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 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.[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 F(U)->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.[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 DX of X.[4] For example, a canonical sheaf (dualizing sheaf) on a normal projective variety is a divisorial sheaf.

## Contents

* 1 See also
* 2 Notes
* 3 References
* 4 Further reading
* 5 External links

## See also

* Torsionless module
* Torsion sheaf
* Twisted sheaf

## 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

* Hartshorne, R. (1980). "Stable reflexive sheaves". Math. Ann. 254 (2): 121–176. doi:10.1007/BF01467074.
* Hartshorne, R. (1982). "Stable reflexive sheaves. II". Invent. Math. 66: 165–190. doi:10.1007/BF01404762. Bibcode: 1982InMat..66..165H.
* Kollár, János. "Chapter 3". Book on Moduli of Surfaces. https://web.math.princeton.edu/~kollar/.

## Further reading

* Greb, Daniel; Kebekus, Stefan; Kovacs, Sandor J.; Peternell, Thomas (2011). "Differential Forms on Log Canonical Spaces". Publications mathématiques de l'IHÉS 114: 87–169. doi:10.1007/s10240-011-0036-0.

## External links

* Reflexive sheaves on singular surfaces
* Push-forward of locally free sheaves
* http://www-personal.umich.edu/~kschwede/GeneralizedDivisors.pdf

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