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?
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2$\begingroup$ For a field $k$ and an element $a$ in $k^\times/(k^\times)^p$, have you considered the cyclic algebra over $\mathbb{A}^1_k$ coming from $a$ and the Artin-Schreier cover of $\mathbb{A}^1_k$ with Galois group $\mathbb{Z}/p\mathbb{Z}$? More generally, have you looked at the $p$-torsion in the Brauer group of $\mathbb{A}^1_k$ realized via Kato's isomorphism (using differential forms and inverse Cartier)? $\endgroup$– Jason StarrCommented Oct 17, 2021 at 23:38
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(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.
answered Oct 18, 2021 at 20:31
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3$\begingroup$ The condition that $A$ is perfect precisely eliminates the elements $a$ in my comment. For a field $k$ of characteristic $p$ such that $k^\times/(k^\times)^p$ is nontrivial, the $p$-torsion in the Brauer group of $\mathbb{A}^1_k$ does not equal the $p$-torsion in the Brauer group of $k$. Kato's isomorphism should also give information about the $p$-torsion in higher degree cohomology of $\mathbb{G}_m$. $\endgroup$ Commented Oct 19, 2021 at 12:48