# Reflexive sheaf

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 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.^{[1]} 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 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.^{[2]} A reflexive sheaf of rank one on an integral locally factorial scheme is invertible.^{[3]}

A **divisorial sheaf** on a scheme *X* is a rank-one reflexive sheaf that is locally free at the generic points of the conductor *D*_{X} of *X*.^{[4]}

## See also

## Notes

- ↑ Hartshorne 1980, Corollary 1.2.
- ↑ Hartshorne 1980, Proposition 1.6.
- ↑ Hartshorne 1980, Proposition 1.9.
- ↑ Kollár, Ch. 3, § 1.

## References

- Hartshorne, R.: Stable reflexive sheaves. Math. Ann.254 (1980), 121–176
- Hartshorne, R.: Stable reflexive sheaves. II, Invent. Math. 66 (1982), 165–190
- Kollár, János, "Chapter 3",
*Book on Moduli of Surfaces*

## Further reading

- Greb, Daniel; Kebekus, Stefan; Kovacs, Sandor J.; Peternell, Thomas (2011). "Differential Forms on Log Canonical Spaces". arXiv:1003.2913v4.

## External links

- http://mathoverflow.net/questions/61806/reflexive-sheaves-on-singular-surfaces
- http://mathoverflow.net/questions/187537/push-forward-of-locally-free-sheaves/187541#187541
- http://www-personal.umich.edu/~kschwede/GeneralizedDivisors.pdf