The Stacks project

Proof. Let $\kappa = \max (|A|, |B|, \aleph _0)$. Set $|M^\bullet | = |\bigcup M^ i|$ and similarly for other complexes. With this notation we have

for the quantity used in the statement of the lemma. We are going to use this and other observations coming from arithmetic of cardinals without further mention.

First, let us show that there are plenty of “small” subcomplexes. For every pair of collections $E = \{ E^ i\}$ and $F = \{ F^ i\}$ of finite subsets $E^ i \subset M^ i$, $i \in \mathbf{Z}$ and $F^ i \subset N^ i$, $i \in \mathbf{Z}$ we can let

be the smallest subcomplexes of $A$ and $B$-modules such that $a(M_1(E, F)^\bullet ) \subset N_1(E, F)^\bullet$ and such that $E^ i \subset M_1(E, F)^ i$ and $F^ i \subset M_2(E, F)^ i$. Then it is easy to see that

Details omitted. It is clear that we have

and the colimits are (termwise) filtered colimits.

There exists a resolution $\ldots \to F^{-1} \to F^0 \to B$ by free $A$-modules $F_ i$ with $|F_ i| \leq \kappa$ (details omitted). The cohomology modules of $M_1^\bullet \otimes _ A^\mathbf {L} B$ are computed by $\text{Tot}(M_1^\bullet \otimes _ A F^\bullet )$. It follows that $|H^ i(M_1^\bullet \otimes _ A^\mathbf {L} B)| \leq \max (\kappa , |M_1^\bullet |)$.

Let $i \in \mathbf{Z}$ and let $\xi \in H^ i(M_1^\bullet \otimes _ A^\mathbf {L} B)$ be an element which maps to zero in $H^ i(N_1^\bullet )$. Then $\xi$ maps to zero in $H^ i(N^\bullet )$ and hence $\xi$ maps to zero in $H^ i(M^\bullet \otimes _ A^\mathbf {L} B)$. Since derived tensor product commutes with filtered colimits, we can find finite collections $E_\xi$ and $F_\xi$ as above such that $\xi$ maps to zero in $H^ i(M_1(E_\xi , F_\xi )^\bullet \otimes _ A^\mathbf {L} B)$.

Let $i \in \mathbf{Z}$ and let $\eta \in H^ i(N_1^\bullet )$. Then the image of $\eta$ in $H^ i(N^\bullet )$ is in the image of $H^ i(M^\bullet \otimes _ A^\mathbf {L} B) \to H^ i(N^\bullet )$. Hence as before, we can find finite collections $E_\eta$ and $F_\eta$ as above such that $\eta$ maps to an element of $H^ i(N_1(E_\eta , F_\eta )$ which is in the image of the map $H^ i(M_1(E_\eta , F_\eta )^\bullet \otimes _ A^\mathbf {L} B) \to H^ i(N_1(E_\eta , F_\eta )$.

Now we simply define

where the sum is over $\xi$ and $\eta$ as in the previous two paragraphs and the sum is taken inside $M^\bullet$. Similarly we set

where the sum is taken inside $N^\bullet$. By construction we will have properties (1) and (2) with these choices. The bound (3) also follows as the set of $\xi$ and $\eta$ has cardinality at most $\max (\kappa , |M_1^\bullet |, |N_1^\bullet |)$. $\square$