Earlier, we have computed the dimension of \( H^i(\mathbb{P} _ k^n, \mathscr{O} _ X(m)) \) for all \( i \) and \( m \). It turned out that it is \( 0 \) for all \( i \neq 0, n \), and at \( i = 0, n \), we had \[ \displaystyle \dim _ k H^0(\mathbb{P} _ k^n, \mathscr{O} _ X(m)) \cong \dim _ k H^n(\mathbb{P} _ k^n, \mathscr{O} _ X(-n-1-m)). \] This phenomenon is an instance of Serre duality. To explain this in detail, we first need to know a bit about \( \mathrm{Ext} \).
12. \( \mathrm{Ext} \) and \( \mathscr{E}xt \)
We have define sheaf cohomology as the right derived functor of \( \Gamma(X,-) \). But we notice that this functor is isomorphic to \( \mathrm{Hom} _ {\mathscr{O} _ X}(\mathscr{O} _ X, -) \). Also, \( \mathrm{Hom} \) is left exact in general. This motivates us to define
Definition. Let \( (X, \mathscr{O} _ X) \) be a ringed space and \( \mathscr{F} \) be an \( \mathscr{O} _ X \)-module. We define \( \mathrm{Ext}^i(\mathscr{F}, -) \) as the right derived functors of \( \mathrm{Hom}(\mathscr{F}, -) \). Also, we define \( \mathscr{E}xt^i(\mathscr{F}, -) \) as the right derived functors of \( \mathscr{H}om^i(\mathscr{F}, -) \).
From our motivation above, we immediately see that \( H^i(X, \mathscr{F}) \cong \mathrm{Ext}^i(\mathscr{O} _ X, \mathscr{F}) \). Here are some basic facts we will need.
Proposition (Hartshorne III.6.8). Let \( (X, \mathscr{O} _ X) \) be a ringed space and \( \mathscr{F}, \mathscr{G} \in \mathsf{Mod} _ {\mathscr{O} _ X} \) with \( \mathscr{F} \) coherent. Then \( \mathscr{E}xt^i(\mathscr{F}, \mathscr{G}) _ x \cong \mathrm{Ext}^i _ {\mathscr{O} _ {X,x}}(\mathscr{F} _ x, \mathscr{G} _ x) \).
Proof.
If we take an injective resolution \( 0 \rightarrow \mathscr{G} \rightarrow \mathscr{J}^\bullet \), the left hand side is taking the sheaf hom \( \mathscr{H}om(\mathscr{F}, -) \) and then taking stalks, while the right hand side is taking the stalk and taking \( \mathrm{Hom}(\mathscr{F} _ x, -) \). Here, we note that \( \mathscr{J} \) being an injective \( \mathscr{O} _ X \)-module implies that \( \mathscr{J} _ x \) is an injective \( \mathscr{O} _ {X,x} \)-module because we can look at skyscraper sheaves. So it suffices to show that \( \mathscr{H}om(\mathscr{F}, \mathscr{G}) _ x \cong \mathrm{Hom} _ {\mathscr{O} _ {X,x}}(\mathscr{F} _ x, \mathscr{G} _ x) \) for any \( \mathscr{O} _ X \)-module \( \mathscr{G} \). To see this, we locally write \( \mathscr{O} _ X^{m} \rightarrow \mathscr{O} _ X^{n} \rightarrow \mathscr{F} \rightarrow 0 \). Then \[ \displaystyle 0 \rightarrow \mathscr{H}om(\mathscr{F}, \mathscr{G}) _ x \rightarrow \mathscr{H}om(\mathscr{O} _ X^n, \mathscr{G}) _ x \rightarrow \mathscr{H}om(\mathscr{O} _ X^m, \mathscr{G}) _ x \] and \[ \displaystyle 0 \rightarrow \mathrm{Hom} _ {\mathscr{O} _ {X,x}}(\mathscr{F} _ x, \mathscr{G} _ x) \rightarrow \mathrm{Hom} _ {\mathscr{O} _ {X,x}}(\mathscr{O} _ {X,x}^n, \mathscr{G} _ x) \rightarrow \mathrm{Hom} _ {\mathscr{O} _ {X,x}}(\mathscr{O} _ {X,x}^m, \mathscr{G} _ x) \] shows what we want.
