# stacks/stacks-project

Small corrections to last two commites

 @@ -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