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. 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 $F(U) \to F(U - Y)$ 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 D_X of X.

See also

• Torsionless module
• Torsion sheaf

References

• Hartshorne, R.: Stable reflexive sheaves. Math. Ann.254 (1980), 121–176
• Hartshorne, R.: Stable reflexive sheaves. II, Invent. Math. 66 (1982), 165–190

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