Proposition. Let \( X = \mathrm{Proj}, S \) where \( S = k[x _ 0, \ldots, x _ n] \). If \( \mathscr{F}, \mathscr{G} \) are coherent sheaves on \( X \), then \( \mathscr{E}xt^i(\mathscr{F}, \mathscr{G}) \cong \mathrm{Ext}^i(\mathscr{F}, \mathscr{G}(\bullet))^{\tilde{}} \).
Proof.
We shall do this, again, by induction on the length of the minimal free resolution of \( \mathscr{F} \). First let us check this for \( \mathscr{F} \) free of finite rank. It suffices to check for \( \mathscr{F} = \mathcal{O} _ X \). In this case, \( \mathscr{H}om(\mathscr{O} _ X, -) \) is the identity morphism, and so \( \mathscr{E}xt^i(\mathscr{O} _ X, \mathscr{G}) = 0 \) vanishes for \( i > 0 \) and \( \mathscr{E}xt^0(\mathscr{O} _ X, \mathscr{G}) = \mathscr{G} \). On the other hand, \( \mathrm{Ext}^i(\mathscr{O} _ X, \mathscr{G}(\bullet)) = H^i(X, \mathscr{G}(\bullet)) \). We have proven in Proposition 6 that \( \Gamma(X, \mathscr{G}(\bullet))^{\tilde{}} \cong \mathscr{G} \) and \( H^i(X, \mathscr{G}(\bullet))^{\tilde{}} = 0 \) for \( i > 0 \).
After checking this, we only need to see that a short exact sequence \( 0 \rightarrow \mathscr{F} _ 1 \rightarrow \mathscr{L} \rightarrow \mathscr{F} \rightarrow 0 \) induces two long exact sequences for \( \mathscr{E}xt^i(\mathscr{F}, \mathscr{G}) \) and for \( \mathrm{Ext}^i(\mathscr{F}, \mathscr{G}(\bullet))^{\tilde{}} \). Then the five lemma takes over. (We actually need maps between the long exact sequences, but there are maps \( \mathrm{Ext}^i(\mathscr{F}, \mathscr{G}(\bullet))^{\tilde{}} \rightarrow \mathscr{E}xt^i(\mathscr{F}, \mathscr{G}) \). Consider \( \beta \) before taking homology of the chain complex.)
As a corollary, we obtain \( \mathrm{Ext} _ {\mathscr{O} _ {X,x}}^i(\mathscr{F} _ x, \mathscr{G} _ x) \cong (\mathrm{Ext}^i(\mathscr{F}, \mathscr{G}(\bullet))^{\tilde{}}) _ x \) for coherent sheaves \( \mathscr{F} \) and \( \mathscr{G} \). This is n73 Proposition 1 of [Ser55], and also what we really needed.
13. Serre duality on projective space
For \( X = \mathbb{P} _ k^n \), we have computed that \( H^n(X, \mathscr{O} _ X(-n-1)) \cong k \). This sheaf \( \mathscr{O} _ X(-n-1) \) is isomorphic to the canonical sheaf, which corresponds to the top exterior power of the cotangent bundle. Let us write \( \omega _ X = \mathscr{O} _ X(-n-1) \).
Proposition. For any coherent sheaf \( \mathscr{F} \) on \( X \), there is a natural bilinear pairing \[ \displaystyle \mathrm{Hom}(\mathscr{F}, \omega _ X) \times H^n(X, \mathscr{F}) \rightarrow H^n(X, \omega _ X) \cong k. \] This pairing is perfect, and in particular, \( \dim _ k \mathrm{Hom}(\mathscr{F}, \omega _ X) = \dim _ k H^n(X, \mathscr{F}) \).
