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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)
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