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 ${𝔾}_m$ 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 Template:Harv.

See also

• Reflexive sheaf
• Torsion sheaf

References

• Template:Cite journal
• Template:Cite journal

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