Twisted sheaf
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 .