Editing Spectral sequence (section)

=== First-quadrant sheet ===
Consider a spectral sequence where <math>E_r^{p,q}</math> vanishes for all <math> p </math> less than some <math> p_0 </math> and for all <math> q </math> less than some <math> q_0 </math>.  If <math> p_0 </math> and <math> q_0 </math> can be chosen to be zero, this is called a '''first-quadrant spectral sequence'''.
The sequence abuts because <math> E_{r+i}^{p,q} = E_r^{p,q} </math> holds for all <math> i\geq 0 </math> if <math> r>p </math> and <math> r>q+1 </math>. To see this, note that either the domain or the codomain of the differential is zero for the considered cases. In visual terms, the sheets stabilize in a growing rectangle (see picture above). The spectral sequence need not degenerate, however, because the differential maps might not all be zero at once. Similarly, the spectral sequence also converges if <math>E_r^{p,q}</math> vanishes for all <math> p </math> greater than some <math> p_0 </math> and for all <math> q </math> greater than some <math> q_0 </math>.