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Reflexive sheaf

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In algebraic geometry, a reflexive sheaf is a coherent sheaf that is isomorphic to its second dual (as a sheaf of modules) via the canonical map. The second dual of a coherent sheaf is called the reflexive hull of the sheaf. A basic example of a reflexive sheaf is a locally free sheaf of finite rank and, in practice, a reflexive sheaf is thought of as a kind of a vector bundle modulo some singularity. The notion is important both in scheme theory and complex algebraic geometry.

For the theory of reflexive sheaves, one works over an integral noetherian scheme.

A reflexive sheaf is torsion-free. The dual of a coherent sheaf is reflexive. Usually, the product of reflexive sheaves is defined as the reflexive hull of their tensor products (so the result is reflexive.)

A coherent sheaf F is said to be "normal" in the sense of Barth if the restriction F ( U ) → F ( U − Y ) {\displaystyle F(U)\to F(U-Y)} is bijective for every open subset U and a closed subset Y of U of codimension at least 2. With this terminology, a coherent sheaf on an integral normal scheme is reflexive if and only if it is torsion-free and normal in the sense of Barth. A reflexive sheaf of rank one on an integral locally factorial scheme is invertible.

A divisorial sheaf on a scheme X is a rank-one reflexive sheaf that is locally free at the generic points of the conductor DX of X. For example, a canonical sheaf (dualizing sheaf) on a normal projective variety is a divisorial sheaf.

See also

Notes

  1. Hartshorne 1980, Corollary 1.2.
  2. Hartshorne 1980, Proposition 1.6.
  3. Hartshorne 1980, Proposition 1.9.
  4. Kollár, Ch. 3, § 1.

References

Further reading

External links


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