Künneth spectral sequence ($E_2^{p,q}={\mathrm{Tor}}_{-p}^R(H^{q}P,M)\Rightarrow H^{p+q}(P{\otimes }_{R}M)$)

For a bounded complex $P$ of projective right modules and a left module $M$ of finite projective dimension, a finite projective resolution of $M$ yields a bounded double cochain complex. Its two spectral sequences give the displayed abutment. Over a noncommutative ring, the handedness of the two tensor factors is essential.

If only the columns $p=-1,0$ are nonzero, a cohomological differential of bidegree $(r,1-r)$ has no possible nonzero source-target pair for $r\ge 2$. Projective boundaries in a complex of projectives give this condition because every cohomology module has projective dimension at most one.