Shapiro's lemma

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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.

Contents 1 Statement for rings 2 Statement for group rings 3 Statement for group cohomology 4 References

[edit] 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:

• If S is projective as a left R-module, then:

See (Benson 1991, p. 47).

[edit] 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 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:

When n = 0, this is called Frobenius reciprocity for completely reducible modules, and Nakayama reciprocity in general. See (Benson 1991, p. 60), which also contains these higher versions of the Mackey decomposition.

[edit] 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 isa representation of H. For N^G the induced representation of N from H to G using the tensor product, and for H_* the group homology:

H_*(G, N^G) = H_*(H, N)

Similarly, for N^G the co-induced representation of N from H to G using the Hom functor, and for H^* the group cohomology:

H^*(G, N^G) = H^*(H, N)

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

See (Weibel, p. 172).

[edit] References

• Benson, D. J. (1991), Representations and cohomology. I, Cambridge Studies in Advanced Mathematics, 30, Cambridge University Press, MR1110581, ISBN 978-0-521-36134-7 
• Page 59 of Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2000), Cohomology of Number Fields, Grundlehren der Mathematischen Wissenschaften, 323, Berlin: Springer-Verlag, MR1737196, ISBN 978-3-540-66671-4 
• Weibel, Charles A. (1994), An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, 38, Cambridge University Press, MR1269324, ISBN 978-0-521-55987-4, OCLC 36131259