Lemma 69.6.5. Let S be a scheme. Let W be an algebraic space over S. Let G be a finite group acting freely on W. Let U = W/G, see Properties of Spaces, Lemma 66.34.1. Let \chi : G \to \{ +1, -1\} be a character. Then there exists a rank 1 locally free sheaf of \mathbf{Z}-modules \underline{\mathbf{Z}}(\chi ) on U_{\acute{e}tale} such that for every abelian sheaf \mathcal{F} on U_{\acute{e}tale} we have
H^0(W, \mathcal{F}|_ W)^\chi = H^0(U, \mathcal{F} \otimes _{\mathbf{Z}} \underline{\mathbf{Z}}(\chi ))
Proof.
The quotient morphism q : W \to U is a G-torsor, i.e., there exists a surjective étale morphism U' \to U such that W \times _ U U' = \coprod _{g \in G} U' as spaces with G-action over U'. (Namely, U' = W works.) Hence q_*\underline{\mathbf{Z}} is a finite locally free \mathbf{Z}-module with an action of G. For any geometric point \overline{u} of U, then we get G-equivariant isomorphisms
(q_*\underline{\mathbf{Z}})_{\overline{u}} = \bigoplus \nolimits _{\overline{w} \mapsto \overline{u}} \mathbf{Z} = \bigoplus \nolimits _{g \in G} \mathbf{Z} = \mathbf{Z}[G]
where the second = uses a geometric point \overline{w}_0 lying over \overline{u} and maps the summand corresponding to g \in G to the summand corresponding to g(\overline{w}_0). We have
H^0(W, \mathcal{F}|_ W) = H^0(U, \mathcal{F} \otimes _\mathbf {Z} q_*\underline{\mathbf{Z}})
because q_*\mathcal{F}|_ W = \mathcal{F} \otimes _\mathbf {Z} q_*\underline{\mathbf{Z}} as one can check by restricting to U'. Let
\underline{\mathbf{Z}}(\chi ) = (q_*\underline{\mathbf{Z}})^\chi \subset q_*\underline{\mathbf{Z}}
be the subsheaf of sections that transform according to \chi . For any geometric point \overline{u} of U we have
\underline{\mathbf{Z}}(\chi )_{\overline{u}} = \mathbf{Z} \cdot \sum \nolimits _ g \chi (g) g \subset \mathbf{Z}[G] = (q_*\underline{\mathbf{Z}})_{\overline{u}}
It follows that \underline{\mathbf{Z}}(\chi ) is locally free of rank 1 (more precisely, this should be checked after restricting to U'). Note that for any \mathbf{Z}-module M the \chi -semi-invariants of M[G] are the elements of the form m \cdot \sum \nolimits _ g \chi (g) g. Thus we see that for any abelian sheaf \mathcal{F} on U we have
\left(\mathcal{F} \otimes _\mathbf {Z} q_*\underline{\mathbf{Z}}\right)^\chi = \mathcal{F} \otimes _\mathbf {Z} \underline{\mathbf{Z}}(\chi )
because we have equality at all stalks. The result of the lemma follows by taking global sections.
\square
Comments (0)