Proof.
Let us first check this for \( \mathscr{F} = \mathscr{O} _ X(m) \). \( \mathrm{Hom}(\mathscr{F}, \omega) \) corresponds to homogeneous degree \( -n-m-1 \) polynomials in variables \( x _ 0, \ldots, x _ n \). \( H^n(X, \mathscr{F}) \) corresponds to degree \( -m \) (Laurent) polynomials in variables \( x _ 0, \ldots, x _ n \) such that each degree is \( \le -1 \). \( H^n(X, \omega _ X) \) corresponds to the line generated by \( x _ 0^{-1} \cdots x _ n^{-1} \), and this shows that the pairing is perfect.
In the general case, we need to show that \( \mathrm{Hom}(\mathscr{F}, \omega _ X) \rightarrow \mathrm{Hom}(H^n(X, \mathscr{F}), H^n(X, \omega _ X)) \) is an isomorphism. Because \( H^{n+1} \) always vanishes (look at the Čech complex), we note that both sides are left exact in \( \mathscr{F} \). Then writing \( \mathscr{L}^1 \rightarrow \mathscr{L}^0 \rightarrow \mathscr{F} \rightarrow 0 \) with \( \mathscr{L}^i \) finite rank free shows that the map on \( \mathscr{F} \) is an isomorphism.
Let us write \( \mathrm{Hom}(A, H^n(X, \mathscr{F})) = A^\prime \). So we have just established a natural isomorphism \[ \displaystyle \mathrm{Hom}(\mathscr{F}, \omega _ X) \cong H^n(X, \mathscr{F})^\prime \] for all coherent \( \mathscr{F} \). But we can say more.
Theorem (Serre duality). For any coherent sheaf \( \mathscr{F} \) on \( X \) and \( i \ge 0 \), there is a natural isomorphism \[ \displaystyle \mathrm{Ext}^i(\mathscr{F}, \omega _ X) \xrightarrow{\cong} H^{n-i}(X, \mathscr{F})^\prime. \]
Proof.
We already know this for \( i = 0 \). For higher \( i \), we first note that if \( \mathscr{F} = \mathscr{O} _ X(-m) \) with \( m > 0 \), then both sides are \( 0 \). Also, every coherent sheaf \( \mathscr{F} \) admits a surjection \( \bigoplus \mathscr{O} _ X(-m) \rightarrow \mathscr{F} \) where \( m > 0 \) is large enough. (This can be seen in the setting of graded modules.) So if we write \( 0 \rightarrow \mathscr{G} \rightarrow \bigoplus \mathscr{O} _ X(-m) \rightarrow \mathscr{F} \rightarrow 0 \), from the long exact sequences we get isomorphisms \[ \displaystyle \mathrm{Ext}^{i+1}(\mathscr{F}, \omega _ X) \cong \mathrm{Ext}^i(\mathscr{G}, \omega _ X), \quad H^{n-i-1}(X, \mathscr{F}) \cong H^{n-i}(X, \mathscr{G}). \] Then we can define these isomorphisms inductively for \( i \). It can be shown that the isomorphisms doesn’t depend on the choice of the free resolution for \( \mathscr{F} \).
Serre duality actually works not only for projective spaces but also in many other cases. Let me write down the more general statement, taken from Hartshorne.
