# Purity of Brauer group for stacks

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 $$\mathrm{H}^2(\mathcal{X},\mathbb{G}_m)\to\mathrm{H}^2(\mathcal{X}-\mathcal{Z},\mathbb{G}_m)$$?

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