# 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

satisfying

- ,

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).

## See also

## 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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