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 1. 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 2 (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 3. 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 4. 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 5 (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 6 (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 7 (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 8 (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 9 (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.