Serre Duality

This post was machine-translated from the Korean original by Marvin (via Kimi). It may contain errors or awkward phrasing — the Korean original is the source of truth.

Geometrically, in favorable cases there is a natural duality between dimension \(k\) cohomology and codimension \(k\) cohomology. To prove this we used the perfect pairing

\[H^k(M;R)\times H^{n-k}(M;R)\rightarrow R\]

and through it obtained results such as [Algebraic Topology] §Poincaré Duality, ⁋Theorem 11. More concretely, since this pairing is constructed via the cap product and the fundamental class \([M] \in H_n(M;R)\), we may say that the source of duality in topology is the orientation class \([M]\).

In this post we examine Serre duality, the algebraic-geometry analogue of duality.

Serre Duality on Projective Space

We first look carefully at the case \(X=\mathbb{P}^n\) only. We know that every line bundle on \(\mathbb{P}^n\) is of the form \(\mathcal{O}(d)\), and in particular we saw in §Canonical Line Bundle, §§Canonical Bundle of \(\mathbb{P}^n\) that this is \(\mathcal{O}(-n-1)\). Then from §Cohomology of Projective Space, ⁋Proposition 1 (Bott) we obtain the following.

Proposition 1 For the canonical line bundle \(\omega_X\) on projective space \(X=\mathbb{P}^n\), there exists an isomorphism

\[H^n(X, \omega_X)\cong \mathbb{K}\]

In general this is understood as the isomorphism that explicitly takes \(\x_0^{-1}\cdots\x_n^{-1}\) as its basis, but it is determined uniquely only up to scalar multiplication. Choosing such a normalization is the same as concretely choosing the trace map \(\tr:H^n(\mathbb{P}^n, \omega_{\mathbb{P}^n}) \rightarrow \mathbb{K}\).

To obtain the duality pairing we now need to define the cup product. For convenience, let us work at the level of Čech cohomology. For any topological space \(X\), an open cover \(\mathcal{U}\) of \(X\), and sheaves \(\mathcal{F}\), \(\mathcal{G}\) on \(X\), the cup product of two Čech cochains \(\alpha \in \check{C}^p(\mathcal{U}, \mathcal{F})\), \(\beta \in \check{C}^q(\mathcal{U}, \mathcal{G})\) is defined by the formula

\[(\alpha \smile \beta)_{i_0, \ldots, i_{p+q}} = \alpha_{i_0,\ldots,i_p}\big\vert_{U_{i_0,\ldots,i_{p+q}}} \otimes \beta_{i_p,\ldots,i_{p+q}}\big\vert_{U_{i_0,\ldots,i_{p+q}}}\in \check{C}^{p+q}(\mathcal{U}, \mathcal{F}\otimes\mathcal{G})\]

We can explicitly compute that this descends to cohomology, and from this the map

\[{\smile}:\check{H}^p(\mathcal{U}, \mathcal{F}) \times \check{H}^q(\mathcal{U}, \mathcal{G}) \rightarrow \check{H}^{p+q}(\mathcal{U}, \mathcal{F} \otimes \mathcal{G})\]

is defined. At the sheaf cohomology level, one can also define this by taking flat resolutions \(\mathcal{I}^\bullet\), \(\mathcal{J}^\bullet\) of \(\mathcal{F}\) and \(\mathcal{G}\) respectively, and then using the tensor product complex of these (that is, the total complex of the double complex with components \(\mathcal{I}^p\otimes \mathcal{J}^q\)).

