Lemma 46.5.12. Let $f : T \to S$ be a quasi-compact and quasi-separated morphism of schemes. For any adequate $\mathcal{O}_ T$-module on $(\mathit{Sch}/T)_\tau $ the pushforward $f_*\mathcal{F}$ and the higher direct images $R^ if_*\mathcal{F}$ are adequate $\mathcal{O}_ S$-modules on $(\mathit{Sch}/S)_\tau $.
Proof. First we explain how to compute the higher direct images. Choose an injective resolution $\mathcal{F} \to \mathcal{I}^\bullet $. Then $R^ if_*\mathcal{F}$ is the $i$th cohomology sheaf of the complex $f_*\mathcal{I}^\bullet $. Hence $R^ if_*\mathcal{F}$ is the sheaf associated to the presheaf which associates to an object $U/S$ of $(\mathit{Sch}/S)_\tau $ the module
\begin{align*} \frac{\mathop{\mathrm{Ker}}(f_*\mathcal{I}^ i(U) \to f_*\mathcal{I}^{i + 1}(U))}{\mathop{\mathrm{Im}}(f_*\mathcal{I}^{i - 1}(U) \to f_*\mathcal{I}^ i(U))} & = \frac{\mathop{\mathrm{Ker}}(\mathcal{I}^ i(U \times _ S T) \to \mathcal{I}^{i + 1}(U \times _ S T))}{\mathop{\mathrm{Im}}(\mathcal{I}^{i - 1}(U \times _ S T) \to \mathcal{I}^ i(U \times _ S T))} \\ & = H^ i(U \times _ S T, \mathcal{F}) \\ & = H^ i((\mathit{Sch}/U \times _ S T)_\tau , \mathcal{F}|_{(\mathit{Sch}/U \times _ S T)_\tau }) \\ & = H^ i(U \times _ S T, \mathcal{F}|_{(U \times _ S T)_{Zar}}) \end{align*}
The first equality by Topologies, Lemma 34.7.12 (and its analogues for other topologies), the second equality by definition of cohomology of $\mathcal{F}$ over an object of $(\mathit{Sch}/T)_\tau $, the third equality by Cohomology on Sites, Lemma 21.7.1, and the last equality by Lemma 46.5.8. Thus by Lemma 46.5.10 it suffices to prove the claim stated in the following paragraph.
Let $A$ be a ring. Let $T$ be a scheme quasi-compact and quasi-separated over $A$. Let $\mathcal{F}$ be an adequate $\mathcal{O}_ T$-module on $(\mathit{Sch}/T)_\tau $. For an $A$-algebra $B$ set $T_ B = T \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(B)$ and denote $\mathcal{F}_ B = \mathcal{F}|_{(T_ B)_{Zar}}$ the restriction of $\mathcal{F}$ to the small Zariski site of $T_ B$. (Recall that this is a “usual” quasi-coherent sheaf on the scheme $T_ B$, see Lemma 46.5.8.) Claim: The functor
\[ B \longmapsto H^ q(T_ B, \mathcal{F}_ B) \]
is adequate. We will prove the lemma by the usual procedure of cutting $T$ into pieces.
Case I: $T$ is affine. In this case the schemes $T_ B$ are all affine and $H^ q(T_ B, \mathcal{F}_ B) = 0$ for all $q \geq 1$. The functor $B \mapsto H^0(T_ B, \mathcal{F}_ B)$ is adequate by Lemma 46.3.18.
Case II: $T$ is separated. Let $n$ be the minimal number of affines needed to cover $T$. We argue by induction on $n$. The base case is Case I. Choose an affine open covering $T = V_1 \cup \ldots \cup V_ n$. Set $V = V_1 \cup \ldots \cup V_{n - 1}$ and $U = V_ n$. Observe that
\[ U \cap V = (V_1 \cap V_ n) \cup \ldots \cup (V_{n - 1} \cap V_ n) \]
is also a union of $n - 1$ affine opens as $T$ is separated, see Schemes, Lemma 26.21.7. Note that for each $B$ the base changes $U_ B$, $V_ B$ and $(U \cap V)_ B = U_ B \cap V_ B$ behave in the same way. Hence we see that for each $B$ we have a long exact sequence
\[ 0 \to H^0(T_ B, \mathcal{F}_ B) \to H^0(U_ B, \mathcal{F}_ B) \oplus H^0(V_ B, \mathcal{F}_ B) \to H^0((U \cap V)_ B, \mathcal{F}_ B) \to H^1(T_ B, \mathcal{F}_ B) \to \ldots \]
functorial in $B$, see Cohomology, Lemma 20.8.2. By induction hypothesis the functors $B \mapsto H^ q(U_ B, \mathcal{F}_ B)$, $B \mapsto H^ q(V_ B, \mathcal{F}_ B)$, and $B \mapsto H^ q((U \cap V)_ B, \mathcal{F}_ B)$ are adequate. Using Lemmas 46.3.11 and 46.3.10 we see that our functor $B \mapsto H^ q(T_ B, \mathcal{F}_ B)$ sits in the middle of a short exact sequence whose outer terms are adequate. Thus the claim follows from Lemma 46.3.16.
Case III: General quasi-compact and quasi-separated case. The proof is again by induction on the number $n$ of affines needed to cover $T$. The base case $n = 1$ is Case I. Choose an affine open covering $T = V_1 \cup \ldots \cup V_ n$. Set $V = V_1 \cup \ldots \cup V_{n - 1}$ and $U = V_ n$. Note that since $T$ is quasi-separated $U \cap V$ is a quasi-compact open of an affine scheme, hence Case II applies to it. The rest of the argument proceeds in exactly the same manner as in the paragraph above and is omitted. $\square$
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