Shapiro's lemma

In mathematics, especially in the areas of abstract algebra dealing with group cohomology or relative homological algebra, Shapiro's lemma, also known as the Eckmann–Shapiro lemma, relates extensions of modules over one ring to extensions over another, especially the group ring of a group and of a subgroup. It thus relates the group cohomology with respect to a group to the cohomology with respect to a subgroup.

Statement for rings

Let "R" → "S" be a ring homomorphism, so that "S" becomes a left and right "R"-module. Let "M" be a left "S"-module and "N" a left "R"-module. By restriction of scalars, "M" is also a left "R"-module.
* If "S" is projective as a right "R"-module, then::$operatorname\{Ext\}^n\_R(N,\; \{\}\_R\; M)\; cong\; operatorname\{Ext\}^n\_S(S\; otimes\_R\; N,\; M)$
* If "S" is projective as a left "R"-module, then::$operatorname\{Ext\}^n\_R(\{\}\_R\; M,N)\; cong\; operatorname\{Ext\}^n\_S(M,operatorname\{Hom\}\_R(S,N))$

See .

References

* | year=1991 | volume=30
* Neukirch, Schmidt, Wingberg: "Cohomology of Number Fields", page 59.
* | year=1994