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 for G m {\displaystyle \mathbb {G} _{m}} on the covering Ui as well as the isomorphisms
g i j : F j | U i j → ∼ F i | U i j {\displaystyle g_{ij}:F_{j}|_{U_{ij}}{\overset {\sim }{\to }}F_{i}|_{U_{ij}}}
satisfying
g i i = id F i {\displaystyle g_{ii}=\operatorname {id} _{F_{i}}} ,
g i j = g j i − 1 , {\displaystyle g_{ij}=g_{ji}^{-1},}
g i j ∘ g j k ∘ g k i = θ i j k id F i . {\displaystyle g_{ij}\circ g_{jk}\circ g_{ki}=\theta _{ijk}\operatorname {id} _{F_{i}}.}
The notion of twisted sheaves was introduced by Jean 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 Torsion sheaf
References Căldăraru, Andrei (2002). "Derived categories of twisted sheaves on elliptic threefolds". Journal für die reine und angewandte Mathematik (Crelle's Journal). 2002 (544): 161–179. arXiv:math/0012083. doi:10.1515/CRLL.2002.022. S2CID 119117575. Lieblich, Max (2007). "Moduli of twisted sheaves". Duke Mathematical Journal. 138. doi:10.1215/S0012-7094-07-13812-2. S2CID 14067307.
