# 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
${R}_{1}\mathrm{Mod}\stackrel{{\mathrm{Hom}}_{{R}_{1}\mathrm{Mod}}\left(-,{R}_{2}\right)}{\to }{R}_{2}\mathrm{Mod}\stackrel{{\mathrm{Hom}}_{{R}_{2}}\left(-,N\right)}{\to }\mathrm{Ab}\phantom{\rule{thinmathspace}{0ex}}.$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 ${\mathrm{Hom}}_{{R}_{2}}\left(-,N\right)$ 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.

Created on October 29, 2012 20:17:14 by Urs Schreiber (131.174.188.167)