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

  1. Serre (1955); Huybrechts (2005), Proposition 4.1.15.
  2. Huybrechts (2005), exercise 3.2.3.
  3. For a curve, Serre duality is simpler but still nontrivial. One proof is given in Tate (1968).
  4. Hartshorne (1977), Theorem III.7.6.
  5. Hartshorne (1977), proof of Proposition III.7.5; Stacks Project, Tag 0A9X.
  6. Hartshorne (1977), Theorem III.7.11; Stacks Project, Tag 0BQZ.
  7. Hartshorne (1966), Corollary VII.3.4(c); Stacks Project, Tag 0B6I; Stacks Project, Tag 0B6S.
  8. Huybrechts (2006), Definition 1.28, Theorem 3.12.
  9. Stacks Project, Tag 0E58.

References

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