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YCor
YCor
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$\mathbf{A}^1$-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_{et}(-,\mathbb{G}_m)$$H^1_{\mathrm{et}}(-,\mathbb{G}_m)$. Are there any similar results for $H_{et}^2(-,\mathbb{G}_m)$$H_{\mathrm{et}}^2(-,\mathbb{G}_m)$ or the Brauer groups?

$\mathbf{A}^1$-invariance of Brauer groups and $H^2_{et}(-;\mathbb{G}_m)$

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 $H^1_{et}(-,\mathbb{G}_m)$. Are there any similar results for $H_{et}^2(-,\mathbb{G}_m)$ or the Brauer groups?

$\mathbf{A}^1$-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?

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Xing Gu
Xing Gu
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$\mathbf{A}^1$-invariance of Brauer groups and $H^2_{et}(-;\mathbb{G}_m)$

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 $H^1_{et}(-,\mathbb{G}_m)$. Are there any similar results for $H_{et}^2(-,\mathbb{G}_m)$ or the Brauer groups?