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Étale cohomology of a DM curve-stack with coefficients in \(\mathbb G _m\)

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Abstract

We compute étale cohomology groups \(H_{\acute{\mathrm{e}}\mathrm{t}}^r(X, \mathbb G _m)\) in several cases, where \(X\) is a connected smooth tame Deligne–Mumford stack of dimension \(1\) over an algebraically closed field. We have complete results for orbicurves (and, more generally, for twisted nodal curves) and in the case all stabilizers are cyclic; we give partial results and examples in the general case. In particular, we show that if the stabilizers are abelian then \(H_{\acute{\mathrm{e}}\mathrm{t}}^2(X, \mathbb{G }_m)\) does not depend on \(X\) but only on the underlying orbicurve \(Y\) and on the generic stabilizer. We show with two examples that, in general, the higher cohomology groups \(H_{\acute{\mathrm{e}}\mathrm{t}}^r(X, \mathbb{G }_m)\) cannot be computed knowing only the base of the gerbe \(X \rightarrow Y\) and the banding group.

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Acknowledgments

I would like to thank my Ph.D. advisor, Barbara Fantechi, for suggesting me this problem and for helpful discussions and suggestions. I’m grateful to Angelo Vistoli for teaching me the key techniques used here. I would also like to thank Andrew Kresch and Matthieu Romagny for corrections and suggestions. I am indebted to the referee for doing a very careful job. I want to acknowledge my host institution SISSA for support; I was partly supported by prin “Geometria delle varietà algebriche e dei loro spazi di moduli”, by Istituto Nazionale di Alta Matematica.

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Correspondence to Flavia Poma.

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Communicated by A. Constantin.

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Poma, F. Étale cohomology of a DM curve-stack with coefficients in \(\mathbb G _m\) . Monatsh Math 169, 33–50 (2013). https://doi.org/10.1007/s00605-012-0450-5

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