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

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.