Change of rings

In algebra, given a ring homomorphism $f: R \to S$ , there are three ways to change the coefficient ring of a module; namely, for a right R-module M and a right S-module N,

$f_! M = M \otimes_R S$ , the induced module.
$f_* M = \operatorname{Hom}_R(S, M)$ , the coinduced module.
$f^* N = N_R$ , the restriction of scalars.

They are related as adjoint functors:

f_! : \text{Mod}_R \leftrightarrows \text{Mod}_S : f^*

and

f^* : \text{Mod}_S \leftrightarrows \text{Mod}_R : f_*.

This is related to Shapiro's lemma.

References

J.P. May, Notes on Tor and Ext

Change of rings

See also

References