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 ref ...
. Precisely, it is specified by: an
open covering 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'' 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 Bandes dessinées, Franco-Belgian ''bandes dessinées'' (BD) tradition. Giraud garnered worldwide acclaim under the pseu ...
. The above definition due to Căldăraru is down-to-earth but is equivalent to a more sophisticated definition in terms of
gerbe
In mathematics, a gerbe (; ) is a construct in homological algebra and topology. Gerbes were introduced by Jean Giraud (mathematician), Jean Giraud following ideas of Alexandre Grothendieck as a tool for non-commutative cohomology in degree 2. Th ...
; 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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