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

$g_{ij}:F_j{|}_{U_{ij}}\stackrel{\sim }{\to }{F}_i{|}_{U_{ij}}$

satisfying

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

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. 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.