In mathematics, the inflation-restriction exact sequence is an
exact sequence occurring in
group cohomology
In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology loo ...
and is a special case of the
five-term exact sequence arising from the study of
spectral sequences.
Specifically, let ''G'' be a
group, ''N'' a
normal subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G i ...
, and ''A'' an
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
which is equipped with an action of ''G'', i.e., a
homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
from ''G'' to the
automorphism group of ''A''. The quotient group ''G''/''N'' acts on
::''A''
''N'' = .
:
Then the inflation-restriction exact sequence is:
::0 → ''H''
1(''G''/''N'', ''A''
''N'') → ''H''
1(''G'', ''A'') → ''H''
1(''N'', ''A'')
''G''/''N'' → ''H''
2(''G''/''N'', ''A''
''N'') →''H''
2(''G'', ''A'')
:
In this sequence, there are maps
* ''inflation'' ''H''
1(''G''/''N'', ''A''
''N'') → ''H''
1(''G'', ''A'')
* ''restriction'' ''H''
1(''G'', ''A'') → ''H''
1(''N'', ''A'')
''G''/''N''
* ''transgression'' ''H''
1(''N'', ''A'')
''G''/''N'' → ''H''
2(''G''/''N'', ''A''
''N'')
* ''inflation'' ''H''
2(''G''/''N'', ''A''
''N'') →''H''
2(''G'', ''A'')
The inflation and restriction are defined for general ''n'':
* ''inflation'' ''H''
''n''(''G''/''N'', ''A''
''N'') → ''H''
''n''(''G'', ''A'')
* ''restriction'' ''H''
''n''(''G'', ''A'') → ''H''
''n''(''N'', ''A'')
''G''/''N''
The transgression is defined for general ''n''
* ''transgression'' ''H''
''n''(''N'', ''A'')
''G''/''N'' → ''H''
''n''+1(''G''/''N'', ''A''
''N'')
only if ''H''
''i''(''N'', ''A'')
''G''/''N'' = 0 for ''i'' ≤ ''n'' − 1.
[Gille & Szamuely (2006) p.67]
The sequence for general ''n'' may be deduced from the case ''n'' = 1 by dimension-shifting or from the
Lyndon–Hochschild–Serre spectral sequence.
[Gille & Szamuely (2006) p. 68]
References
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Homological algebra
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