In any case, by the cup product pairing we obtain for any locally free sheaf \(\mathcal{E}\) the bilinear map

\[H^k(\mathbb{P}^n, \mathcal{E})\times H^{n-k}(\mathbb{P}^n, \omega_{\mathbb{P}^n}\otimes \mathcal{E}^\vee)\rightarrow H^n(\mathbb{P}^n, \mathcal{E}\otimes \omega_{\mathbb{P}^n}\otimes \mathcal{E}^\vee)\]

and then, using the evaluation map \(\mathcal{E}\otimes \mathcal{E}^\vee\rightarrow \mathcal{O}_{\mathbb{P}^n}\) and the trace map above, we obtain the bilinear form

\[H^k(\mathbb{P}^n, \mathcal{E})\times H^{n-k}(\mathbb{P}^n, \omega_{\mathbb{P}^n}\otimes\mathcal{E}^\vee)\rightarrow \mathbb{K}\]

We show non-degeneracy for \(\mathcal{O}(d)\) by direct computation in §Cohomology of Projective Space, ⁋Proposition 1 (Bott), and extend this non-degeneracy to general locally free sheaves \(\mathcal{E}\) using the syzygy theorem.

From the discussion so far we obtain the following.

Proposition 2 (Serre duality pairing, projective case) For a locally free sheaf \(\mathcal{E}\) on \(\mathbb{P}^n\), the bilinear form

\[H^k(\mathbb{P}^n, \mathcal{E}) \times H^{n-k}(\mathbb{P}^n, \omega_{\mathbb{P}^n} \otimes \mathcal{E}^\vee) \rightarrow \mathbb{K};\quad (\alpha, \beta) \mapsto \tr(\alpha \smile \beta)\]

is a perfect pairing.

More explicitly, Serre duality in general means the following isomorphism obtained from this:

\[H^k(\mathbb{P}^n, \mathcal{E})\cong H^{n-k}(\mathbb{P}^n, \omega_{\mathbb{P}^n}\otimes\mathcal{E}^\vee)^\ast\]

More generally, by the Noether normalization theorem, for any \(n\)-dimensional smooth projective variety \(X\) there exists a finite surjective morphism \(f: X \rightarrow \mathbb{P}^n\). Then we can pull back Serre duality, proved on \(\mathbb{P}^n\), to \(X\) via this finite morphism \(f\), and in this setting Serre duality means the isomorphism

\[H^i(X, \mathcal{E}) \cong H^{n-i}(X, \omega_X \otimes \mathcal{E}^\vee)^\ast\]

Example 3 Let us look concretely at Proposition 2 (Serre duality pairing, projective case) on \(\mathbb{P}^2\). Here \(\omega_{\mathbb{P}^2} \cong \mathcal{O}(-3)\), so what Serre duality asserts is the isomorphism \(H^k(\mathbb{P}^2, \mathcal{O}(d)) \cong H^{2-k}(\mathbb{P}^2, \mathcal{O}(-d-3))^\ast\).

First, for \(d=0\), by §Cohomology of Projective Space, ⁋Proposition 1 (Bott) we have

\[H^0(\mathbb{P}^2, \mathcal{O}) = \mathbb{K},\qquad H^1(\mathbb{P}^2, \mathcal{O}) = 0, \qquad H^2(\mathbb{P}^2, \mathcal{O}) = 0\]

and the cohomology of \(\mathcal{O}(-3)\) is

\[H^0(\mathbb{P}^2, \mathcal{O}(-3)) = 0, \qquad H^1(\mathbb{P}^2, \mathcal{O}(-3)) = 0,\qquad H^2(\mathbb{P}^2, \mathcal{O}(-3)) = \mathbb{K}\]

so we see that Serre duality holds. Similarly, for \(d=1\), the only nonzero cohomology is

\[H^0(\mathbb{P}^2, \mathcal{O}(1)) = \mathbb{K}^3\]

and by Serre duality we must have \(H^0(\mathcal{O}(1)) \cong H^2(\mathcal{O}(-4))^\ast\), so \(\dim H^2(\mathcal{O}(-4))\) should be \(3\). Applying §Cohomology of Projective Space, ⁋Proposition 1 (Bott) again, the \(H^2\) of \(\mathcal{O}(-4)\) indeed has dimension

\[\binom{2+(-4)}{2}=\binom{-2}{2} = 3\]

so we verify that they match.

