Homology with coefficients and direct Limits

Question

I am trying to see that homology with coefficients commutes with direct limits, that is let $(G_{\alpha})_{\alpha \in D}$ be a direct system of groups and $G :=\lim_{\longrightarrow}G_{\alpha}$. I'm trying to see that $\lim H_*(G_{\alpha},M)\cong H_*(G,M)$. If we take the standard resolution $F_{\alpha}$, we know that $\lim H_*(G_{\alpha},M)=\lim H_*(F_{\alpha}\otimes_{G_{\alpha}} M) = H_*(\lim(F_{\alpha}\otimes_{G_{\alpha}} M))$, so what I am trying to see is that $\lim(F_{\alpha}\otimes_{G_{\alpha}} M) \cong (\lim F_{\alpha})\otimes_{G}M$ to get what I want, but I am not seeing how to do this, do I create an explicit isomorphism between them both and try using restriction of scalars maybe, it just a bit confusing for me because we are in the category of chain complexes. Thanks in advance.

Answer 1

(As a general note, be careful with the notation $\lim$ : in your situation you want to use \varinjlim, which gives $\varinjlim$, otherwise there's a pretty strong conflict of notation with other notions of limit - the best, of course, being to use colimits, although this may not be the most common thing in algebra)

The result is true; here's a "categorical" proof :

you have, for each $n$, a direct system of abelian groups : $F_{\alpha,n}\otimes_{G_\alpha}M$ and want to compare it to $F_n\otimes_G M$.

Note that $\mathbb Z[-]$ commutes with direct limits : if you don't already know this, you should try and prove it. This allows you (using the standard resolution, as you mentioned), to have $\varinjlim F_\alpha = F$ (where $F$ is the standard resolution over $G$)

Now the key point is to notice that $F_{\alpha,n}\otimes_{G_\alpha}M$ is the coequalizer of $$F_{\alpha,n}\otimes \mathbb Z[G_\alpha]\otimes M \rightrightarrows F_{\alpha,n}\otimes M$$

where the two arrows are $x\otimes g \otimes m\mapsto (xg)\otimes m$ and $x\otimes g\otimes m \mapsto x\otimes (gm)$ respectively.

This is interesting because these maps are compatible with the maps in the direct system: for $\alpha \leq \beta$, the two obvious squares commute :

$\require{AMScd}\begin{CD}F_{\alpha,n}\otimes \mathbb Z[G_\alpha]\otimes M @>>> F_{\alpha,n}\otimes M\\ @VVV @VVV \\ F_{\beta,n}\otimes \mathbb Z[G_\beta]\otimes M @>>> F_{\beta,n}\otimes M\end{CD}$

Now a general property is that colimits commute, so taking the coequalizer and then the direct limit is the same thing as taking first the direct limit and then the coequalizer.

Since $-\otimes M$ commutes with colimits, you are left with examining the direct limit of $F_{\alpha,n}\otimes \mathbb Z[G_\alpha]\otimes M$. But that one is easy to identify with $F_n\otimes \mathbb Z[G]\otimes M$, for the same kind of reason.

So your direct limit is just the coequalizer of $$F_n\otimes \mathbb Z[G]\otimes M \rightrightarrows F_n\otimes M$$

i.e. $F_n\otimes_G M$.

Note that under suitable hypotheses on $(M_\alpha)$, this can easily be adapted for a direct system where $M_\alpha$ is a $G_\alpha$-module (and you have to define a suitable $G$-module structure on $M:=\varinjlim M_\alpha$)