The $\mathbf{A}^1$-invariance of vector bundles have been discussed in, for example, [this paper](https://projecteuclid.org/journals/duke-mathematical-journal/volume-166/issue-10/Affine-representability-results-in-A1-homotopy-theory-I--Vector/10.1215/00127094-0000014X.short) 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?



$\mathbf{A}^1$-invariance of Brauer groups and $H^2_{\mathrm{et}}(-;\mathbb{G}_m)$
(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](https://www.ams.org/journals/tran/1960-097-03/S0002-9947-1960-0121392-6/S0002-9947-1960-0121392-6.pdf)) 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](https://core.ac.uk/download/pdf/82736669.pdf)) Theorem 3.6 and I'd be interested in a more complete characterization.


