In mathematics, a twisted sheaf is a variant of a coherent sheaf. Precisely, it is specified by: an open covering in the étale topology Ui, coherent sheaves Fi over Ui, a Čech 2-cocycle θ on the covering Ui as well as the isomorphisms
\( {\displaystyle g_{ij}:F_{j}|_{U_{ij}}{\overset {\sim }{\to }}F_{i}|_{U_{ij}}} \)
satisfying
\( {\displaystyle g_{ii}=\operatorname {id} _{F_{i}}}, \)
\( {\displaystyle g_{ij}=g_{ji}^{-1},} \)
\( {\displaystyle g_{ij}\circ g_{jk}\circ g_{ki}=\theta _{ijk}\operatorname {id} _{F_{i}}.} \)
The notion of twisted sheaves was introduced by Giraud. 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 (Lieblich 2007).
See also
reflexive sheaf
References
Căldăraru, A.: Derived categories of twisted sheaves on Calabi-Yau manifolds. Thesis, Cornell Univ., May 2000.
Lieblich, M.: Moduli of twisted sheaves, Duke Math. J. 138 (2007), no. 1, 23–118.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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