$, which is not just any sheaf, not even merely a coherent sheaf, but a locally free sheaf, which it isn’t on every scheme (not even every variety). So we take $X$ to be a proper scheme over $k$, where a proper scheme is one where the map to $\mathrm{Spec}(k)$ is separated, of finite type, and takes closed sets to closed sets, even after base change. Now, a dualizing sheaf is a coherent sheaf $\omega^\circ_X$ with a morphism $t:H^n(X,\omega_X^\circ)\to k$ which has for every coherent sheaf $\mathscr{F}$ on $X$, a pairing $\hom(\mathscr{F},\omega_X^\circ)\times H^n(X,\mathscr{F})\to H^n(X,\omega_X^\circ)$ which, when followed by $t$, gives an isomorphism $\hom(\mathscr{F},\omega_X^\circ)\to H^n(X,\mathscr{F})'$.

Now, it’s not too hard to prove that every projective scheme over a field has a dualizing sheaf, but it requires Ext-sheaves, which we’re not going to get into. The point here isn’t the blood and guts of how to prove Serre Duality, it’s the statement and an important corollary which allows us to actually do things. So now, the full version of Serre duality:

Theorem: Let $X$ be a projective scheme of dimension $n$ over an algebraically closed field $k$. Let $\omega_X^\circ$ be dualizing. Then for all $i\geq 0$ and $\mathscr{F}$ coherent, there are natural maps $\mathrm{Ext}^i(\mathscr{F},\omega_X^\circ)\to H^{n-1}(X,\mathscr{F})'$, and under reasonable conditions, they are isomorphisms.

Now, the REAL key is the corollary, which is often called Serre Duality itself, which states that for $X$ a projective scheme which is nice enough (ie, nonsingular, or not too badly singular) with all components of dimension $n$, then for any locally free sheaf $\mathscr{F}$, we have $H^i(X,\mathscr{F})\cong H^{n-i}(X,\mathscr{F}^\vee\otimes \omega_X^\circ)'$. This helps us compute the cohomology of vector bundles more easily.

We stop this rather dense entry with a quick comment that will be exploited next time. Let $C$ be a smooth curve, that is, a nonsingular variety of dimension 1. Now let $\mathscr{L}$ be a line bundle on $C$. Then Serre Duality says that $H^1(X,\mathscr{L})\cong H^0(X,\mathscr{L}^\vee\otimes \omega_C)$. And, being coherent on a one dimensional scheme, all higher cohomology vanishes. So all cohomology is then the global sections of SOME line bundle. This will allow us to do quite a bit with divisors in the future, which will let us gear up for doing similar stuff with a surface.