In
algebraic geometry, a reflexive sheaf is a
coherent sheaf
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with ref ...
that is isomorphic to its second dual (as a
sheaf of modules In mathematics, a sheaf of ''O''-modules or simply an ''O''-module over a ringed space (''X'', ''O'') is a sheaf (mathematics), sheaf ''F'' such that, for any open subset ''U'' of ''X'', ''F''(''U'') is an ''O''(''U'')-module and the restriction map ...
) 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
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refer ...
of finite rank and, in practice, a reflexive sheaf is thought of as a kind of a
vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...
modulo some singularity. The notion is important both in
scheme theory
In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations ''x'' = 0 and ''x''2 = 0 define the same algebraic variety but different sc ...
and
complex algebraic geometry
In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of spaces such as complex manifolds and c ...
.
For the theory of reflexive sheaves, one works over an
integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along wit ...
noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite lengt ...
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
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 In algebraic geometry, an algebraic variety or scheme ''X'' is normal if it is normal at every point, meaning that the local ring at the point is an integrally closed domain. An affine variety ''X'' (understood to be irreducible) is normal if and ...
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''.
For example, a
canonical sheaf (
dualizing sheaf
In algebraic geometry, the dualizing sheaf on a proper scheme ''X'' of dimension ''n'' over a field ''k'' is a coherent sheaf \omega_X together with a linear functional
:t_X: \operatorname^n(X, \omega_X) \to k
that induces a natural isomorphism of ...
) on a normal projective variety is a divisorial sheaf.
See also
*
Torsionless module
In abstract algebra, a module ''M'' over a ring ''R'' is called torsionless if it can be embedded into some direct product ''R'I''. Equivalently, ''M'' is torsionless if each non-zero element of ''M'' has non-zero image under some ''R''-linear ...
*
Torsion sheaf
*
Twisted sheaf
Notes
References
*
*
*
Further reading
*
External links
Reflexive sheaves on singular surfacesPush-forward of locally free sheaves
*http://www-personal.umich.edu/~kschwede/GeneralizedDivisors.pdf
Sheaf theory
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