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 $\mathbb {G} _{m}$ on the covering U_i as well as the isomorphisms

$g_{ij}:F_{j}|_{U_{ij}}{\overset {\sim }{\to }}F_{i}|_{U_{ij}}$

satisfying

• $g_{ii}=\operatorname {id} _{F_{i}}$,
• $g_{ij}=g_{ji}^{-1},$
• $g_{ij}\circ g_{jk}\circ g_{ki}=\theta _{ijk}\operatorname {id} _{F_{i}}.$

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.