# Serre duality

**Serre duality** is a duality theorem in algebraic geometry, due to [Jean-Pierre Serre](https://www.edgepedia.org/jean-pierre-serre), that identifies the top cohomology of a coherent sheaf (algebraic object assigning vector-space-like data locally to a variety) on a projective variety with the dual of global sections of a twisted Hom sheaf. In its basic form on projective space, it states that for a coherent sheaf \( \mathcal{F} \) on \( \mathbb{P}^n_k \), the groups \( \operatorname{Ext}^i(\mathcal{F}, \mathcal{O}_{\mathbb{P}^n}(-n-1)) \) and \( H^{n-i}(\mathbb{P}^n, \mathcal{F})^\vee \) are naturally isomorphic, so that the otherwise mysterious higher cohomology \( H^n \) becomes computable as ordinary Hom groups.<sup>[1](https://ocw.mit.edu/courses/18-726-algebraic-geometry-spring-2009/3e1b3cc3a5b2d36c6d0bbfe4730101ce_MIT18_726s09_lec23_serre_dual_proj.pdf)</sup> Serre proved the theorem in 1955, in the same period as his foundational paper *Faisceaux algébriques cohérents* (FAC).<sup>[2](https://smf.emath.fr/system/files/filepdf/Gaz-99.pdf)</sup>

| Key fact | Detail |
|---|---|
| Origin | Serre's duality theorem, 1955, alongside FAC (*Faisceaux algébriques cohérents*, Annals of Math. 61)<sup>[2](https://smf.emath.fr/system/files/filepdf/Gaz-99.pdf)</sup><sup> • </sup><sup>[3](https://stacks.math.columbia.edu/bibliography/FAC)</sup> |
| Projective space | For \( \mathcal{F} \) coherent on \( \mathbb{P}^n_k \), \( \operatorname{Ext}^i(\mathcal{F}, \mathcal{O}(-n-1)) \cong H^{n-i}(\mathbb{P}^n, \mathcal{F})^\vee \); \( H^n(\mathcal{O}(-n-1)) \) is one-dimensional over \( k \)<sup>[1](https://ocw.mit.edu/courses/18-726-algebraic-geometry-spring-2009/3e1b3cc3a5b2d36c6d0bbfe4730101ce_MIT18_726s09_lec23_serre_dual_proj.pdf)</sup> |
| Dualizing sheaf | A pair \( (\omega_X, t) \) with \( t: H^n(\omega_X) \to k \) such that \( \operatorname{Hom}(\mathcal{F}, \omega_X) \cong H^n(\mathcal{F})^\vee \) for all coherent \( \mathcal{F} \); unique up to unique isomorphism when it exists<sup>[4](https://wstein.org/hartnotes/hartnotes.pdf)</sup> |
| Existence | Every projective scheme of dimension \( n \) over a field \( k \) has a dualizing sheaf<sup>[4](https://wstein.org/hartnotes/hartnotes.pdf)</sup> |
| General duality | \( \operatorname{Ext}^i_X(\mathcal{F}, \omega_X) \cong H^{n-i}(X, \mathcal{F})^\vee \); the pairing is perfect when \( X \) is projective, equidimensional, and Cohen-Macaulay<sup>[4](https://wstein.org/hartnotes/hartnotes.pdf)</sup><sup> • </sup><sup>[5](https://www.math.columbia.edu/~dejong/courses/schemes-spring-2020/lecture-notes-b.pdf)</sup> |
| Smooth case | For \( X \) proper and smooth of dimension \( r \) over \( k \) and \( \mathcal{F} \) locally free: \( H^i(X, \mathcal{F})' \cong H^{r-i}(\operatorname{Hom}_{\mathcal{O}_X}(\mathcal{F}, \Omega^r_{X/k})) \)<sup>[6](https://jcreinhold.github.io/sga/ii/12-projective-algebraic-schemes.html)</sup> |
| Grothendieck extension | Generalized to the relative case for proper morphisms of schemes via derived categories (Séminaire Bourbaki, May 1957; Séminaire Hartshorne, fall 1963)<sup>[7](https://smf.emath.fr/publications/residus-et-dualite-prenotes-pour-un-seminaire-hartshorne?language=en)</sup> |

