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 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 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).

References

Căldăraru, A.: Derived categories of twisted sheaves on Calabi-Yau manifolds. Thesis, Cornell Univ., May 2000.
Lieblich, M.: Moduli of twisted sheaves, Duke Math. J. 138 (2007), no. 1, 23–118.

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Twisted sheaf

See also

References