Small fix

etale-cohomology.tex

@@ -9311,7 +9311,7 @@ \section{Traces}
 $$
 (\mathbf{Z}/\ell^n\mathbf{Z})[G]^\natural
 =
-\bigoplus_{\frac{conjugacy }{classes of G}}
+\bigoplus\nolimits_{\text{conjugacy classes of }G}
 \mathbf{Z}/\ell^n\mathbf{Z}.
 $$
 For a free $\Lambda$-module, we have $\text{End}_\Lambda(\Lambda^{\oplus m}) =
@@ -9728,8 +9728,10 @@ \section{Perfectness}
 We denote by $K_{perf}(\Lambda)$ the category whose objects are bounded
 complexes of finite projective $\Lambda$-modules, and whose morphisms are
 morphisms of complexes up to homotopy. The functor $K_{perf}(\Lambda)\to
-D(\Lambda)$ is fully faithful, and we denote $D_{perf}(\Lambda)$ its essential
-image. An object of $D(\Lambda)$ is called {\it perfect} if it is in
+D(\Lambda)$ is fully faithful (Derived Categories, Lemma
+\ref{derived-lemma-morphisms-from-projective-complex}).
+Denote $D_{perf}(\Lambda)$ its essential image.
+An object of $D(\Lambda)$ is called {\it perfect} if it is in
 $D_{perf}(\Lambda)$.
 \end{definition}
 
@@ -9740,17 +9742,21 @@ \section{Perfectness}
 \end{proposition}
 
 \begin{proof}
-Let $P^\bullet$ be a bounded complex of finite projective $\Lambda$-modules and
-$\alpha: P^\bullet \cong K$ be an isomorphism in $D(\Lambda)$. Then
-$\alpha^{-1}\circ f\circ \alpha$ is the class of some morphism of complexes
-$f^\bullet: P^\bullet \to P^\bullet$ by (insert reference here).
+We will use Derived Categories, Lemma
+\ref{derived-lemma-morphisms-from-projective-complex}
+without further mention in this proof.
+Let $P^\bullet$ be a bounded complex of finite projective $\Lambda$-modules
+and let $\alpha: P^\bullet \to K$ be an isomorphism in $D(\Lambda)$. Then
+$\alpha^{-1}\circ f\circ \alpha$ corresponds to a morphism of complexes
+$f^\bullet: P^\bullet \to P^\bullet$ well defined up to homotopy.
 Set
 $$
 \text{Tr}(f) = \sum_i (-1)^i \text{Tr}(f^i: P^i \to P^i) \in \Lambda^\natural.
 $$
 Given $P^\bullet$ and $\alpha$, this is independent of the choice of
-$f^\bullet$: any other choice is of the form $\tilde{f}^\bullet = f^\bullet +
-dh +hd$ for some $h^i: P^i \to P^{i-1}(i\in \mathbf{Z})$. But
+$f^\bullet$. Namely, any other choice is of the form
+$\tilde{f}^\bullet = f^\bullet + dh +hd$ for some
+$h^i: P^i \to P^{i-1}(i\in \mathbf{Z})$. But
 \begin{eqnarray*}
 \text{Tr}(dh) & = & \sum_i (-1)^i \text{Tr}(P^i\xrightarrow{dh} P^i) \\
 & = & \sum_i (-1)^i \text{Tr}(P^{i-1}\xrightarrow{hd} P^{i-1}) \\
@@ -9759,7 +9765,7 @@ \section{Perfectness}
 \end{eqnarray*}
 and so $\sum_i (-1)^i \text{Tr} ((dh+hd)|_{P^i})=0$.
 Furthermore, this is independent of the choice of $(P^\bullet , \alpha)$:
-suppose $(Q^\bullet, \beta)$ is another choice. Then by ???, the compositions
+suppose $(Q^\bullet, \beta)$ is another choice. The compositions
 $$
 Q^\bullet \xrightarrow{\beta} K \xrightarrow{\alpha^{-1}} P^\bullet
 \quad\text{and}\quad
@@ -9777,7 +9783,7 @@ \section{Perfectness}
 & = & \text{Tr}(f^\bullet|_{P^\bullet})
 \end{eqnarray*}
 by the fact that $\gamma_1^\bullet \circ \gamma_2^\bullet$ is homotopic to the
-identity and the independence from $(P^\bullet, \alpha)$ already proved.
+identity and the independence of the choice of $f^\bullet$ we saw above.
 \end{proof}