## Duality on projective space

The theorem is easiest to state on \( X = \mathbb{P}^n_k \) over a field \( k \), which need not be algebraically closed. The key input is Serre's explicit computation of the cohomology of twisting sheaves, from which \( H^n(X, \mathcal{O}_X(-n-1)) \) is one-dimensional over \( k \).<sup>[1](https://ocw.mit.edu/courses/18-726-algebraic-geometry-spring-2009/3e1b3cc3a5b2d36c6d0bbfe4730101ce_MIT18_726s09_lec23_serre_dual_proj.pdf)</sup> The duality theorem then has two parts:<sup>[1](https://ocw.mit.edu/courses/18-726-algebraic-geometry-spring-2009/3e1b3cc3a5b2d36c6d0bbfe4730101ce_MIT18_726s09_lec23_serre_dual_proj.pdf)</sup>

1. The pairing \( \operatorname{Hom}_X(\mathcal{F}, \mathcal{O}_X(-n-1)) \times H^n(X, \mathcal{F}) \to H^n(X, \mathcal{O}_X(-n-1)) \) is a perfect pairing of finite-dimensional \( k \)-vector spaces.
2. For each \( i \ge 0 \) there is a natural isomorphism \( \operatorname{Ext}^i(\mathcal{F}, \mathcal{O}_X(-n-1)) \to H^{n-i}(X, \mathcal{F})' \), compatible with short exact sequences, which reproduces the pairing for \( i = 0 \).

The proof reduces to the case of twisting sheaves \( \mathcal{O}(m) \), where the computation is explicit, and then descends to a general coherent \( \mathcal{F} \) by writing a presentation \( E_1 \to E_0 \to \mathcal{F} \to 0 \) with \( E_0, E_1 \) direct sums of twisting sheaves; on \( \mathbb{P}^n \) such resolutions exist by Serre's theorem.<sup>[1](https://ocw.mit.edu/courses/18-726-algebraic-geometry-spring-2009/3e1b3cc3a5b2d36c6d0bbfe4730101ce_MIT18_726s09_lec23_serre_dual_proj.pdf)</sup> The same reduction underlies the SGA 2 account: every coherent module is a quotient of a direct sum of modules \( \mathcal{O}(-m) \) with \( m \) large, and one argues by descending induction on \( i \).<sup>[6](https://jcreinhold.github.io/sga/ii/12-projective-algebraic-schemes.html)</sup>

The sheaf \( \omega_{\mathbb{P}^n} = \mathcal{O}_{\mathbb{P}^n}(-n-1) \) is canonically the top exterior power of the differentials, \( \omega_X = \bigwedge^n \Omega_{X/k} \cong \mathcal{O}_X(-n-1) \), and \( H^n(X, \omega_X) \cong k \).<sup>[8](https://math.ug/alggeom3-ws2324/sect0002.html)</sup>

## The dualizing sheaf

For an arbitrary projective scheme \( X \) of dimension \( n \) over \( k \), the role of \( \mathcal{O}(-n-1) \) is played by a [dualizing sheaf](https://www.edgepedia.org/dualizing-sheaf). A dualizing sheaf is a pair \( (\omega_X, t) \), where \( \omega_X \) is a coherent sheaf and \( t: H^n(X, \omega_X) \to k \) is a homomorphism, such that for every coherent sheaf \( \mathcal{F} \) the natural map \( \operatorname{Hom}_{\mathcal{O}_X}(\mathcal{F}, \omega_X) \to H^n(\mathcal{F})^\vee \) is an isomorphism.<sup>[4](https://wstein.org/hartnotes/hartnotes.pdf)</sup> On \( \mathbb{P}^n \) the map \( t \) is an isomorphism \( H^n(\omega_{\mathbb{P}^n}) \cong k \), but on a general scheme it need not be.<sup>[4](https://wstein.org/hartnotes/hartnotes.pdf)</sup>

