In this notes (pg 4): http://www.math.uchicago.edu/~may/MISC/SpecSeqPrimer.pdf, it is written that "complete knowledge of the Bockstein spectral sequences of $C$ for all primes $p$ allows a complete description of $H_*(C)$ as a graded abelian group."
I don't really understand this part. The Bockstein spectral sequence converges to $E^\infty\cong(H_*(C)/torsion)\otimes F_p$ for each prime $p$. Hence, I can see how knowing the Bockstein spectral sequence can allow us to know the free part of $H_*(C)$, but how can it recover the torsion part since it is not reflected in the $E^\infty$?
Thanks for any enlightenment.
What the author wants is really a complete knowledge of the Bockstein Spectral sequence. That is, all the differentials on all the pages for each prime $p$, rather than just its abutment.
The module to which the sequence converges for each prime really only contains the rational homology of the complex, albeit stated in a manner specific to the given prime.
The point is that for each prime $p$ the BSS takes as input the module $H_*(C;\mathbb{Z}_p)$ and a complete knowledge of its differentials will determine the orders of the $p$-torsion in $H_*(C;\mathbb{Z}_p)$. The first miracle is that this information can be lifted to determine the orders of the $p$-torsion in the integral complex $H_*(C)$. The second miracle is that the information on the torsion orders given for two distinct primes $p$, $p'$ is compatible, and we can take the information from each of their BSSs independently to get the $p$- and $p'$-torsion in $H_*(C)$.
Since the abutement of any BSS essentially tells us the rational homology $H_*(C;\mathbb{Q})$, we get the number of free summands in the complex. Using the information from the BSS for each prime $p$ we get the order of the p-torsion, and in which degrees it lies in the complex. We then assemble all this information to get the integral complex $H_*(C)$. Hence the author's statement.