Spectral sequence $B^1_{p,q}= B^\infty_{p,q}$ definition, nlab

In an introductory notes to spectral sequences, nlab, Definition 1.26 we define

$$B^r_{p,q} = \partial(F_{p+r-1} C_{p+q+1})$$

and

$$B^\infty_{p,q}= \partial(F_{p}C_{p+q+1})$$

So what is the difference between $$B^1_{p,q}$$ and $$B^\infty_{p,q}$$? Or any references would help!

• They must want $\cup_{r = 0}^{\infty} B_{p,q}^{r}$ or something for $B_{p,q}^{\infty}$. I agree that I can't make sense of the part you've included in the question (although this is not my area). – user113102 Mar 29 at 20:16
• I don't know what the nlab meant. I won't answer definitively because I'm not an expert and some people will probably understand better what the nlab is saying, but one definition is to go back to $E^1_{p,q}$ : you can somehow see every $B^r_{p,q}$ as a quotient of a submodule (or subobject if you're woriking in an abelian category) of $E^1_{p,q}$, and so you can lift it to the corresponding submodule of $E^1_{p,q}$ and put $B^\infty_{p,q}$ as the union of all of these (similarly for $Z^\infty_{p,q}$ but you're taking the intersection) – Max Mar 30 at 17:49
• Yea, of the references I know $B^\infty_{p,q}$ is the union, but this doesn't coincide with this one in nlab... – CL. Mar 30 at 17:57
• Based on consulting Weibel p.133 (supposedly the source of this), it should be $B^\infty_{p,q} = \partial (C_{p+q+1}) \cap F_pC/F_{p-1}C$. The definition given above is imprecise. $B^r_{p,q}$ lives inside $F_pC/F_{p-1}C$. – Justin Young Apr 2 at 16:57
• It is confirmed (nforum.ncatlab.org/discussion/2402/…) that it is a typo and that the page needs cleaning up. – user113102 Apr 8 at 14:32