Cohomological Serre spectral sequence ($E_2^{p,q}=H^p(B;H^q(F;A))\Rightarrow H^{p+q}(E;A)$)

For a Serre fibration, this first-quadrant spectral sequence has differential of bidegree $(r,1-r)$ and a fiber-cohomology local coefficient system on the base. A simply connected base makes that system constant. Its edge map is restriction to the fiber; a surviving fiber class is the restriction of a total-space class. For commutative ring coefficients it is multiplicative and its differentials are graded derivations.