Theorem (Hartshorne III.7.6). Let \( X \) be a projective scheme of dimension \( n \) over an algebraically closed \( k \). There exists a coherent sheaf \( \omega _ X^\circ \) with a map \( t : H^n(X, \omega _ X^\circ) \rightarrow k \) such that for all coherent sheaves \( \mathscr{F} \) on \( X \), the pairing \[ \displaystyle \mathrm{Hom}(\mathscr{F}, \omega _ X^\circ) \times H^n(X, \mathscr{F}) \rightarrow H^n(X, \omega _ X^\circ) \] induces isomorphisms \( \theta _ 0 : \mathrm{Hom}(\mathscr{F}, \omega _ X^\circ) \rightarrow H^n(X, \mathscr{F})^\prime \). There are also natural maps \[ \displaystyle \theta _ i : \mathrm{Ext}^i(\mathscr{F}, \omega _ X^\circ) \rightarrow H^{n-i}(X, \mathscr{F})^\prime \] for all \( i \ge 0 \) and coherent \( \mathscr{F} \). All of these \( \theta _ i \) are isomorphisms for all coherent \( \mathscr{F} \) if and only if \( X \) is Cohen–Macaulay and equidimensional (all irreducible components have the same dimension).
14. Application to nonsingular and normal varieties
Putting together everything we have discussed so far, we can prove the following theorem.
Theorem (Serre n74 Theorem 1). Let \( \mathscr{F} \) be a coherent sheaf on \( \mathbb{P} _ k^n \), and let \( i \ge 0 \). The following two conditions are equivalent:
- \( H^i(X, \mathscr{F}(-m)) = 0 \) for all sufficiently large \( m \).
- \( \mathrm{Ext}^{n-i} _ {\mathscr{O} _ {X,x}}(\mathscr{F} _ x, \mathscr{O} _ {X,x}) = 0 \) for all \( x \in X \).
Proof.
By Serre duality, \( H^i(X, \mathscr{F}(-m)) \cong \mathrm{Ext}^{n-i}(\mathscr{F}(-m), \omega _ X) \), so the first condition is equivalent to \( \mathrm{Ext}^{n-i}(\mathscr{F}, \omega _ X(m)) = 0 \) for all sufficiently large \( m \). This just means that \( \mathrm{Ext}^{n-i}(\mathscr{F}, \omega _ X(\bullet)) \) is in \( \mathcal{C} \), as a graded module, and is equivalent to \( \mathrm{Ext}^{n-i}(\mathscr{F}, \omega _ X(\bullet))^{\tilde{}} = 0 \). Because a sheaf is \( 0 \) if and only if all the stalks are \( 0 \), this is equivalent to \( (\mathrm{Ext}^{n-i}(\mathscr{F}, \omega _ X(\bullet))^{\tilde{}}) _ x = 0 \) for all \( x \in X \). But the point of the section on Ext was to show that the left hand side is equal to \( \mathrm{Ext}^{n-i} _ {\mathscr{O} _ {X,x}}(\mathscr{F} _ x, (\omega _ X) _ x) \). Because \( (\omega _ X) _ x \cong \mathscr{O} _ {X,x} \), we get the second condition.
Now this can be applied to different situations. Let \( V \) be a closed subvariety of \( X = \mathbb{P} _ k^n \), and let \( \mathscr{F} \) be a locally free sheaf (i.e., a vector bundle) on \( V \). We want to say something about \( H^i(V, \mathscr{F}(-m)) \) as \( m \rightarrow \infty \). Here, we don’t have a good tool for computing cohomology over quasi-projective schemes, so we push-forward to sheaf and compute \( H^i(X, i _ \ast \mathscr{F}(-m)) \) instead, where \( i : V \rightarrow \mathbb{P} _ k^n \) is the embedding. Note that \( H^i(V, \mathscr{F}(-m)) \cong H^i(X, i _ \ast \mathscr{F}(-m)) \) because we can compute cohomology using Čech cohomology.
Proposition (Serre n76 Theorem 3). Let \( V \) be a nonsingular closed subvariety of \( \mathbb{P} _ k^n \) of dimension \( d \). For a finite rank locally free sheaf \( \mathscr{F} \) over \( V \), we have \( H^i(V, \mathscr{F}(-m)) = 0 \) for \( 0 \le i < d \) and \( m \) sufficiently large.
Proof.
