After many years of avoiding spectral sequences I have run into one that I can’t seem to avoid. I have very limited understanding of what convergence of spectral sequences really means and I am hoping somebody can give me a concrete description of what this spectral sequence is telling me in my particular setup.
The spectral sequence is Theorem 10.62 in Rotman’s book on homological algebra (modulo correcting what I believe is a typo). I think it can also be found in a slight variant in Cartan-Eilenberg. My setup is more specialized, which I am hoping will make it easier to understand what this means.
Let $R$ and $S$ be finite dimensional algebras over a field $K$. I work with left modules. $S$-$R$-bimodules are assumed to have the same $K$-vector space structure from the action on both sides (i.e., are $S\otimes_K R^{op}$-modules). Also all modules and bimodules finite dimensional.
Then, according to Rotman, if $B$ is an $S$-$R$-bimodule and $C$ is an $S$-module such that $\mathrm{Ext}^i_S(B\otimes_R P,C)=0$ for every $i\geq 1$ when $P$ is a projective $R$-module, then there is a convergent 3rd quadrant spectral sequence
$$\mathrm{Ext}^p_S(\mathrm{Tor}^R_q(B,A),C)\Rightarrow_p \mathrm{Ext}^n_R(A,\mathrm{Hom}_S(B,C))$$ for every $R$-module $A$.
Since all these $\mathrm{Ext}$ are $K$-vector spaces all filtrations can be turned into vector space direct sums and so my vague understanding is that I should be able to express $\mathrm{Ext}^n_R(A,\mathrm{Hom}_S(B,C))$ as some sort of vector space direct sum related to $\mathrm{Ext}^p_S(\mathrm{Tor}^R_q(B,A),C)$. But what exactly do I get? Is it the case $$\mathrm{Ext}^n_R(A,\mathrm{Hom}_S(B,C))\cong \bigoplus_{p+q=n}\mathrm{Ext}^p_S(\mathrm{Tor}^R_q(B,A),C)$$ at least under some weaker hypotheses than $C$ is injective (for which I don’t need spectral sequences)? If it helps, in my particular set up $B$ is free as a left $S$-module but not even flat as a right $R$-module.
I'd be happy to understand the special case where there is a surjective ring homomorphism $R\to S$ and $B=S$ so that the bimodule structure is the obvious one and this is the change of ring spectral sequence.