# stacks/stacks-project

Invertible objects of D(O) for a ringed site

 @@ -11951,6 +11951,210 @@ \section{Duals} \section{Invertible objects in the derived category} \label{section-invertible-D-or-R} \noindent We characterize invertible objects in the derived category of a ringed space (both in the case of a locally ringed topos and in the general case). \begin{lemma} \label{lemma-category-summands-finite-free} Let $(\mathcal{C}, \mathcal{O})$ be a ringed space. Set $R = \Gamma(\mathcal{C}, \mathcal{O})$. The category of $\mathcal{O}$-modules which are summands of finite free $\mathcal{O}$-modules is equivalent to the category of finite projective $R$-modules. \end{lemma} \begin{proof} Observe that a finite projective $R$-module is the same thing as a summand of a finite free $R$-module. The equivalence is given by the functor $\mathcal{E} \mapsto \Gamma(\mathcal{C}, \mathcal{E})$. The inverse functor is given by the following construction. Consider the morphism of topoi $f : \Sh(\mathcal{C}) \to \Sh(\text{pt})$ with $f_*$ given by taking global sections and $f^{-1}$ by sending a set $S$, i.e., an object of $\Sh(\text{pt})$, to the constant sheaf with value $S$. We obtain a morphism $(f, f^\sharp) : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\text{pt}), R)$ of ringed topoi by using the identity map $R \to f_*\mathcal{O}$. Then the inverse functor is given by $f^*$. \end{proof} \begin{lemma} \label{lemma-invertible-derived} Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $M$ be an object of $D(\mathcal{O})$. The following are equivalent \begin{enumerate} \item $M$ is invertible in $D(\mathcal{O})$, see Categories, Definition \ref{categories-definition-invertible}, and \item there is a locally finite\footnote{This means that for every object $U$ of $\mathcal{C}$ there is a covering $\{U_i \to U\}$ such that for every $i$ the sheaf $\mathcal{O}_n|_{U_i}$ is nonzero for only a finite number of $n$.} direct product decomposition $$\mathcal{O} = \prod\nolimits_{n \in \mathbf{Z}} \mathcal{O}_n$$ and for each $n$ there is an invertible $\mathcal{O}_n$-module $\mathcal{H}^n$ (Modules, Definition \ref{modules-definition-invertible}) 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 $(\mathcal{C}, \mathcal{O})$ is a locally ringed site these condition are also equivalent to \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 $n_i$. \end{enumerate} \end{lemma} \begin{proof} Assume (2). Consider the object $R\SheafHom(M, \mathcal{O})$ and the composition map $$R\SheafHom(M, \mathcal{O}) \otimes_\mathcal{O}^\mathbf{L} M \to \mathcal{O}$$ To prove this is an isomorphism, we may work locally. Thus we may assume $\mathcal{O} = \prod_{a \leq n \leq b} \mathcal{O}_n$ and $M = \bigoplus_{a \leq n \leq b} \mathcal{H}^n[-n]$. Then it suffices to show that $$R\SheafHom(\mathcal{H}^m, \mathcal{O}) \otimes_\mathcal{O}^\mathbf{L} \mathcal{H}^n$$ is zero if $n \not = m$ and equal to $\mathcal{O}_n$ if $n = m$. The case $n \not = m$ follows from the fact that $\mathcal{O}_n$ and $\mathcal{O}_m$ 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 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. \medskip\noindent Assume (1). The description in (2) shows that we have a candidate for $\mathcal{O}_n$, namely, $\SheafHom_\mathcal{O}(H^n(M), H^n(M))$. If this is a locally finite family of sheaves of rings and if $\mathcal{O} = \prod \mathcal{O}_n$, then we immediately obtain the direct sum decomposition $M = \bigoplus H^n(M)[-n]$ using the idempotents in $\mathcal{O}$ coming from the product decomposition. This shows that in order to prove (2) we may work locally in the following sense. Let $U$ be an object of $\mathcal{C}$. We have to show there exists a covering $\{U_i \to U\}$ of $U$ such that with $\mathcal{O}_n$ as above we have the statements above and those of (2) after 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 $X$. \medskip\noindent Choose an object $N$ of $D(\mathcal{O})$ and an isomorphism $M \otimes_\mathcal{O}^\mathbf{L} N \cong \mathcal{O}$. Then $N$ is a left dual for $M$ in the monoidal category $D(\mathcal{O})$ and we conclude that $M$ is perfect by Lemma \ref{lemma-left-dual-derived}. By symmetry we see that $N$ is perfect. After replacing $X$ by the members of a covering, we may assume $M$ and $N$ are represented by a strictly perfect complexes $\mathcal{E}^\bullet$ and $\mathcal{F}^\bullet$. Then $M \otimes_\mathcal{O}^\mathbf{L} N$ is represented by $\text{Tot}(\mathcal{E}^\bullet \otimes_\mathcal{O} \mathcal{F}^\bullet)$. After replacing $X$ by the members of a covering of $X$ we may assume the mutually inverse isomorphisms $\mathcal{O} \to M \otimes_\mathcal{O}^\mathbf{L} N$ and $M \otimes_\mathcal{O}^\mathbf{L} N \to \mathcal{O}$ are given by maps of complexes $$\alpha : \mathcal{O} \to \text{Tot}(\mathcal{E}^\bullet \otimes_\mathcal{O} \mathcal{F}^\bullet) \quad\text{and}\quad \beta : \text{Tot}(\mathcal{E}^\bullet \otimes_\mathcal{O} \mathcal{F}^\bullet) \to \mathcal{O}$$ See Lemma \ref{lemma-local-actual}. Then $\beta \circ \alpha = 1$ as maps of complexes and $\alpha \circ \beta = 1$ as a morphism in $D(\mathcal{O})$. After replacing $X$ by the members of a covering of $X$ we may assume the composition $\alpha \circ \beta$ is homotopic to $1$ by some homotopy $\theta$ with components $$\theta^n : \text{Tot}^n(\mathcal{E}^\bullet \otimes_\mathcal{O} \mathcal{F}^\bullet) \to \text{Tot}^{n - 1}( \mathcal{E}^\bullet \otimes_\mathcal{O} \mathcal{F}^\bullet)$$ by the same lemma as before. Set $R = \Gamma(\mathcal{C}, \mathcal{O})$. By Lemma \ref{lemma-category-summands-finite-free} we find that we obtain \begin{enumerate} \item $M^\bullet = \Gamma(X, \mathcal{E}^\bullet)$ is a bounded complex of finite projective $R$-modules, \item $N^\bullet = \Gamma(X, \mathcal{F}^\bullet)$ is a bounded complex of finite projective $R$-modules, \item $\alpha$ and $\beta$ correspond to maps of complexes $a : R \to \text{Tot}(M^\bullet \otimes_R N^\bullet)$ and $b : \text{Tot}(M^\bullet \otimes_R N^\bullet) \to R$, \item $\theta^n$ corresponds to a map $h^n : \text{Tot}^n(M^\bullet \otimes_R N^\bullet) \to \text{Tot}^{n - 1}(M^\bullet \otimes_R N^\bullet)$, and \item $b \circ a = 1$ and $b \circ a - 1 = dh + hd$, \end{enumerate} It follows that $M^\bullet$ and $N^\bullet$ define mutually inverse objects of $D(R)$. By More on Algebra, Lemma \ref{lemma-invertible-derived} we find a product decomposition $R = \prod_{a \leq n \leq b} R_n$ and invertible $R_n$-modules $H^n$ such that $M^\bullet \cong \bigoplus_{a \leq n \leq b} H^n[-n]$. This isomorphism in $D(R)$ can be lifted to an morphism $$\bigoplus H^n[-n] \longrightarrow M^\bullet$$ of complexes because each $H^n$ is projective as an $R$-module. Correspondingly, using Lemma \ref{lemma-category-summands-finite-free} again, we obtain an morphism $$\bigoplus H^n \otimes_R \mathcal{O}[-n] \to \mathcal{E}^\bullet$$ which is an isomorphism in $D(\mathcal{O})$. Here $M \otimes_R \mathcal{O}$ denotes the functor from finite projective $R$-modules to $\mathcal{O}$-modules constructed in the proof of Lemma \ref{lemma-category-summands-finite-free}. Setting $\mathcal{O}_n = R_n \otimes_R \mathcal{O}$ we conclude (2) is true. \medskip\noindent If $(\mathcal{C}, \mathcal{O})$ is a locally ringed site, then given an object $U$ and a finite product decomposition $\mathcal{O}|_U = \prod_{a \leq n \leq b} \mathcal{O}_n|_U$ we can find a covering $\{U_i \to U\}$ such that for every $i$ there is at most one $n$ with $\mathcal{O}_n|_{U_i}$ nonzero. This follows readily from part (2) of Modules on Sites, Lemma \ref{sites-modules-lemma-locally-ringed} and the definition of locally ringed sites as given in Modules on Sites, Definition \ref{sites-modules-definition-locally-ringed}. From this the implication (2) $\Rightarrow$ (3) is easily seen. The implication (3) $\Rightarrow$ (2) holds without any assumptions on the ringed site. We omit the details. \end{proof} \section{Projection formula} \label{section-projection-formula}