The definition is representability of the functor \( \mathcal{F} \mapsto H^n(\mathcal{F})^\vee \) on the category of coherent sheaves, and this viewpoint gives uniqueness: if \( X \) admits a dualizing sheaf \( (\omega_X, t) \), the pair is unique up to unique isomorphism.<sup>[4](https://wstein.org/hartnotes/hartnotes.pdf)</sup> [Existence](https://www.edgepedia.org/existence) is a theorem: every projective scheme of dimension \( n \) over \( k \) has a dualizing sheaf. One construction embeds \( X \) in \( \mathbb{P}^N_k \) and sets \( \omega_X = \operatorname{Ext}^{N-n}_{\mathcal{O}_{\mathbb{P}^N}}(\mathcal{O}_X, \omega_{\mathbb{P}^N}) \); another uses a finite morphism \( f: X \to \mathbb{P}^n \) and defines \( \omega_X = f^! \omega_{\mathbb{P}^n} \).<sup>[4](https://wstein.org/hartnotes/hartnotes.pdf)</sup>

With a dualizing sheaf in hand, the general duality theorem reads \( \operatorname{Ext}^i_X(\mathcal{F}, \omega_X) \cong H^{n-i}(X, \mathcal{F})^\vee \).<sup>[4](https://wstein.org/hartnotes/hartnotes.pdf)</sup> A qualification matters here: for \( X = \mathbb{P}^n_k \) the natural maps are always isomorphisms, but for an arbitrary projective scheme they need not be unless \( X \) is locally Cohen-Macaulay, meaning the local ring at each point is Cohen-Macaulay (depth equals dimension).<sup>[4](https://wstein.org/hartnotes/hartnotes.pdf)</sup> In that language, if \( X \) is projective, equidimensional, and Cohen-Macaulay, the pairings \( \operatorname{Ext}^i_X(\mathcal{F}, \omega_X) \times H^{\dim X - i}(X, \mathcal{F}) \to k \) are perfect; more general statements are Hartshorne's Theorem III 7.6.<sup>[5](https://www.math.columbia.edu/~dejong/courses/schemes-spring-2020/lecture-notes-b.pdf)</sup>

## The smooth case and the canonical sheaf

For \( X \) proper and smooth of dimension \( r \) over \( k \), the theorem takes the form \( H^i(X, \mathcal{F})' \cong H^{r-i}(\operatorname{Hom}_{\mathcal{O}_X}(\mathcal{F}, \Omega^r_{X/k})) \) for \( \mathcal{F} \) locally free; this is the statement one recovers from the general duality theorem, and it remains true in this form for proper smooth schemes.<sup>[6](https://jcreinhold.github.io/sga/ii/12-projective-algebraic-schemes.html)</sup> The homomorphism is deduced from the Yoneda pairing \( H^i(X, \mathcal{F}) \times \operatorname{Ext}^{r-i}(X; \mathcal{F}, \Omega^r_{X/k}) \to H^r(X, \Omega^r_{X/k}) \), together with the computation \( H^r(X, \Omega^r_{X/k}) = H^r(\mathbb{P}^r_k, \mathcal{O}_{\mathbb{P}^r}(-r-1)) \cong k \).<sup>[6](https://jcreinhold.github.io/sga/ii/12-projective-algebraic-schemes.html)</sup>

