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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 (mathematics)|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;<ref>{{citation|title=Differential algebra and algebraic groups|volume=54|series=Pure and applied mathematics|first=Ellis Robert|last=Kolchin|publisher=Academic Press|year=1973|isbn=978-0-12-417650-8|page=53|url=http://books.google.com/books?id=yDCfhIjka-8C&pg=PA53}}.</ref> however, [[Beno Eckmann]] had discovered it earlier, in 1953.<ref>{{citation|series=Lectures Notes in Mathematics|volume=1758|year=2001|title=Continuous Bounded Cohomology of Locally Compact Groups|first=Nicolas|last=Monod|publisher=Springer-Verlag|contribution=Cohomological techniques|pages=129–168|doi=10.1007/3-540-44962-0_5|isbn=978-3-540-42054-5}}.</ref>


==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:
:<math>\operatorname{Ext}^n_R(N, {}_R M) \cong \operatorname{Ext}^n_S(S \otimes_R N, M)</math>
* If ''S'' is projective as a left ''R''-module, then:
:<math>\operatorname{Ext}^n_R({}_R M,N) \cong \operatorname{Ext}^n_S(M,\operatorname{Hom}_R(S,N))</math>


See {{harv|Benson|1991|p=47}}.  The projectivity conditions can be weakened into conditions on the vanishing of certain Tor- or Ext-groups: see {{harv|Cartan|Eilenberg|1956|p=118|loc=VI.&sect;5}}.
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==Statement for group rings==
When ''H'' is a subgroup of finite [[Index of a subgroup|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'' ⊗<sub>''R''</sub> ''N'' is called the [[induced representation]] of ''N'' from ''H'' to ''G'', and <sub>''R''</sub>''M'' is called the [[restricted representation]] of ''M'' from ''G'' to ''H''. One has that:
:<math>\operatorname{Ext}^n_G( M, N\uparrow_H^G) \cong \operatorname{Ext}^n_H( M\downarrow_H^G, N)</math>
 
When ''n'' = 0, this is called [[Frobenius reciprocity]] for completely reducible modules, and Nakayama reciprocity in general. See {{harv|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''<sup>''G''</sup> the [[induced representation]] of ''N'' from ''H'' to ''G'' using the [[tensor product]], and for H<sub>*</sub> the [[group homology]]:
:H<sub>*</sub>(''G'', ''N''<sup>''G''</sup>) = H<sub>*</sub>(''H'', ''N'')
Similarly, for ''N''<sup>''G''</sup> the co-induced representation of ''N'' from ''H'' to ''G'' using the [[Hom functor]], and for H<sup>*</sup> the [[group cohomology]]:
:H<sup>*</sup>(''G'', ''N''<sup>''G''</sup>) = H<sup>*</sup>(''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 {{harv|Weibel||p=172}}.
 
==Notes==
{{reflist}}
 
==References==
* {{Citation | last1=Benson | first1=D. J. | title=Representations and cohomology. I | publisher=[[Cambridge University Press]] | series=Cambridge Studies in Advanced Mathematics | isbn=978-0-521-36134-7 | id={{MathSciNet | id = 1110581}} | year=1991 | volume=30}}
* {{Citation | last1=Cartan | first1=H. |last2=Eilenberg | first2=S. | title=Homological Algebra | publisher=[[Princeton University Press]] | year=1956}}
*{{citation
| last = Eckmann | first = Beno | authorlink = Beno Eckmann
| mr = 0058600
| journal = Annals of Mathematics | series = 2nd ser.
| pages = 481–493
| title = Cohomology of groups and transfer
  | volume = 58
| year = 1953
| doi = 10.2307/1969749
| issue = 3}}.
* Page 59 of {{Neukirch et al. CNF}}
* {{Weibel IHA}}
 
[[Category:Homological algebra]]
[[Category:Representation theory]]

Latest revision as of 04:10, 12 May 2014


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