Five-term exact sequence
Encyclopedia
In mathematics, five-term exact sequence or exact sequence of low-degree terms is a sequence
of terms related to the first step of a spectral sequence
.
More precisely, let
be a spectral sequence, whose terms are non-trivial only for p, q ≥ 0.
Then there is an exact sequence
Here, the map E20,1 → E22,0 is the differential of the E2-term of the spectral sequence.
Exact sequence
An exact sequence is a concept in mathematics, especially in homological algebra and other applications of abelian category theory, as well as in differential geometry and group theory...
of terms related to the first step of a spectral sequence
Spectral sequence
In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations...
.
More precisely, let
- E2p,q ⇒ H n(A)
be a spectral sequence, whose terms are non-trivial only for p, q ≥ 0.
Then there is an exact sequence
- 0 → E21,0 → H 1(A) → E20,1 → E22,0 → H 2(A).
Here, the map E20,1 → E22,0 is the differential of the E2-term of the spectral sequence.
Example
- The inflation-restriction exact sequenceInflation-restriction exact sequenceIn mathematics, the inflation-restriction exact sequence is an exact sequence occurring in group cohomology and is a special case of the five-term exact sequence arising from the study of spectral sequences....
-
- 0 → H 1(G/N, AN) → H 1(G, A) → H 1(N, A)G/N → H 2(G/N, AN) →H 2(G, A)
- in group cohomologyGroup cohomologyIn abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper, group cohomology is a way to study groups using a sequence of functors H n. The study of fixed points of groups acting on modules and quotient modules...
arises as the five-term exact sequence associated to the Lyndon–Hochschild–Serre spectral sequenceLyndon–Hochschild–Serre spectral sequenceIn mathematics, especially in the fields of group cohomology, homological algebra and number theory the Lyndon spectral sequence or Hochschild–Serre spectral sequence is a spectral sequence relating the group cohomology of a normal subgroup N and the quotient group G/N to the cohomology of the...
- H p(G/N, H q(N, A)) ⇒ H p+q(G, A)
- where G is a profinite group, N is a closed normal subgroupNormal subgroupIn abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
, and A is a G-moduleG-moduleIn mathematics, given a group G, a G-module is an abelian group M on which G acts compatibly with the abelian group structure on M. This widely applicable notion generalizes that of a representation of G...
.