On curves the statement becomes concrete. For a smooth projective curve \( C \), Serre duality gives \( H^1(C, \mathcal{F}) \cong H^0(C, \mathcal{F}^\vee \otimes \omega_C)^\vee \) and \( H^0(C, \mathcal{F}) \cong H^1(C, \mathcal{F}^\vee \otimes \omega_C)^\vee \), so the top cohomology \( H^1 \) is always the dual of an ordinary space of sections.<sup>[9](https://fiveable.me/algebraic-geometry/unit-5/serre-duality-riemann-roch-theorem/study-guide/C5NQ9agbGHAbJ3AS)</sup> This identification is the engine behind the Riemann-Roch theorem for curves, where the term \( h^1 \) is computed as a space of sections of a twisted canonical bundle.<sup>[10](https://math.uchicago.edu/~may/REU2013/REUPapers/Fung.pdf)</sup>

## Grothendieck's generalization

The modern form of the theorem is due to [Alexander Grothendieck](https://www.edgepedia.org/alexander-grothendieck). In a letter to Serre of January 12, 1956, he wrote of Serre's formula \( H^{n-p}(X, F)' = \operatorname{Ext}^p_{\mathcal{O}}(X, F, \Omega^n) \): "I find your formula very exciting, as I am quite convinced that it is the right way to state the duality theorem in both the analytic case and the algebraic case."<sup>[11](https://mattbaker.blog/2014/11/15/excerpts-from-the-grothendieck-serre-correspondence/)</sup> Grothendieck announced his duality theorems for coherent algebraic sheaves in a Séminaire Bourbaki talk of May 1957.<sup>[6](https://jcreinhold.github.io/sga/ii/12-projective-algebraic-schemes.html)</sup>

In the fall of 1963, a seminar at Harvard on duality theorems in algebraic geometry aimed to understand how the systematic use of derived categories, then a new method, had enabled Grothendieck to generalize Serre's duality theorems for projective varieties over a field to the relative case, for a proper morphism of schemes; Grothendieck prepared typewritten prenotes to guide [Robin Hartshorne](https://www.edgepedia.org/robin-hartshorne) in running the seminar.<sup>[7](https://smf.emath.fr/publications/residus-et-dualite-prenotes-pour-un-seminaire-hartshorne?language=en)</sup> In the derived-category formulation, for a proper morphism \( f: X \to \operatorname{Spec}(k) \), the dualizing complex is \( \omega_X^\bullet = f^\times \mathcal{O}_{\operatorname{Spec} k} \), where \( f^\times \) is right adjoint to \( Rf_* \), and Grothendieck-Serre duality follows formally from this adjunction.<sup>[8](https://math.ug/alggeom3-ws2324/sect0002.html)</sup> The dualizing sheaf of the classical theory captures only one cohomology object of this complex, so when the complex is not concentrated in a single degree, the sheaf alone does not express the full duality.<sup>[8](https://math.ug/alggeom3-ws2324/sect0002.html)</sup>

## Applications

Serre duality is a working tool across algebraic geometry and neighboring fields:<sup>[10](https://math.uchicago.edu/~may/REU2013/REUPapers/Fung.pdf)</sup>

