$\mathbf{A}^1$-invariance of Brauer groups and $H^2_{\mathrm{et}}(-;\mathbb{G}_m)$ The $\mathbf{A}^1$-invariance of vector bundles have been discussed in, for example, this paper by Asok, Hoyois and Wendt. This of course implies storng $\mathbf{A}^1$-invariance results for the first etale cohomology group $H^1_{\mathrm{et}}(-,\mathbb{G}_m)$. Are there any similar results for $H_{\mathrm{et}}^2(-,\mathbb{G}_m)$ or the Brauer groups?
 A: (For $i=0$, the map $H_{\mathrm{et}}^{0}(\operatorname{Spec} A,\mathbb{G}_{m}) \to H_{\mathrm{et}}^{0}(\operatorname{Spec} A[t],\mathbb{G}_{m})$ is an isomorphism if and only if $A$ is reduced.)
For $i=1$, it is a theorem of Traverso that $H_{\mathrm{et}}^{1}(\operatorname{Spec} A,\mathbb{G}_{m}) \to H_{\mathrm{et}}^{1}(\operatorname{Spec} A[t],\mathbb{G}_{m})$ is an isomorphism if and only if $A$ is seminormal.
For $i=2$, at least when $A$ is a field, the map $H_{\mathrm{et}}^{2}(\operatorname{Spec} A,\mathbb{G}_{m}) \to H_{\mathrm{et}}^{2}(\operatorname{Spec} A[t],\mathbb{G}_{m})$ is an isomorphism if and only if $A$ is perfect; thus if $A$ is regular with perfect fraction field we have a similar positive result (see Auslander, Goldman, The Brauer group of a commutative ring, (link) 7.5, 7.7 respectively). In general (for the torsion at least) we only have to worry about the $p$-torsion for primes $p$ that are not invertible in $A$. There are some additional positive results in Knus, Ojanguren, A Mayer-Vietoris sequence for the Brauer group (link) Theorem 3.6 and I'd be interested in a more complete characterization.
