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 for $\backslash mathbb\{G\}\_m$ on the covering U_i as well as the isomorphisms

:$g\_\{ij\}:\; F\_j|\_\{U\_\{ij\}\}\; \backslash overset\{\backslash sim\}\backslash to\; F\_i|\_\{U\_\{ij\}\}$

satisfying

• $g\_\{ii\}\; =\; \backslash operatorname\{id\}\_\{F\_i\}$,
• $g\_\{ij\}\; =\; g\_\{ji\}^\{-1\},$
• $g\_\{ij\}\; \backslash circ\; g\_\{jk\}\; \backslash circ\; g\_\{ki\}\; =\; \backslash theta\_\{ijk\}\; \backslash operatorname\{id\}\_\{F\_i\}.$

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 {{harv|Lieblich|2007}}.