A mistake in Grothendieck's Tôhoku paper? Theorem 5.2.1 I was reading the ``Sur quelques points d'alegbre homologique'' English translation when I came across the spectral sequences for equivariant cohomology shown in Theorem 5.2.1 (in the original French document). I checked the original version to see if it had been a translator's error, but no, it is not, so Theorem 5.2.1 says the following:


There exists on the category of Abelian $G$-sheaves on $X$ two co-homological spectral functors abutting on the graded functor $H^n(X,G;\mathscr F)$, whose initial
terms are, respectively:
$$
I_2^{p,q} = H^p(Y;\mathscr{H}^q(X;\mathscr F))
$$
and
$$
II_2^{p,q} = \mathscr{H}^p(G,H^q(X;\mathscr F)).
$$
(Here $Y$ denotes the quotient $X/G$).


I think there are two mistakes:

In $I_2$, I think $\mathscr H^q(X;\mathscr F) $ actually means
$$
\mathscr H^q(X;\mathscr F)
$$
which is the sheaf on $Y$ associated to the presheaf $V\mapsto H^q(f^{-1}(V),G;\mathscr F)$ (here $f:X\rightarrow Y$ is the quotient map).


In $II_2$, $\mathscr H^p(G,H^q(X;\mathscr F))$ is meaningless for me, and I think it should be
$$
H^p(G, H^q(X;\mathscr F)).
$$

How could a sheaf converge to an Abelian group?
Finally, notice that, with my corrections, Theorem 5.2.1 looks pretty much like Theorem 4.4.1 (always referring to the French edition).
Am I right then?
 A: As Grothendieck writes, the $G$-invariant global sections functor $\Gamma^G_X$ can be factored as the global sections functor $\Gamma_X$ followed by the ordinary $G$-invariants functor $\Gamma^G$, or as the the relative $G$-invariants functor $f_*^G$ followed by the global sections functor $\Gamma_Y$.
(Also recall that the relative $G$-invariants functor $f_*^G$ can be factored as the direct image functor $f_*$ followed by the $G$-invariants functor $\Gamma^G$.)
The right derived functors of $\Gamma_X$ are the usual sheaf cohomology functors $H^* (X, -)$.
You can check that for any $G$-sheaf $A$, $H^* (X, A)$ has an induced $G$-action, so it makes sense to take its $G$-invariants.
The right derived functors of $\Gamma^G$ are the usual group cohomology functors $H^* (G, -)$.
Thus, the Grothendieck spectral sequence applied to the factorisation $\Gamma^G_X \cong \Gamma^G \Gamma_X$ gives us a spectral sequence relating $H^* (G, H^* (X, A))$ to $H^* (X; G, A)$.
On the other hand, we also have the factorisation $\Gamma^G_X \cong \Gamma_Y f^G_*$.
The right derived functors of $f^G_*$ are what you might call $\mathscr{H}^* (G, -)$.
The right derived functors of $\Gamma_Y$ are $H^* (Y, -)$, of course.
Thus, the Grothendieck spectral sequence relates $H^* (Y, \mathscr{H}^* (G, A))$ to $H^* (X; G, A)$.
So I think there are two typographical errors: $\mathscr{H}^q (X, A)$ in the expression for $\mathrm{I}^{p, q}_2 (A)$ should instead be $\mathscr{H}^q (G, A)$, and $\mathscr{H}^p$ in the expression for $\mathrm{II}^{p, q}_2 (A)$ should instead be $H^p$.
This is consistent with the later remarks about edge homomorphisms and 5-term exact sequences.
