Serre duality
In algebraic geometry, a branch of mathematics, Serre duality is a duality for the coherent sheaf cohomology of algebraic varieties, proved by Jean-Pierre Serre. The basic version applies to vector bundles on a smooth projective variety, but Alexander Grothendieck found wide generalizations, for example to singular varieties. On an n-dimensional variety, the theorem says that a cohomology group is the dual space of another one, . Serre duality is the analog for coherent sheaf cohomology of Poincaré duality in topology, with the canonical line bundle replacing the orientation sheaf.
Serre duality for vector bundles
Let X be a smooth variety of dimension n over a field k. Define the canonical line bundle to be the bundle of n-forms on X, the top exterior power of the cotangent bundle:
Suppose in addition that X is proper (for example, projective) over k. Then Serre duality says: for an algebraic vector bundle E on X and an integer i, there is a natural isomorphism
of finite-dimensional k-vector spaces. Here denotes the tensor product of vector bundles. It follows that the dimensions of the two cohomology groups are equal:
Serre also proved the same duality statement for X a compact complex manifold and E a holomorphic vector bundle.[1]
As in Poincaré duality, the isomorphism in Serre duality comes from the cup product in sheaf cohomology. Namely, the composition of the cup product with a natural trace map on is a perfect pairing:
The trace map is the analog for coherent sheaf cohomology of integration in de Rham cohomology. There is also a direct connection between Serre duality and Poincaré duality, via Hodge theory.[2]
Algebraic curves
A fundamental application of Serre duality is to algebraic curves. (Over the complex numbers, it is equivalent to consider compact Riemann surfaces.) For a line bundle L on a smooth projective curve X over a field k, the only possibly nonzero cohomology groups are and . Serre duality describes the group in terms of an group (for a different line bundle).[3] That is more concrete, since of a line bundle is simply its space of sections.
Serre duality is especially relevant to the Riemann-Roch theorem for curves. For a line bundle L of degree d on a curve X of genus g, the Riemann-Roch theorem says that
Using Serre duality, this can be restated in more elementary terms:
The latter statement (expressed in terms of divisors) is in fact the original version of the theorem from the 19th century. This is the main tool used to analyze how a given curve can be embedded into projective space and hence to classify algebraic curves.
Example: Every global section of a line bundle of negative degree is zero. Moreover, the degree of the canonical bundle is . Therefore, Riemann-Roch implies that for a line bundle L of degree , is equal to . When the genus g is at least 2, it follows by Serre duality that . Here is the first-order deformation space of X. This is the basic calculation needed to show that the moduli space of curves of genus g has dimension .
Serre duality for coherent sheaves
Another formulation of Serre duality holds for all coherent sheaves, not just vector bundles. As a first step in generalizing Serre duality, Grothendieck showed that this version works for schemes with mild singularities, Cohen–Macaulay schemes, not just smooth schemes.
Namely, for a Cohen-Macaulay scheme X of pure dimension n over a field k, Grothendieck defined a coherent sheaf on X called the dualizing sheaf. (Some authors call this sheaf .) Suppose in addition that X is proper over k. For a coherent sheaf E on X and an integer i, Serre duality says that there is a natural isomorphism
of finite-dimensional k-vector spaces.[4] Here the Ext group is taken in the abelian category of -modules. This includes the previous statement, since is isomorphic to when E is a vector bundle.
In order to use this result, one has to determine the dualizing sheaf explicitly, at least in special cases. When X is smooth over k, is the canonical line bundle defined above. More generally, if X is a Cohen–Macaulay subscheme of codimension r in a smooth scheme Y over k, then the dualizing sheaf can be described as an Ext sheaf:[5]
When X is a local complete intersection of codimension r in a smooth scheme Y, there is a more elementary description: the normal bundle of X in Y is a vector bundle of rank r, and the dualizing sheaf of X is given by[6]
In this case, X is a Cohen–Macaulay scheme with a line bundle, which says that X is Gorenstein.
Example: Let X be a complete intersection in projective space over a field k, defined by homogeneous polynomials of degrees . (To say that this is a complete intersection means that X has dimension .) There are line bundles O(d) on for integers d, with the property that homogeneous polynomials of degree d can be viewed as sections of O(d). Then the dualizing sheaf of X is the line bundle
by the adjunction formula. For example, the dualizing sheaf of a plane curve X of degree d is .
Grothendieck duality
Grothendieck's theory of coherent duality is a broad generalization of Serre duality, using the language of derived categories. For any scheme X of finite type over a field k, there is an object of the bounded derived category of coherent sheaves on X, , called the dualizing complex of X over k. Formally, is the exceptional inverse image , where f is the given morphism . When X is Cohen–Macaulay of pure dimension n, is ; that is, it is the dualizing sheaf discussed above, viewed as a complex in (cohomological) degree −n. In particular, when X is smooth over k, is the canonical line bundle placed in degree −n.
Using the dualizing complex, Serre duality generalizes to any proper scheme X over k. Namely, there is a natural isomorphism of finite-dimensional k-vector spaces
for any object E in .[7]
More generally, for a proper scheme X over k, an object E in , and F a perfect complex in , one has the elegant statement:
Here the tensor product means the derived tensor product, as is natural in derived categories. (To compare with previous formulations, note that can be viewed as .) When X is also smooth over k, every object in is a perfect complex, and so this duality applies to all E and F in . The statement above is then summarized by saying that is a Serre functor on for X smooth and proper over k.[8]
Serre duality holds more generally for proper algebraic spaces over a field.[9]
Notes
- Serre (1955); Huybrechts (2005), Proposition 4.1.15.
- Huybrechts (2005), exercise 3.2.3.
- For a curve, Serre duality is simpler but still nontrivial. One proof is given in Tate (1968).
- Hartshorne (1977), Theorem III.7.6.
- Hartshorne (1977), proof of Proposition III.7.5; Stacks Project, Tag 0A9X.
- Hartshorne (1977), Theorem III.7.11; Stacks Project, Tag 0BQZ.
- Hartshorne (1966), Corollary VII.3.4(c); Stacks Project, Tag 0B6I; Stacks Project, Tag 0B6S.
- Huybrechts (2006), Definition 1.28, Theorem 3.12.
- Stacks Project, Tag 0E58.
References
- Hartshorne, Robin (1977), Algebraic geometry, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157, OCLC 13348052
- Hartshorne, Robin (1966), Residues and duality, Lecture Notes in Mathematics, 20, Berlin, New York: Springer-Verlag, ISBN 978-3-540-03603-6, MR 0222093
- Hazewinkel, Michiel, ed. (2001) [1994], "Duality", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
- Huybrechts, Daniel (2005), Complex geometry, Berlin: Springer-Verlag, ISBN 3-540-21290-6, MR 2093043
- Huybrechts, Daniel (2006), Fourier–Mukai transforms in algebraic geometry, Oxford University Press, ISBN 978-0199296866, MR 2244106
- Serre, Jean-Pierre (1955), "Un théorème de dualité", Commentarii Mathematici Helvetici, 29: 9–26, doi:10.1007/BF02564268, MR 0067489
- Tate, John (1968), "Residues of differentials on curves" (PDF), Annales Scientifiques de l'École Normale Supérieure, Série 4, 1: 149–159, doi:10.24033/asens.1162, ISSN 0012-9593, MR 0227171
External links
- The Stacks Project Authors, The Stacks Project