The Stacks project

Lemma 21.52.7. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Assume $\mathcal{C}$ has the following properties

  1. $\mathcal{C}$ has a quasi-compact final object $X$,

  2. every quasi-compact object of $\mathcal{C}$ has a cofinal system of coverings which are finite and consist of quasi-compact objects,

  3. for a finite covering $\{ U_ i \to U\} _{i \in I}$ with $U$, $U_ i$ quasi-compact the fibre products $U_ i \times _ U U_ j$ are quasi-compact.

Let $K$ be a perfect object of $D(\mathcal{O})$. Then

  1. $K$ is a compact object of $D^+(\mathcal{O})$ in the following sense: if $M = \bigoplus _{i \in I} M_ i$ is bounded below, then $\mathop{\mathrm{Hom}}\nolimits (K, M) = \bigoplus _{i \in I} \mathop{\mathrm{Hom}}\nolimits (K, M_ i)$.

  2. If $(\mathcal{C}, \mathcal{O})$ has finite cohomological dimension, i.e., if there exists a $d$ such that $H^ i(X, \mathcal{F}) = 0$ for $i > d$ for any $\mathcal{O}$-module $\mathcal{F}$, then $K$ is a compact object of $D(\mathcal{O})$.

Proof. Let $K^\vee $ be the dual of $K$, see Lemma 21.48.4. Then we have

\[ \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O})}(K, M) = H^0(X, K^\vee \otimes _\mathcal {O}^\mathbf {L} M) \]

functorially in $M$ in $D(\mathcal{O})$. Since $K^\vee \otimes _\mathcal {O}^\mathbf {L} -$ commutes with direct sums it suffices to show that $R\Gamma (X, -)$ commutes with the relevant direct sums.

Proof of (a). After reformulation as above this is a special case of Lemma 21.52.4 with $U = X$.

Proof of (b). Since $R\Gamma (X, K) = R\mathop{\mathrm{Hom}}\nolimits (\mathcal{O}, K)$ and since $H^ p(X, -)$ commutes with direct sums by Lemma 21.16.1 this is a special case of Lemma 21.52.5. $\square$

Comments (2)

Comment #4081 by on

This lemma suffers from the same problem as the one I just commented on. We need to add an hypothesis of finite coh dim. I will fix this soon.

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