Generalizations of Serre Duality

We now generalize the preceding discussion. The first thing one can do is to extend from locally free sheaves to arbitrary coherent sheaves. This is not as difficult as it might seem, because on a smooth variety any coherent sheaf has a finite locally free resolution, so one can inductively carry Serre duality along the terms of the resolution. (§Canonical Line Bundle) However, the statement obtained this way is not a duality between \(H^i(X,\mathcal{E})\) and \(H^{n-i}(X,\omega_X\otimes\mathcal{E}^\vee)\), but rather takes the form using \(\Ext\) that we shall see below.

Next we drop the smoothness condition on \(X\). In this case there are two major problems. The first visible one is that \(X\) need not have a canonical line bundle. Another problem is somewhat more subtle: when we obtained the explicit isomorphism from the perfect pairing, we somewhat implicitly used the isomorphism

\[\mathcal{H}om(\mathcal{E}, \mathcal{F})\cong \mathcal{E}^\vee\otimes \mathcal{F}\]

but in fact this is possible only because \(\mathcal{E}\) is locally free, and for a coherent sheaf that is not locally free this isomorphism does not generally hold even if \(X\) is smooth. For example, for the skyscraper sheaf \(\mathcal{E}=\mathcal{O}_X/\mathfrak{m}_0\) at the origin on \(X=\mathbb{A}^1\) we have \(\mathcal{E}^\vee=0\) but \(\mathcal{H}om(\mathcal{E},\mathcal{E})\cong\mathcal{E}\) is not \(0\), and if \(X\) is singular we cannot even use the argument of carrying along a finite locally free resolution as before. Therefore we introduce derived functors again, and call \(\omega_X\) the dualizing sheaf of \(X\) if it satisfies

\[\Ext^i_X(\mathcal{F},\omega_X)\cong H^{n-i}(X,\mathcal{F})^\ast\]

for all coherent sheaves \(\mathcal{F}\) on \(X\) and all \(i\). In general, its existence is guaranteed for Cohen-Macaulay varieties of pure dimension \(n\), and although we will not give the definition, one can think of the Cohen-Macaulay condition intuitively as a notion encompassing singular varieties that do not cause dimension problems.

A somewhat less intuitive generalization is relative Serre duality. In fact, we have not been paying attention to the underlying field \(\mathbb{K}\) of the variety, but in this context it helps to make its role explicit.

That an affine variety \(X\) is defined over \(\mathbb{K}\) means that its coordinate ring \(A\) is a \(\mathbb{K}\)-algebra, so there exists a ring homomorphism \(\mathbb{K}\rightarrow A\) encoding this structure. Viewing these as the coordinate rings of a point \(\Spec\mathbb{K}\) and of \(X\) respectively, this structure morphism is geometrically the map \(X\rightarrow \Spec\mathbb{K}\), that is, the map to a point.

Relative Serre duality generalizes this setting by replacing the target point \(\Spec\mathbb{K}\) with another variety. First, for arbitrary varieties \(X,Y\), we say that a morphism \(f:X\rightarrow Y\) is a smooth projective morphism if \(f\) is a flat proper morphism and each fiber \(f^{-1}(y)\) over \(y\in Y\) is a smooth projective variety. Then in this case, each fiber \(f^{-1}(y)\) is a smooth projective variety so it has a canonical line bundle \(\omega_{X_y}\), and these are consistently assembled into the relative dualizing sheaf \(\omega_{X/Y}\) on \(X\). That is, \(\omega_{X/Y}\) is a sheaf satisfying \(\omega_{X/Y}\vert_{X_y}\cong\omega_{X_y}\) for each \(y\). The generalization in this case is as follows.

Proposition 4 (Relative Serre duality) For a smooth projective morphism \(f \colon X \rightarrow Y\), let \(n = \dim X - \dim Y\), and assume that \(R^j f_\ast \mathcal{O}_X\) is locally free for all \(j\).¹ Then for all \(i\) there exists an isomorphism

\[R^i f_\ast \omega_{X/Y} \cong (R^{n-i} f_\ast \mathcal{O}_X)^\vee\]

In particular, since each fiber is a variety it is connected, and therefore \(f_\ast \mathcal{O}_X \cong \mathcal{O}_Y\), so for \(i = n\) we have \(R^n f_\ast \omega_{X/Y} \cong \mathcal{O}_Y\).

