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Small corrections to last two commites

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aisejohan committed Nov 6, 2019
1 parent 862896c commit 801072a44962bc78f957e4f87d19bd03f8ac8c57
Showing with 9 additions and 7 deletions.
  1. +2 −2 cohomology.tex
  2. +7 −5 sites-cohomology.tex
@@ -11128,8 +11128,8 @@ \section{Invertible objects in the derived category}
\begin{enumerate}
\item[(3)] there exists an open covering $X = \bigcup U_i$
and for each $i$ an integer $n_i$ such that $M|_{U_i}$
is represented by an invertible $\mathcal{O}_{U_i}$
module placed in degree $n_i$.
is represented by an invertible $\mathcal{O}_{U_i}$-module
placed in degree $n_i$.
\end{enumerate}
\end{lemma}

@@ -12001,7 +12001,8 @@ \section{Invertible objects in the derived category}
\mathcal{O} = \prod\nolimits_{n \in \mathbf{Z}} \mathcal{O}_n
$$
and for each $n$ there is an invertible $\mathcal{O}_n$-module
$\mathcal{H}^n$ (Modules, Definition \ref{modules-definition-invertible})
$\mathcal{H}^n$
(Modules on Sites, Definition \ref{sites-modules-definition-invertible-sheaf})
and $M = \bigoplus \mathcal{H}^n[-n]$ in $D(\mathcal{O})$.
\end{enumerate}
If (1) and (2) hold, then $M$ is a perfect object of $D(\mathcal{O})$. If
@@ -12010,8 +12011,8 @@ \section{Invertible objects in the derived category}
\begin{enumerate}
\item[(3)] for every object $U$ of $\mathcal{C}$ there exists a
covering $\{U_i \to U\}$ and for each $i$ an integer $n_i$ such that
$M|_{U_i}$ is represented by an invertible $\mathcal{O}_{U_i}$
module placed in degree $n_i$.
$M|_{U_i}$ is represented by an invertible $\mathcal{O}_{U_i}$-module
placed in degree $n_i$.
\end{enumerate}
\end{lemma}

@@ -12034,7 +12035,8 @@ \section{Invertible objects in the derived category}
$\mathcal{O}_m$ are flat $\mathcal{O}$-algebras with
$\mathcal{O}_n \otimes_\mathcal{O} \mathcal{O}_m = 0$.
Using the local structure of invertible $\mathcal{O}$-modules
(Modules, Lemma \ref{modules-lemma-invertible}) and working locally
(Modules on Sites, Lemma \ref{sites-modules-lemma-invertible})
and working locally
the isomorphism in case $n = m$ follows in a straightforward manner;
we omit the details. Because $D(\mathcal{O})$ is symmetric monoidal,
we conclude that $M$ is invertible.
@@ -12055,7 +12057,7 @@ \section{Invertible objects in the derived category}
restriction to $\mathcal{C}/U_i$.
Thus we may assume $\mathcal{C}$ has a final object $X$
and during the proof of (2) we may finitely many times
repplace $X$ by the members of a covering of $X$.
replace $X$ by the members of a covering of $X$.

\medskip\noindent
Choose an object $N$ of $D(\mathcal{O})$ and an isomorphism

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