Not supported by the ICS, CIA, NRO, DSRP, NSA, DIA, USAF, Army, ONI, TIARA, INR, DOE, FBI, or any other intelligence agency of the United States government.Also numerous changes were made to the article by a copy editor who neither understood the text nor recognized that even a minor change can affect the meaning.
Finally, the author's standard (Chicago Manual of Style) method of citation was changed by the AMS to a barbaric numbering system, and (of course) they got it wrong. 1
I recommend reading the original manuscripts, available on my web site, rather than the published version.
p6$^{4}$. $C^{d}(V\times V)$ should be $A^{d}(V\times V)$.
p6. The formula should read ("rank" has been omitted) \[ \text{rank }h(V)=\sum\text{rank }h^{i}(V)\quad\quad(\text{rather than }% \sum(-1)^{i}\text{rank }h^{i}(V)). \]
p11 In the proof of Proposition 1.15, "Milne 1986" should refer to my article in the Amer. J. Math. 1986, not to ADT (in the published version, this is p411, and the reference is [22, 8.6]).
p38. The proof of Proposition 3.11 is garbled.
Let $L$ be a CM-subfield of $\mathbb{Q}{}^{\text{al}}$, finite and Galois over
$\mathbb{Q}{}$. Then
\[
H^{1}(\mathbb{Q}{},P_{0}^{L})=\mathrm{Br}(E/F)
\]
where $E$ is the fixed field of the decomposition group $D(w_{0})$
and $F$ is the largest real subfield of $E$. This implies (a).
For a torus $T$, let $\tilde{T}$ be the universal covering of $T$, i.e.,
$\tilde{T}$ is the projective system $(T_{n},T_{mn}\overset{m}{\rightarrow
}T_{n})$ with $T_{n}=T$ for all integers $n\geq1$. For any covariant functor
$H$ to abelian groups, $H(\tilde{T})=\mathrm{Hom}(\mathbb{Q}{},H(T))$; in particular,
$H(\tilde{T})=0$ if $H(T)$ is torsion. Thus,
\[
H^{1}(\mathbb{Q}{},\widetilde{(P^{L})^{0}})=\mathrm{Hom}(\mathbb{Q},\mathrm{Br}(E/F))=0.
\]
The map $P^{0}\rightarrow(P^{L})^{0}$ factors through the map
$\widetilde{(P^{L})^{0}}$, and so the map $H^{1}(\mathbb{Q}{},P^{0}%
)\rightarrow H^{1}(\mathbb{Q}{},(P^{L})^{0})$ is zero. A similar argument
applies to show the other groups are zero.
p38 In the diagram, replace $\oplus_{v}H^{2}(\mathbb{Q}_{p},\ldots$ with $\oplus_{\ell}H^{2}(\mathbb{Q}_{\ell},\ldots$.
p47. Replace the last sentence of the first proof with: According to (3.11b), $H^{1}(\mathbb{Q}_{\ell},P(p^{\infty}))=0$ torsor is trivial: $\alpha\circ\zeta_{\ell}\approx\zeta_{\ell}^{\prime}$.