In mathematics, a twisted sheaf is a variant of 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 refer ...
. Precisely, it is specified by: an
open covering
In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a family of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\subs ...
in the
étale topology
In algebraic geometry, the étale topology is a Grothendieck topology on the category of schemes which has properties similar to the Euclidean topology, but unlike the Euclidean topology, it is also defined in positive characteristic. The étale ...
''U''
''i'', coherent sheaves ''F''
''i'' over ''U''
''i'', a
Čech 2-cocycle ''θ'' for
on the covering ''U''
''i'' as well as the isomorphisms
:
satisfying
*
,
*
*
The notion of twisted sheaves was introduced by
Jean Giraud
Jean Henri Gaston Giraud (; 8 May 1938 – 10 March 2012) was a French artist, cartoonist, and writer who worked in the Franco-Belgian comics, Franco-Belgian ''bandes dessinées'' (BD) tradition. Giraud garnered worldwide acclaim predomin ...
. The above definition due to Căldăraru is down-to-earth but is equivalent to a more sophisticated definition in terms of
gerbe; see § 2.1.3 of .
See also
*
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 re ...
*
Torsion sheaf
References
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*
Geometry
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