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base change spectral sequence

For RR a ring write RRMod for its category of modules. Given a homomorphism of ring f:R 1R 2f : R_1 \to R_2 and an R 2R_2-module NN there are composites of base change along ff with the hom-functor and the tensor product functor

R 1Mod R 1R 2R 2Mod R 2NAb R_1 Mod \stackrel{\otimes_{R_1} R_2}{\to} R_2 Mod \stackrel{\otimes_{R_2} N}{\to} Ab
R 1ModHom R 1Mod(,R 2)R 2ModHom R 2(,N)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 Hom R 2(,N)Hom_{R_2}(-,N) and R 2N\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)