Lemma 20.47.5 (09J3)—The Stacks project

Lemma 20.47.5. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $K, L$ be objects of $D(\mathcal{O}_ X)$ .

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 $K \otimes _{\mathcal{O}_ X}^\mathbf {L} L$ is $t$ -pseudo-coherent with $t = \max (m + a, n + b)$ .
If $K$ and $L$ are pseudo-coherent, then $K \otimes _{\mathcal{O}_ X}^\mathbf {L} L$ is pseudo-coherent.

Proof. Proof of (1). By replacing $X$ by the members of an open covering 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

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

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). We may first replace $X$ by the members of an open covering to reduce to the case that $K$ and $L$ are bounded above. Then the statement follows immediately from case (1). $\square$

The Stacks project

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