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.
When H is a subgroup of finite index in G, then the group ring R[G] is finitely generated projective as a left and right R[H] module, so the previous theorem applies in a simple way.
See (Benson 1991, p. 42), which also contains these higher versions of the Mackey decomposition.
Specializing M to be the trivial module produces the familiar Shapiro's lemma.
the group homology: Similarly, for NG the co-induced representation of N from H to G using the Hom functor, and for H