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. Shapiro's lemma is named after Arnold Shapiro, who proved it in 1961; however, Beno Eckmann had discovered it earlier, in 1953.

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.

See (Benson 1991, p. 47). The projectivity conditions can be weakened into conditions on the vanishing of certain Tor- or Ext-groups: see (Cartan & Eilenberg 1956, p. 118, VI.§5).

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 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 ⊗_RN 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:

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.

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