I have a fairly rudimentary understanding of spectral sequences. I have a couple questions though.
In Algebra, Lang states what seems to me, to be two slightly different notion of convergence of a spectral sequence.
He says that if $E_r^{p,q}=E_{r+1}^{p,q}=...$ then the spectral sequence converges to $E_{\infty}^{p,q}$.
But later he says that in the case of a double complex, or more generally a filtered complexe we usually say that the associated spectral sequence converges to the homology of the complex, to mean that the spectral sequence converges to the graded groups associated with the filtration induced on the homology of the complex. In other words we say that $E_r^{p,q}$ converges to $H^{p+q}(C_\bullet)$ to mean that $E_r^{p,q}$ converges to $Gr^p(H^{p+q}(C_\bullet)$.
This has caused me no trouble up until now, as I have only encountered spectral sequences coming from double complexes.
However I'm a bit puzzeled when I encounter that kind of sentence (this is the first example that comes to my mind, maybe that's not the most elementary).
There is a spectral sequence whose 2nd sheet is $H^{p-q, -q}(F, \mathbb{Z})$ that converges to $K_{-p-q}(F)$ for a field $F$.
Does that mean that $E_{\infty}^{p,q}=K_{-p-q}(F)$ (which I don't believe), or that there exist a filtration on $K_{-p-q}(F)$ whose associated graded groups are $E_{\infty}^{p,q}$. And if it's the latter, how do I find the filtration?
I'm a bit lost because in the proof, it is done by exhibiting a so-called "exact couple", and I don't see where the filtration comes into play, and how I should interpret the general statement of the theorem.