The Stacks project

Lemma 21.49.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $M$ be an object of $D(\mathcal{O})$. The following are equivalent

  1. $M$ is invertible in $D(\mathcal{O})$, see Categories, Definition 4.43.4, and

  2. there is a locally finite1 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 on Sites, Definition 18.32.1) and $M = \bigoplus \mathcal{H}^ n[-n]$ in $D(\mathcal{O})$.

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

  1. 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 $n_ i$.

Proof. Assume (2). Consider the object $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (M, \mathcal{O})$ and the composition map

\[ R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (M, \mathcal{O}) \otimes _\mathcal {O}^\mathbf {L} M \to \mathcal{O} \]

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

\[ R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{H}^ m, \mathcal{O}) \otimes _\mathcal {O}^\mathbf {L} \mathcal{H}^ n \]

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 on Sites, Lemma 18.32.2) 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.

Assume (1). The description in (2) shows that we have a candidate for $\mathcal{O}_ n$, namely, $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(H^ n(M), H^ n(M))$. 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 replace $X$ by the members of a covering of $X$.

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 21.48.7. 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 21.44.8. 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 21.49.1 we find that we obtain

  1. $M^\bullet = \Gamma (X, \mathcal{E}^\bullet )$ is a bounded complex of finite projective $R$-modules,

  2. $N^\bullet = \Gamma (X, \mathcal{F}^\bullet )$ is a bounded complex of finite projective $R$-modules,

  3. $\alpha $ and $\beta $ correspond to maps of complexes $a : R \to \text{Tot}(M^\bullet \otimes _ R N^\bullet )$ and $b : \text{Tot}(M^\bullet \otimes _ R N^\bullet ) \to R$,

  4. $\theta ^ n$ 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

  5. $b \circ a = 1$ and $b \circ a - 1 = dh + hd$,

It follows that $M^\bullet $ and $N^\bullet $ define mutually inverse objects of $D(R)$. By More on Algebra, Lemma 15.126.4 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 21.49.1 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 21.49.1. Setting $\mathcal{O}_ n = R_ n \otimes _ R \mathcal{O}$ we conclude (2) is true.

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 18.40.1 and the definition of locally ringed sites as given in Modules on Sites, Definition 18.40.4. 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. $\square$

[1] 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 $n$.

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