Clean up and give a name

We are going to use QC(O) for this category of "cartesian" modules on a
category. Please complain now if you object!

more-algebra.tex

@@ -28730,6 +28730,125 @@ \section{Miscellany}
 This concludes the proof.
 \end{proof}
 
+\begin{lemma}
+\label{lemma-enlarge}
+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:
+\begin{enumerate}
+\item $\Ker(H^i(M_1^\bullet \otimes_A^\mathbf{L} B) \to H^i(N_1^\bullet))$
+maps to zero in $H^i(M_2^\bullet \otimes_A^\mathbf{L} B)$,
+\item $\Im(H^i(N_1^\bullet) \to H^i(N_2^\bullet))$ is contained in
+$\Im(H^i(M_2^\bullet \otimes_A^\mathbf{L} B) \to H^2(N_2^\bullet))$,
+\item $|\bigcup M_2^i \cup \bigcup N_2^i| \leq
+\max(\kappa, |\bigcup M_1^i \cup \bigcup N_1^i|)$.
+\end{enumerate}
+\end{lemma}
+
+\begin{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.
+
+\medskip\noindent
+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
+$$
+M^\bullet = \colim_{(E, F)} M_1(E, F)^\bullet
+\quad\text{and}\quad
+N^\bullet = \colim_{(E, F)} N_1(E, F)^\bullet
+$$
+and the colimits are (termwise) filtered colimits.
+
+\medskip\noindent
+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
+$|H^i(M_1^\bullet \otimes_A^\mathbf{L} B)| \leq \max(\kappa, |M_1^\bullet|)$.
+
+\medskip\noindent
+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
+$H^i(M_1(E_\xi, F_\xi)^\bullet \otimes_A^\mathbf{L} B)$.
+
+\medskip\noindent
+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
+$H^i(M_1(E_\eta, F_\eta)^\bullet \otimes_A^\mathbf{L} B) \to
+H^i(N_1(E_\eta, F_\eta)$.
+
+\medskip\noindent
+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
+$\max(\kappa, |M_1^\bullet|, |N_1^\bullet|)$.
+\end{proof}
+
+
+