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
Gm
gij:Fj|
Uij |
\overset{\sim}\toFi|
Uij |
gii=
\operatorname{id} | |
Fi |
gij=
-1 | |
g | |
ji |
,
gij\circgjk\circgki=\thetaijk
\operatorname{id} | |
Fi |
.
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 .