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 [math]\displaystyle{ \mathbb{G}_m }[/math] on the covering U_i as well as the isomorphisms

[math]\displaystyle{ g_{ij}: F_j|_{U_{ij}} \overset{\sim}\to F_i|_{U_{ij}} }[/math]

satisfying

• [math]\displaystyle{ g_{ii} = \operatorname{id}_{F_i} }[/math],
• [math]\displaystyle{ g_{ij} = g_{ji}^{-1}, }[/math]
• [math]\displaystyle{ g_{ij} \circ g_{jk} \circ g_{ki} = \theta_{ijk} \operatorname{id}_{F_i}. }[/math]

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. doi:10.1515/CRLL.2002.022. 
• Lieblich, Max (2007). "Moduli of twisted sheaves". Duke Mathematical Journal 138. doi:10.1215/S0012-7094-07-13812-2.