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).

• Reflexive sheaf
• Torsion sheaf

• 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.