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# 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.

A reflexive sheaf is torsion-free. The dual of a coherent sheaf is reflexive.[1]

A coherent sheaf F is said to be "normal" in the sense of Barth if the restriction \( F(U) \to F(U - Y) \) 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.[2] A reflexive sheaf of rank one on an integral locally factorial scheme is invertible.[3]

See also

Torsionless module

Torsion sheaf

Divisorial sheaf

Notes

Hartshorne 1980, Corollary 1.2.

Hartshorne 1980, Proposition 1.6.

Hartshorne 1980, Proposition 1.9.

References

Hartshorne, R.: Stable reflexive sheaves. Math. Ann.254 (1980), 121–176

Hartshorne, R.: Stable reflexive sheaves. II, Invent. Math. 66 (1982), 165–190

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

Undergraduate Texts in Mathematics

Graduate Studies in Mathematics

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