$\newcommand{\Tor}{\mathrm{Tor}}$ I am working on exercise A3.47 of Eisenbud’s commutative algebra text, which asks us to use the change-of-rings spectral sequence for Tor to show that, given a map of rings $R\to S$ and a field $k$ that is a quotient of $S$, we have the exact sequence $$ \Tor_2^R(k,k)\to \Tor_2^S(k,k)\to\Tor_1^R(S,k)\to\Tor_1^R(k,k)\to\Tor_1^S(k,k)\to 0. $$ The change-of-rings spectral sequence that Eisenbud refers to is obtained as follows: Given $R\to S$ a map of rings, $B$ an $R$-module, and $A$ an $S$-module, take a free resolution $P_*\to B$ of $B$ as an $R$-module and a free resolution $Q_*\to A$ of $A$ as an $S$-module. Then take the (third-quadrant) double complex $E^0_{**}$ given by $E^0_{-p,-q}=P_{q}\otimes_R Q_{p}$. Noting that $P_q\otimes_R Q_p\cong (P_q\otimes_R S)\otimes_S Q_p$, we can see that the spectral sequence of this double complex whose differentials on the 0th page are vertical has second page $$ E^2_{-p,-q}=\Tor_p^S(\Tor_q^R(B,S),A) $$ and this spectral sequence converges to $\Tor^R_{p+q}(B,A)$. (This is all from Exercise A3.45 of Eisenbud, although my indexing might be a little off).
Towards obtaining the above five-term exact sequence, I took $A=k$ and $B=k$ in the above setup and wrote down the corresponding double complex and the first few pages of the spectral sequence. I then wrote down the five-term exact sequence that we can read off the second page of a spectral sequence for a double complex (this is Eisenbud Proposition A3.25). In this case, we get the exact sequence $$ \Tor_2^R(k,k)\to\Tor_2^S(k\otimes_R S,k)\to\Tor_1^R(S,k)\otimes_S k\to\Tor_1^R(k,k)\to\Tor_1^S(k\otimes_R S,k)\to 0. $$ But I can’t see how to show that this should be the same as the five-term exact sequence that I’m supposed to be deriving, unless we can somehow show that (among other things) $k=k\otimes_R S$, or at the very least, say, that $\Tor_2^S(k\otimes_R S,k)=\Tor_2^S(k,k)$. It is worth noting, however, that so far I haven’t made any use of the fact that $S$ surjects onto $k$, so it seems likely that this will be an important component of whatever argument it is that I’m missing.
Is there any way of getting from the sequence that I’ve written down to the one I’m supposed to be getting? Or do I need another approach/have I written down the wrong spectral sequence?