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

$R_1 Mod \stackrel{\otimes_{R_1} R_2}{\to} R_2 Mod \stackrel{\otimes_{R_2} N}{\to} Ab$

$R_1 Mod \stackrel{Hom_{R_1 Mod}(-,R_2)}{\to}
R_2 Mod
\stackrel{Hom_{R_2}(-,N)}{\to}
Ab
\,.$

The derived functors of $Hom_{R_2}(-,N)$ 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)