we find that: In general, the two gradings on Hp+q(T(C•,•)) are distinct. Despite this, it is still possible to gain useful information from these two spectral sequences. Commutativity of Tor Let R be a ring, let M be a right R-module and N a left R-module. Recall that the derived functors of the tensor product are denoted Tor. Tor is defined using a projective resolution of its first argument. However, it turns out that . While this can be verified without a spectral sequence, it is very easy with spectral sequences. Choose projective resolutions and of M and N, respectively. Consider these as complexes which vanish in negative degree having differentials d and e, respectively. We can construct a double complex whose terms are and whose differentials are and . (The factor of −1 is so that the differentials anticommute.) Since projective modules are flat, taking the tensor product with a projective module commutes with taking homology, so we get: Since the two complexes are resolutions, their homology vanishes outside of degree zero. In degree zero, we are left with In particular, the terms vanish except along the lines q = 0 (for the I spectral sequence) and p = 0 (for the II spectral sequence). This implies that the spectral sequence degenerates at the second sheet, so the E∞ terms are isomorphic to the E2 terms: Finally, when p and q are equal, the two right-hand sides are equal, and the commutativity of Tor follows. Worked-out examples First-quadrant sheet Consider a spectral sequence where vanishes for all less than some and for all less than some . If and can be chosen to be zero, this is called a first-quadrant spectral sequence. The sequence abuts because holds for all if and . 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 vanishes for all greater than some and for all greater than some . 2 non-zero adjacent columns Let be a homological spectral sequence such that for all p other than 0, 1. Visually, this is the spectral sequence with -page The differentials on the second page have degree (-2,