In their famous paper "On the differentiation of De Rham cohomology classes..." Katz and Oda construct the spectral sequence for de Rham cohomology for the situation of a smooth morphism


$\pi: X \rightarrow S$


of smooth $k$-schemes ($k$ a field), where $S$ is assumed affine.


There is a step which is not clear to me: in Lemma 8 of this paper they say that the equality


$(\*) \qquad \mathbb R^0\Gamma\_X =\Gamma\_S\circ \mathbb R^0\pi\_\*$


yields a spectral sequence of composition


$(\*\*) \qquad E\_2^{a,b}=R^a \Gamma\_S\circ \mathbb R^b\pi\_\* \Rightarrow \mathbb R^{a+b}\Gamma\_X$.


They seem to consider $(\*)$ as an equality of functors on the category of complexes of abelian sheaves on $X$ - Arguments involving quasi-coherence and affineness of $S$ only appear in the next step.


My question is a very simple one which I nevertheless can't figure out:


Why do they have $(\*\*)$?


It is a spectral sequence of composition, hence only exists when one knows that $\mathbb R^0 \pi\_\* $ sends injective objects in the category of complexes of sheaves on $X$ to $\Gamma\_S$-acyclic ones. I don't see why this is true: imagine a two-term complex $I^0\rightarrow I^1$ of injective abelian sheaves on $X$, then $\mathbb R^0 \pi\_\*$ of it is $ker(\pi\_\*I^0 \rightarrow \pi\_\*I^1)$. This has no reason to be acyclic for $\Gamma\_S$, hasn't it?


A similar problem arises a bit later in Lemma 9 when they consider the equality


$(+) \qquad \mathbb R^0\Gamma\_S = H^0\circ \Gamma\_S$


which I see as equality of functors on the category of complexes of abelian sheaves on $S$.


Can anybody give a hint how to cleanly resolve these two problems?



Construction of the spectral sequence of Katz/Oda
Let me try to say the same thing as Will but in a different way. Let $f \colon A \to B$ be left exact between abelian categories and assume there are enough injectives. Then one can distinguish between *derived functors* 
$$\newcommand{\R}{\mathrm R} \R^i f \colon A \to B,$$
as well as *hyper-derived functors* 
$$\newcommand{\RR}{\mathbb R} \RR^i f \colon \mathrm{Kom}^+(A) \to B,$$
and the *total derived functor*
$$ \newcommand{\RRR}{\mathbf R} \RRR f \colon D^+(A) \to D^+(B).$$
Now suppose we also have $g \colon B \to C$ with the same hypotheses and that $f$ maps injective objects of $A$ to $g$-acyclic objects of $B$. Then there is a natural isomorphism 
$$ \RRR(g \circ f) \cong \RRR g \circ \RRR f.$$
This isomorphism specializes not only to the usual spectral sequence 
$$ E\_2^{p,q} = \R^p g( \R^q f(X)) \implies \R^{p+q}(g \circ f)(X),$$
for $X$ any object of $A$, but also to the spectral sequence
$$ E\_2^{p,q} = \R^p g( \RR^q f(X^\bullet)) \implies \RR^{p+q}(g \circ f)(X^\bullet),$$
for $X^\bullet$ any object of $\mathrm{Kom}^+(A)$, in a similar way. In particular one needs no extra hypotheses to get the latter spectral sequence; even though the domain of $\RR^i f$ is $\mathrm{Kom}^+(A)$, one should not assume that complexes of injectives are sent to acyclics.


