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If $X$ is a smooth projective variety over a field, $Z\subset X$ a smooth closed subvariety of codimension $d$, $X'\to X$ the blowup of $X$ along $Z$, there's the blowup formula

$$H^j(X'_{et},\mathbf{Z}(n)) = H^j(X_{et},\mathbf{Z}(n))\oplus\bigoplus_{r=1}^{d-1}H^{j-2r}(Z_{et},\mathbf{Z}(j-r)).$$

I must be missing something. For $j = 2n+1$ and $n = 1$, we get

$$Br(X') = Br(X) \oplus(things)$$

and it isn't clear to me the "things" vanish. But we know the cohomological Brauer group is a birational invariant of $X$, so they must vanish.

What do I not know about $H^{j-2r}(Z_{et},\mathbf{Z}(j-r))$?

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There is simply a typo in this formula, the expression $H^{j - 2r}(Z_{et},j-r)$ should be replaced by $H^{j - 2r}(Z_{et},n-r)$. This is closesly related to the fact that as motives $M(\mathbb{P}(E)) = \oplus_{r = 0}^{rank(E)} M(X)(r)[2r]$ for a vector bundle $E$ over a smooth variety $E$. In this case the sum in the RHS correspond to the exceptional divisor, which is the projectivisation of the normal bundle for a smooth blow up, the the shift in the cohomology and the weight correspond to the $"(r)"$ and the $"[2r]"$ in the formula for the motive of this vector bundle.

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