Shapiro's lemma

Statement for group rings

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. Let M be a finite-dimensional representation of G and N a finite-dimensional representation of H. In this case, the module S ⊗_R N is called the induced representation of N from H to G, and _RM is called the restricted representation of M from G to H. One has that:

${\displaystyle {\mathrm{Ext}}_G^n(M,N{\uparrow }_H^G)\cong {\mathrm{Ext}}_H^n(M{\downarrow }_H^G,N)}$

When n = 0, this is called Frobenius reciprocity for completely reducible modules, and Nakayama reciprocity in general. See (Benson 1991, p. 42), which also contains these higher versions of the Mackey decomposition.

Statement for group cohomology

Specializing M to be the trivial module produces the familiar Shapiro's lemma. Let H be a subgroup of G and N a representation of H. For N^G the induced representation of N from H to G using the tensor product, and for H_{${\displaystyle \ast }$} the group homology:

H_{${\displaystyle \ast }$}(G, N^G) = H_{${\displaystyle \ast }$}(H, N)

Similarly, for N^G the co-induced representation of N from H to G using the Hom functor, and for H^{${\displaystyle \ast }$} the group cohomology:

H^{${\displaystyle \ast }$}(G, N^G) = H^{${\displaystyle \ast }$}(H, N)

When H has finite index in G, then the induced and coinduced representations coincide and the lemma is valid for both homology and cohomology.

See (Weibel 1994, p. 172).