The Stacks project

Proposition 15.64.3. Let $R$ be a ring. Let $K^\bullet $ and $L^\bullet $ be filtered complexes of $R$-modules. Then there exists a filtered complex $T^\bullet $ representing $K^\bullet \otimes _ R^\mathbf {L} L^\bullet $ in $D(R)$ such that the associated spectral sequence has $E_1$-page

\[ E_1^{p, q} = \bigoplus \nolimits _{i + j = p} H^{p + q}(\text{gr}^ iK^\bullet \otimes _ R^\mathbf {L} \text{gr}^ jL^\bullet ) \]

If

  1. $F^ iK^\bullet $ is acyclic for $i \gg 0$,

  2. $F^ iK^\bullet \to K^\bullet $ is a quasi-isomorphism for $i \ll 0$,

  3. $F^ jL^\bullet $ is acyclic for $j \gg 0$, and

  4. $F^ jL^\bullet \to L^\bullet $ is a quasi-isomorphism for $j \ll 0$.

then the spectral sequence is bounded, the associated filtration on each $H^ n(K^\bullet \otimes _ R^\mathbf {L} L^\bullet )$ is finite and the spectral sequence convergences.

Proof. Choose $P^\bullet \to K^\bullet $ and $Q^\bullet \to L^\bullet $ as in Lemma 15.64.2. Then we use the spectral sequence for the filtered complex

\[ T^\bullet = \text{Tot}(P^\bullet \otimes _ R Q^\bullet ) \]

described in the text of this section and in Lemma 15.64.1. $\square$

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