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A note on finite abelian gerbes over toric Deligne-Mumford stacks. (English) Zbl 1165.14007
Let $${\pmb \Sigma}$$ be a stacky fan as introduced by L. A. Borisov, L. Chen and G. G. Smith [J. Am. Math. Soc. 18, No 1, 193–215 (2005; Zbl 1178.14057)]. The stacky fan $${\pmb \Sigma}$$ encodes a group $$G$$ action on a quasi-affine variety $$Z$$, and the quotient stack $$[Z/G]$$ is the toric Deligne-Mumford stack associated to the stacky fan $${\pmb \Sigma}$$.
The observations of the paper under review concerns finite abelian gerbes over $$[Z/G]$$. These gerbes are classified by the second cohomology group with coefficients in the finite abelian group $$\nu$$. A neat calculation shows that $$H^2([Z/G],{\nu})=H^2({\mathcal B}G,{\nu})$$, and consequently one obtains the main result saying that $$\nu$$-gerbes over the toric Deligne-Mumford stack $$[Z/G]$$ are given by central extensions of $$G$$ and $$\nu$$. Such a gerbe is not always a toric Deligne-Mumford stack. However, the author shows that if the extension is abelian then the corresponding gerbe is the toric Deligne-Mumford stack associated to a stacky fan obtained from the extension.

##### MSC:
 14A20 Generalizations (algebraic spaces, stacks)
##### Keywords:
gerbes; toric Deligne-Mumford stacks
Full Text:
##### References:
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