- **Riemann-Roch for curves.** The duality isomorphism converts the defect term \( h^1 \) in Riemann-Roch into a dimension of sections, making the theorem provable and usable.<sup>[10](https://math.uchicago.edu/~may/REU2013/REUPapers/Fung.pdf)</sup><sup> • </sup><sup>[9](https://fiveable.me/algebraic-geometry/unit-5/serre-duality-riemann-roch-theorem/study-guide/C5NQ9agbGHAbJ3AS)</sup>
- **Bott's theorem.** In representation theory, the algebraic-geometric machinery built around Serre duality and dualizing sheaves yields Bott's theorem on the cohomology of homogeneous vector bundles.<sup>[10](https://math.uchicago.edu/~may/REU2013/REUPapers/Fung.pdf)</sup>
- **Ext computations.** On \( \mathbb{P}^n_k \), the theorem converts Ext groups between coherent sheaves into duals of cohomology groups, and for locally free sheaves \( \mathcal{F} \) one has \( \operatorname{Ext}^i(\mathcal{F}, \omega_X) = H^i(\mathcal{F}^\vee \otimes \omega_X) \).<sup>[1](https://ocw.mit.edu/courses/18-726-algebraic-geometry-spring-2009/3e1b3cc3a5b2d36c6d0bbfe4730101ce_MIT18_726s09_lec23_serre_dual_proj.pdf)</sup><sup> • </sup><sup>[4](https://wstein.org/hartnotes/hartnotes.pdf)</sup>
- **Further settings.** The theorem extends beyond schemes: Serre duality has been established for tame Deligne-Mumford stacks using the exceptional inverse image functor for quasi-finite proper morphisms, and analogs appear for supercurves, where the proof runs by compatibility with [Poincaré duality](https://www.edgepedia.org/poincare-duality) on the reduced curve.<sup>[12](https://link.springer.com/article/10.1007/s40687-022-00367-7)</sup><sup> • </sup><sup>[13](https://arxiv.org/abs/1401.5738)</sup>

## References

1. [Duality on projective space, MIT OpenCourseWare 18.726 Lecture 23 (Algebraic Geometry, Spring 2009)](https://ocw.mit.edu/courses/18-726-algebraic-geometry-spring-2009/3e1b3cc3a5b2d36c6d0bbfe4730101ce_MIT18_726s09_lec23_serre_dual_proj.pdf)
2. [Le prix Abel décerné à Jean-Pierre Serre, Gazette de la Société Mathématique de France](https://smf.emath.fr/system/files/filepdf/Gaz-99.pdf)
3. [Bibliography entry: Serre, Faisceaux algébriques cohérents (1955), The Stacks Project](https://stacks.math.columbia.edu/bibliography/FAC)
4. [William Stein, Hartshorne lecture notes, Chapters 24–27: Serre Duality for Arbitrary Projective Schemes](https://wstein.org/hartnotes/hartnotes.pdf)
5. [Lecture Notes B, Algebraic Geometry (Schemes), Columbia University, Spring 2020](https://www.math.columbia.edu/~dejong/courses/schemes-spring-2020/lecture-notes-b.pdf)
6. [SGA 2, Exposé XII: Applications to projective algebraic schemes, English translation](https://jcreinhold.github.io/sga/ii/12-projective-algebraic-schemes.html)
7. [Résidus et dualité, Prénotes pour un "Séminaire Hartshorne", Société Mathématique de France](https://smf.emath.fr/publications/residus-et-dualite-prenotes-pour-un-seminaire-hartshorne?language=en)
8. [Algebraic Geometry 3, Winter term 2023/24, lecture notes: Serre duality, Universität Göttingen](https://math.ug/alggeom3-ws2324/sect0002.html)
9. [Serre duality and Riemann-Roch theorem, Algebraic Geometry class notes, Fiveable](https://fiveable.me/algebraic-geometry/unit-5/serre-duality-riemann-roch-theorem/study-guide/C5NQ9agbGHAbJ3AS)
10. [Jun Hou Fung, Serre Duality and Applications, REU paper, University of Chicago (2013)](https://math.uchicago.edu/~may/REU2013/REUPapers/Fung.pdf)
11. [Excerpts from the Grothendieck-Serre Correspondence, Matt Baker's Math Blog](https://mattbaker.blog/2014/11/15/excerpts-from-the-grothendieck-serre-correspondence/)
12. [Serre duality for tame Deligne–Mumford stacks, Research in the Mathematical Sciences (2022)](https://link.springer.com/article/10.1007/s40687-022-00367-7)
13. [Serre Duality, Abel's Theorem, and Jacobi Inversion for Supercurves Over a Thick Superpoint, arXiv:1401.5738](https://arxiv.org/abs/1401.5738)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Advanced algebraic structures › Algebraic geometry › Schemes, stacks, and morphisms › Cohomology of schemes and formal functions*

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