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 U_i, coherent sheaves F_i over U_i, a Čech 2-cocycle θ for ${\displaystyle \mathbb {G} _{m}}$ on the covering U_i 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 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).