# Serre duality

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In algebraic geometry, a branch of mathematics, Serre duality is a duality present on non-singular projective algebraic varieties V of dimension n (and in greater generality for vector bundles and further, for coherent sheaves). It shows that a cohomology group Hi is the dual space of another one, Hni.

In the case for holomorphic vector bundle E over a smooth compact complex manifold V, the statement is in the form:

$H^{q}(V,E)\cong H^{n-q}(V,K\otimes E^{\ast })^{\ast },$ in which V is not necessarily projective.

## Algebraic curve

The case of algebraic curves was already implicit in the Riemann-Roch theorem. For a curve C the coherent groups Hi vanish for i > 1; but H1 does enter implicitly. In fact, the basic relation of the theorem involves l(D) and l(KD), where D is a divisor and K is a divisor of the canonical class. After Serre we recognise l(KD) as the dimension of H1(D), where now D means the line bundle determined by the divisor D. That is, Serre duality in this case relates groups H1(D) and H0(KD*), and we are reading off dimensions (notation: K is the canonical line bundle, D* is the dual line bundle, and juxtaposition is the tensor product of line bundles).

In this formulation the Riemann-Roch theorem can be viewed as a computation of the Euler characteristic of a sheaf

h0(D) − h1(D),

in terms of the genus of the curve, which is

h1(C,OC),

and the degree of D. It is this expression that can be generalised to higher dimensions.

Serre duality of curves is therefore something very classical; but it has an interesting light to cast. For example, in Riemann surface theory, the deformation theory of complex structures is studied classically by means of quadratic differentials (namely sections of L(K2)). The deformation theory of Kunihiko Kodaira and D. C. Spencer identifies deformations via H1(T), where T is the tangent bundle sheaf K*. The duality shows why these approaches coincide.

## Equidimensional projective schemes

There is an analogue of Serre duality for equidimensional projective schemes $X$ over an algebraically closed field $k$ : if we suppose ${\mathcal {O}}_{X}(1)$ is very ample (meaning it gives the embedding of $X$ into $\mathbb {P} ^{m}$ ) there is a sheaf $\omega _{X}^{\circ }$ called the dualizing sheaf such that for every coherent sheaf ${\mathcal {F}}\in {\textbf {Coh}}(X)$ there are functorial maps

$\theta ^{i}:{\text{Ext}}^{i}({\mathcal {F}},\omega _{X}^{\circ })\to H^{n-i}(X,{\mathcal {F}})^{*}$ which are isomorphisms. These dualizing sheaves can be computed as

$\omega _{X}^{\circ }={\mathcal {Ext}}_{{\mathcal {O}}_{\mathbb {P} ^{m}}}^{r}({\mathcal {O}}_{X},\omega _{\mathbb {P} ^{m}}),$ where $r$ is the codimension of $X$ in $\mathbb {P} ^{m}$ . If $X$ is a local complete intersection, then the Koszul complex of ${\mathcal {O}}_{X}$ can be used to compute this Ext group (since it is a locally-free resolution of ${\mathcal {O}}_{X}$ ), giving

${\mathcal {Ext}}_{{\mathcal {O}}_{\mathbb {P} ^{m}}}^{r}({\mathcal {O}}_{X},\omega _{\mathbb {P} ^{m}})=\omega _{X}^{\circ }\cong \omega _{\mathbb {P} ^{m}}\otimes \bigwedge ^{r}({\mathcal {I}}/{\mathcal {I}}^{2})^{*},$ where ${\mathcal {I}}$ is the ideal sheaf of $X$ .

### Example

Consider a local complete intersection $X\subset \mathbb {P} ^{m}$ with presentation given by the ideal ${\mathcal {I}}=(f_{1},\ldots ,f_{r})$ generated by forms of degrees $d_{1},\ldots ,d_{r}$ . Then,

${\mathcal {I}}/{\mathcal {I}}^{2}\cong {\mathcal {O}}_{X}(-d_{1})\oplus \cdots \oplus {\mathcal {O}}_{X}(-d_{r})$ giving us that

$\bigwedge ^{r}({\mathcal {I}}/{\mathcal {I}}^{2})^{*}\cong {\mathcal {O}}_{X}(d_{1}+\cdots +d_{r}).$ Since $\omega _{\mathbb {P} ^{m}}|_{X}\cong {\mathcal {O}}_{\mathbb {P} ^{m}}(-m-1)|_{X}$ we get

$\omega _{X}^{\circ }\cong {\mathcal {O}}_{X}(d_{1}+\cdots +d_{r}-m-1)$ as the dualizing sheaf. For example, the dualizing sheaf of a plane curve of degree $d$ is given by ${\mathcal {O}}_{X}(d-3)$ . This is a special case of the proof of the adjunction formula, which uses the second fundamental (conormal) exact sequence. This sequence is exact on the left when $X$ is (locally) a complete intersection.

## Origin and generalisations

The origin of the theory lies in Serre's earlier work on several complex variables. In the generalisation of Alexander Grothendieck, Serre duality becomes a part of coherent duality in a much broader setting. While the role of K above in general Serre duality is played by the determinant line bundle of the cotangent bundle, when V is a manifold, in full generality K cannot merely be a single sheaf in the absence of some hypothesis of non-singularity on V. The formulation in full generality uses a derived category and Ext functors, to allow for the fact that K is now represented by a chain complex of sheaves, namely, the dualizing complex. Nevertheless, the statement of the theorem is recognisably Serre's.