Add lemma in perfect.tex

perfect.tex

@@ -3322,28 +3322,19 @@ \section{Cohomology and base change, IV}
 \label{lemma-cohomology-base-change}
 Let $f : X \to Y$ be a quasi-compact and quasi-separated morphism
 of schemes. For $E$ in $D_\QCoh(\mathcal{O}_X)$ and
-$K$ in $D_\QCoh(\mathcal{O}_Y)$ we have
+$K$ in $D_\QCoh(\mathcal{O}_Y)$ the map
 $$
-Rf_*(E) \otimes_{\mathcal{O}_Y}^\mathbf{L} K =
+Rf_*(E) \otimes_{\mathcal{O}_Y}^\mathbf{L} K
+\longrightarrow
 Rf_*(E \otimes_{\mathcal{O}_X}^\mathbf{L} Lf^*K)
 $$
+defined in
+Cohomology, Equation (\ref{cohomology-equation-projection-formula-map})
+is an isomorphism.
 \end{lemma}
 
 \begin{proof}
-Without any assumptions there is a map
-$Rf_*(E) \otimes_{\mathcal{O}_Y}^\mathbf{L} K \to
-Rf_*(E \otimes_{\mathcal{O}_X}^\mathbf{L} Lf^*K)$.
-Namely, it is the adjoint to the canonical map
-$$
-Lf^*(Rf_*(E) \otimes_{\mathcal{O}_Y}^\mathbf{L} K) =
-Lf^*(Rf_*(E)) \otimes_{\mathcal{O}_X}^\mathbf{L} Lf^*K
-\longrightarrow
-E \otimes_{\mathcal{O}_X}^\mathbf{L} Lf^*K
-$$
-coming from the map $Lf^*Rf_*E \to E$. See
-Cohomology, Lemmas \ref{cohomology-lemma-pullback-tensor-product} and
-\ref{cohomology-lemma-adjoint}.
-To check it is an isomorphism we may work locally on $Y$.
+To check the map is an isomorphism we may work locally on $Y$.
 Hence we reduce to the case that $Y$ is affine.
 
 \medskip\noindent
@@ -3717,6 +3708,27 @@ \section{Producing perfect complexes}
 has cohomology in the range $[a, \infty)$ and we win.
 \end{proof}
 
+\noindent
+We will generalize the following lemma to perfect proper morphisms in
+More on Morphisms, Lemma
+\ref{more-morphisms-lemma-perfect-proper-perfect-direct-image}.
+
+\begin{lemma}
+\label{lemma-flat-proper-perfect-direct-image}
+Let $S$ be a Noetherian scheme. Let $f : X \to S$ be a flat proper
+morphism of schemes. Let $E \in D(\mathcal{O}_X)$ be perfect. Then
+$Rf_*E$ is a perfect object of $D(\mathcal{O}_S)$.
+\end{lemma}
+
+\begin{proof}
+We claim that Lemma \ref{lemma-perfect-direct-image} applies.
+Conditions (1) and (2) are immediate. Condition (3) is local
+on $X$. Thus we may assume $X$ and $S$ affine and $E$
+represented by a strictly perfect complex of $\mathcal{O}_X$-modules.
+Since $\mathcal{O}_X$ is flat as a sheaf of $f^{-1}\mathcal{O}_S$-modules
+we find that condition (3) is satisfied.
+\end{proof}
+