The Stacks project

Lemma 15.102.5. Let $A \to B$ be a ring map. There exists a cardinal $\kappa = \kappa (A \to B)$ with the following property: Let $M^\bullet $, resp. $N^\bullet $ be a complex of $A$-modules, resp. $B$-modules. Let $a : M^\bullet \to N^\bullet $ be a map of complexes of $A$-modules which induces an isomorphism $M^\bullet \otimes _ A^\mathbf {L} B \to N^\bullet $ in $D(B)$. Let $M_1^\bullet \subset M^\bullet $, resp. $N_1^\bullet \subset N^\bullet $ be a subcomplex of $A$-modules, resp. $B$-modules such that $a(M_1^\bullet ) \subset N_1^\bullet $. Then there exist subcomplexes

\[ M_1^\bullet \subset M_2^\bullet \subset M^\bullet \quad \text{and}\quad N_1^\bullet \subset N_2^\bullet \subset N^\bullet \]

such that $a(M_2^\bullet ) \subset N_2^\bullet $ with the following properties:

  1. $\mathop{\mathrm{Ker}}(H^ i(M_1^\bullet \otimes _ A^\mathbf {L} B) \to H^ i(N_1^\bullet ))$ maps to zero in $H^ i(M_2^\bullet \otimes _ A^\mathbf {L} B)$,

  2. $\mathop{\mathrm{Im}}(H^ i(N_1^\bullet ) \to H^ i(N_2^\bullet ))$ is contained in $\mathop{\mathrm{Im}}(H^ i(M_2^\bullet \otimes _ A^\mathbf {L} B) \to H^2(N_2^\bullet ))$,

  3. $|\bigcup M_2^ i \cup \bigcup N_2^ i| \leq \max (\kappa , |\bigcup M_1^ i \cup \bigcup N_1^ i|)$.

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

\[ \max (\kappa , |\bigcup M_1^ i \cup \bigcup N_1^ i|) = \max (\kappa , |M_1^\bullet |, |M_2^\bullet |) \]

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

\[ M_1^\bullet \subset M_1(E, F)^\bullet \subset M^\bullet \quad \text{and}\quad N_1^\bullet \subset N_1(E, F)^\bullet \subset N^\bullet \]

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

\[ |M_1(E, F)^\bullet | \leq \max (\kappa , |M_1^\bullet |) \quad \text{and}\quad |M_2(E, F)^\bullet | \leq \max (\kappa , |M_2^\bullet |) \]

Details omitted. It is clear that we have

\[ M^\bullet = \mathop{\mathrm{colim}}\nolimits _{(E, F)} M_1(E, F)^\bullet \quad \text{and}\quad N^\bullet = \mathop{\mathrm{colim}}\nolimits _{(E, F)} N_1(E, F)^\bullet \]

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

\[ M_2^\bullet = \sum \nolimits _\xi M_1(E_\xi , F_\xi )^\bullet + \sum \nolimits _\eta M_1(E_\eta , F_\eta )^\bullet \]

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

\[ N_2^\bullet = \sum \nolimits _\xi N_1(E_\xi , F_\xi )^\bullet + \sum \nolimits _\eta N_1(E_\eta , F_\eta )^\bullet \]

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 $\max (\kappa , |M_1^\bullet |, |N_1^\bullet |)$. $\square$

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