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 Ui, coherent sheaves Fi over Ui, a Čech 2-cocycle θ for $$\mathbb{G}_m$$ on the covering Ui 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.