Grothendieck Duality

Let us retrace the process of generalizing Serre duality. We first proved Serre duality on \(\mathbb{P}^n\) using the trace map and cup product (Proposition 2 (Serre duality pairing, projective case)), and extended this to arbitrary smooth projective varieties via a finite morphism. The extension to coherent sheaves was handled by induction through locally free resolutions, and the extension to singular varieties was handled by introducing the dualizing sheaf. Proposition 4 (Relative Serre duality) was the generalization replacing the target point with an arbitrary variety.

The most modern interpretation of Serre duality is Grothendieck duality, which is formulated in the language of derived categories. ([Homological Algebra] §Derived Categories, ⁋Definition 2) This generalization has quite a convincing motivation: for example, when we defined sheaf cohomology we already had to think about injective resolutions, and when we generalized Serre duality to arbitrary coherent sheaves above we also had to think about locally free resolutions, so we know that the derived category is where everything actually happens. In particular, the key point is that the perfect pairing in Serre duality encodes exactly the same information as the choice of a concrete isomorphism

\[H^n(X, \omega_X) \cong \mathbb{K}\]

and lifting this to the derived category yields the observation that it is a special case of the adjunction between the derived pushforward \(R f_\ast\) and its right adjoint. Concretely, the Serre duality isomorphism

\[H^i(X, \mathcal{E}) \cong H^{n-i}(X, \omega_X \otimes \mathcal{E}^\vee)^\ast\]

is derived from the following adjunction isomorphism in the derived category:

\[\operatorname{Hom}_{D(X)}(\mathcal{F}, f^! \mathcal{G}) \cong \operatorname{Hom}_{D(Y)}(R f_\ast \mathcal{F}, \mathcal{G})\]

Here the exceptional inverse image \(f^!\) is the functor defined as the right adjoint of \(R f_\ast\) in the derived category, and to define this properly one must necessarily work in the derived category.

As mentioned earlier, Grothendieck duality is a result that includes relative Serre duality. To see this, consider the case of a smooth morphism \(f:X\rightarrow Y\); then \(f^! \mathcal{O}_Y \cong \omega_{X/Y}[n]\) holds, and from this we see that \(\omega_{X/Y}\) placed in the correct degree is exactly \(f^!\mathcal{O}_Y\).

Proposition 5 (Grothendieck Duality) For a proper morphism \(f \colon X \rightarrow Y\) and a coherent sheaf \(\mathcal{F}\) on \(X\), the following isomorphism holds in the derived category:

\[R f_\ast R\mathcal{H}om_{\mathcal{O}_X}(\mathcal{F}, f^! \mathcal{G}) \cong R\mathcal{H}om_{\mathcal{O}_Y}(R f_\ast \mathcal{F}, \mathcal{G})\]

Here \(R\mathcal{H}om\) is derived Hom ([Homological Algebra] §Derived Categories, ⁋Proposition 10), and \(\mathcal{G}\) is a bounded complex of coherent sheaves on \(Y\).

Intuitively, this theorem says that ‘Hom after pushforward’ and ‘pushforward after Hom’ agree. That is, computing the Hom between \(\mathcal{F}\) and \(f^! \mathcal{G}\) on \(X\) and then pushing down to \(Y\) is the same as first pushing \(\mathcal{F}\) down to \(Y\) and then computing the Hom with \(\mathcal{G}\).

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References

[Hart] R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, Springer, 1977.
[Ser] J.-P. Serre, Faisceaux algébriques cohérents, Annals of Mathematics, 1955.

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1. This condition always holds in characteristic zero by the degeneration of the Hodge-de Rham spectral sequence. It can fail in characteristic \(p\). ↩