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Invertible objects of D(O) for a ringed site

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aisejohan committed Nov 6, 2019
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@@ -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}

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