5
$\begingroup$

Let $k$ be a field, let $X$ be a smooth quasi-projective $k$-variety, let $Z\subset X$ be a closed subscheme of codimension at least $2$, it is shown that the restriction map $\mathrm{H}^2(X,\mathbb{G}_m)\to\mathrm{H}^2(X-Z,\mathbb{G}_m)$ is an isomorphism.

Let $\mathcal{X}$ be a smooth Deligne-mumford stack over $k$, let $\mathcal{Z}$ be a closed substack of codimension at least $2$, does it still hold that $\mathrm{H}^2(\mathcal{X},\mathbb{G}_m)\to\mathrm{H}^2(\mathcal{X}-\mathcal{Z},\mathbb{G}_m)$?

$\endgroup$
6
$\begingroup$

The answer seems to be positive and actually at least in the context of regular (locally) noetherian Deligne--Mumford stacks. (Actually, Artin stack should also be enough as we can compute the Brauer group also as the fppf-cohomology of $\mathbb{G}_m$.)

Let $p\colon X \to \mathcal{X}$ be an \'etale cover. We obtain a descent spectral sequence

$$H^i(X^{\times_{\mathcal{X}}j}; \mathbb{G}_m) \Rightarrow H^{i+j}(\mathcal{X}; \mathbb{G}_m).$$

Likewise we obtain a spectral sequence

$$H^i((X-p^{-1}(\mathcal{Z}))^{\times_{\mathcal{X}}j}; \mathbb{G}_m) \Rightarrow H^{i+j}(\mathcal{X}-\mathcal{Z}; \mathbb{G}_m).$$

As \'etale maps are smooth of relative dimension zero, $X - p^{-1}(\mathcal{Z})$ has still codimension at least $2$ in $X$ and likewise for all higher fiber products. Moreover, under your regularity hypotheses purity also holds for the group of units and for the Picard group. Thus, the map $\mathcal{X} - \mathcal{Z} \hookrightarrow \mathcal{X}$ induces an isomorphism of $E_2$-terms of spectral sequence for $i\leq 2$ and all $j$. Thus, it also induces an isomorphism on convergenda for $i+j\leq 2$, which is exactly the statement we want.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.