This question was asked on math.stackexchange and didn't receive any traction, so I'm turning to the MO community. Hello!

Let $C_{\bullet}$ and $D_{\bullet}$ be chain complexes of $R$-modules for some ring $R$. My core problem is that I'd like to know how to compute $\mathrm{Ext}_{R}^{n}(C_{\bullet},D_{\bullet})$ in terms of the homologies $h_{i}(C_{\bullet})$ and $h_{j}(D_{\bullet})$.

In my specific circumstances, $C_{\bullet}$ is bounded below by $0$ ($C_{i}=0$ for $i<0$) and $D_{\bullet}$ is bounded, supported in $[0,1]$ ($D_{j}=0$ for $j\not\in\{0,1\}$), but if you're interested in saying something more general, that is interesting to me too.

I'm familiar with related results. For instance, if $M$ is merely an $R$-module and not a complex, then there is a convergent spectral sequence \begin{align*} E_{2}^{pq}=\mathrm{Ext}_{R}^{p}(h_{q}(C_{\bullet}),M)\Rightarrow\mathrm{Ext}_{R}^{p+q}(C_{\bullet},M). \end{align*} Additionally, I'm aware of spectral sequences computing Tor from Weibel's book: \begin{align*} ^{II}E_{pq}^{2}&&=&&\bigoplus_{q'+q''=q}\mathrm{Tor}_{p}^{R}(h_{q'}(C_{\bullet}),h_{q''}(D_{\bullet}))&&\Rightarrow&&\mathrm{Tor}_{p+q}^{R}(C_{\bullet},D_{\bullet})\\ ^{I}E_{pq}^{2}&&=&&h_{p}\mathrm{Tot}^{\oplus}\mathrm{Tor}_{q}(C_{\bullet},D_{\bullet})&&\Rightarrow&&\mathrm{Tor}_{p+q}^{R}(C_{\bullet},D_{\bullet}) \end{align*} but Weibel skips an analogous statement for Ext (and even for the Tor I'm also not sure how to parse the $^{I}E_{pq}^{2}$ sequence, since it seems like you already have to know how to compute "$\mathrm{Tor}_{q}(C_{\bullet},D_{\bullet})$", but unless that's something that pops up in Ext as well, I guess that's ancillary to my core question).

My questions are these:

  1. What are the spectral sequences computing $\mathrm{Ext}_{R}^{n}(C_{\bullet},D_{\bullet})$ for chain complexes $C_{\bullet}$ and $D_{\bullet}$?
  2. In the specific case mentioned above ($C_{i}=0$ for $i<0$ and $D_{j}=0$ for $j\not\in\{0,1\}$), how does the boundedness affect the spectral sequences? Can we get an explicit isomorphism for low values of $n$? That is, what, explicitly, is say $\mathrm{Ext}_{R}^{0}(C_{\bullet},D_{\bullet})$, $\mathrm{Ext}_{R}^{1}(C_{\bullet},D_{\bullet})$, and $\mathrm{Ext}_{R}^{2}(C_{\bullet},D_{\bullet})$ in terms of $h_{i}(C_{\bullet})$ and $h_{j}(D_{\bullet})$? Those seem low dimensional enough that you shouldn't have to turn a bunch of pages.

I am in general familiar with the machinery of spectral sequences, but I perhaps treat them a little too indelicately -- I was told in passing that notions of convergence can be fiddly with regards to my question, so I'm in particular looking for the details involved in answering #2.

I've had trouble throwing together the right keywords in google to find good reading about this, so in addition I'm tagging this question reference-request. I'd like a resource that doesn't skip too many details and preferably does some examples computing Ext of chain complexes, if such references exist. Targeted at capable and motivated grad students would be especially cool 🙂



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