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 G m {\displaystyle \mathbb {G} _{m}} on the covering Ui as well as the isomorphisms

g i j : F j | U i j → ∼ F i | U i j {\displaystyle g_{ij}:F_{j}|_{U_{ij}}{\overset {\sim }{\to }}F_{i}|_{U_{ij}}}

satisfying

  • g i i = id F i {\displaystyle g_{ii}=\operatorname {id} _{F_{i}}} ,
  • g i j = g j i − 1 , {\displaystyle g_{ij}=g_{ji}^{-1},}
  • g i j ∘ g j k ∘ g k i = θ i j k id F i . {\displaystyle 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 (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. arXiv:math/0012083. doi:10.1515/CRLL.2002.022. S2CID 119117575.
  • Lieblich, Max (2007). "Moduli of twisted sheaves". Duke Mathematical Journal. 138. doi:10.1215/S0012-7094-07-13812-2. S2CID 14067307.


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