Lemma 21.45.5. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $K, L$ be objects of $D(\mathcal{O})$.

1. If $K$ is $n$-pseudo-coherent and $H^ i(K) = 0$ for $i > a$ and $L$ is $m$-pseudo-coherent and $H^ j(L) = 0$ for $j > b$, then $K \otimes _\mathcal {O}^\mathbf {L} L$ is $t$-pseudo-coherent with $t = \max (m + a, n + b)$.

2. If $K$ and $L$ are pseudo-coherent, then $K \otimes _\mathcal {O}^\mathbf {L} L$ is pseudo-coherent.

Proof. Proof of (1). Let $U$ be an object of $\mathcal{C}$. By replacing $U$ by the members of a covering and replacing $\mathcal{C}$ by the localization $\mathcal{C}/U$ we may assume there exist strictly perfect complexes $\mathcal{K}^\bullet$ and $\mathcal{L}^\bullet$ and maps $\alpha : \mathcal{K}^\bullet \to K$ and $\beta : \mathcal{L}^\bullet \to L$ with $H^ i(\alpha )$ and isomorphism for $i > n$ and surjective for $i = n$ and with $H^ i(\beta )$ and isomorphism for $i > m$ and surjective for $i = m$. Then the map

$\alpha \otimes ^\mathbf {L} \beta : \text{Tot}(\mathcal{K}^\bullet \otimes _\mathcal {O} \mathcal{L}^\bullet ) \to K \otimes _\mathcal {O}^\mathbf {L} L$

induces isomorphisms on cohomology sheaves in degree $i$ for $i > t$ and a surjection for $i = t$. This follows from the spectral sequence of tors (details omitted).

Proof of (2). Let $U$ be an object of $\mathcal{C}$. We may first replace $U$ by the members of a covering and $\mathcal{C}$ by the localization $\mathcal{C}/U$ to reduce to the case that $K$ and $L$ are bounded above. Then the statement follows immediately from case (1). $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).