nLab
base change spectral sequence

For $R$ a ring write $R$Mod for its category of modules. Given a homomorphism of ring $f:R_1\to R_2$ and an $R_2$-module $N$ there are composites of base change along $f$ with the hom-functor and the tensor product functor

$$R_1\mathrm{Mod}\stackrel{{\otimes }_{R_1}{R}_2}{\to }{R}_2\mathrm{Mod}\stackrel{{\otimes }_{R_2}N}{\to }\mathrm{Ab}$$R_1 Mod \stackrel{\otimes_{R_1} R_2}{\to} R_2 Mod \stackrel{\otimes_{R_2} N}{\to} Ab

$$R_1\mathrm{Mod}\stackrel{{\mathrm{Hom}}_{R_1\mathrm{Mod}}(-,R_2)}{\to }{R}_2\mathrm{Mod}\stackrel{{\mathrm{Hom}}_{R_2}(-,N)}{\to }\mathrm{Ab}.$$R_1 Mod \stackrel{Hom_{R_1 Mod}(-,R_2)}{\to} R_2 Mod \stackrel{Hom_{R_2}(-,N)}{\to} Ab \,.

The derived functors of ${\mathrm{Hom}}_{R_2}(-,N)$ and ${\otimes }_{R_2}N$ are the Ext- and the Tor-functors, respectively, so the Grothendieck spectral sequence applied to these composites is the base change spectral sequence for these.