Differentiable stack

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.

See also

• Six operations

References

• J.P. May, Notes on Tor and Ext

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