\( V \) nonsingular means that for each \( x \in V \), the ring \( \mathscr{O} _ {V,x} \) is a regular local ring. Also, \( X = \mathbb{P} _ k^n \), so the ring \( \mathscr{O} _ {X,x} \) is a regular local ring, and \( \mathscr{O} _ {V,x} \) is some quotient of it. Let us write \( \mathscr{O} _ {V,x} = \mathscr{O} _ {X,x} / I _ x \). Here is one property of regular local rings (Stacks Lemma 10.105.4): if \( \mathscr{O} _ {X,x} \) and \( \mathscr{O} _ {X,x} / I _ x \) are regular local, there exists a regular sequence \( f _ 1, \ldots, f _ n \in \mathfrak{m} _ {X,x} \) such that \( I _ x = (f _ 1, \ldots, f _ d) \). This means that, for each \( 1 \le k \le n \), \[ \displaystyle f _ k \cdot m \in (f _ 1, \ldots, f _ {k-1}) \subseteq \mathfrak{m} _ {X,x} \] for \( m \in \mathscr{O} _ {X,x} \) implies \( m \in (f _ 1, \ldots, f _ {k-1}) \). We want to check is that \( \mathrm{Ext} _ {\mathscr{O} _ {X,x}}^i(\mathscr{O} _ {V,x}, \mathscr{O} _ {X,x}) \) vanishes for \( i < d \). (This is because \( \mathscr{F} _ x \) will be a finite direct sum of \( \mathscr{O} _ {V,x} \).) In this case, we have a finite free resolution \[ \displaystyle 0 \rightarrow L _ {n-d} \rightarrow L _ {n-d-1} \rightarrow \cdots \rightarrow L _ 1 \rightarrow L _ 0 \rightarrow \mathscr{O} _ {V,x} \rightarrow 0 \] called the Koszul complex. This immediately shows that \( \mathrm{Ext}^{n-i} _ {\mathscr{O} _ {X,x}}(\mathscr{O} _ {V,x}, \mathscr{O} _ {X,x}) = 0 \) for \( i < d \). Then the previous theorem implies that \( H^i(V, \mathscr{F}(-m)) = 0 \) for sufficiently large \( m \) and \( 0 \le i < d \).
Here is another similar theorem.
Theorem (Enriques–Severi–Zariski). Let \( V \) be a irreducible, normal, projective scheme of dimension at least \( 2 \). Then for any finite locally free sheaf on \( V \), \( H^1(V, \mathscr{F}(-m)) = 0 \) for \( m \) sufficiently large.
Proof.
Here, we need to prove that \( \mathrm{Ext}^{n-1} _ {\mathscr{O} _ {X,x}}(\mathscr{O} _ {V,x}, \mathscr{O} _ {X,x}) = 0 \). The given condition that \( V \) is normal can be rephrased as each \( \mathscr{O} _ {V,x} \) is integrally closed. Then by Serre’s criterion for normality, the depth of \( \mathscr{O} _ {V,x} \) (as a ring) is at least \( 2 \) for any closed point \( x \). The depth of \( \mathscr{O} _ {V,x} \) doesn’t change if we compute it as a \( \mathscr{O} _ {X,x} \)-module, so this is at least \( 2 \) as well. Then the Auslander–Buchsbaum formula tells us that the projective dimension of \( \mathscr{O} _ {V,x} \) over \( \mathscr{O} _ {X,x} \) is at most \( n-2 \), and hence \( \mathrm{Ext}^{n-1}(\mathscr{O} _ {V,x}, \mathscr{O} _ {X,x}) = 0 \).
Honestly, I don’t know what’s going on in the last commutative algebra part, but the point is that proving vanishing of cohomology reduces to showing that certain \( \mathrm{Ext} \) groups vanish.
References
[Har77] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1997, Graduate Texts in Mathematics, No. 52.
[Ser55] Jean-Pierre Serre, Faisceaux Algébriques Cohérents, Ann. of Math. (2) 61 (1955), 197–278.