Five-term exact sequence

The 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

E₂^p,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 → E₂^1,0 → H ¹(A) → E₂^0,1 → E₂^2,0 → H ²(A).

Here, the map E₂^0,1 → E₂^2,0 is the differential of the E₂-term of the spectral sequence.

Example

• The inflation-restriction exact sequence

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 group cohomology arises as the five-term exact sequence associated to the Lyndon–Hochschild–Serre spectral sequence

H ^p(G/N, H ^q(N, A)) ⇒ H ^p+q(G, A)

where G is a profinite group, N is a closed normal subgroup, and A is a G-module.

References

• Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2000), Cohomology of Number Fields, Grundlehren der Mathematischen Wissenschaften, vol. 323, Berlin: Springer-Verlag, ISBN 978-3-540-66671-4, MR 1737196, Zbl 0948.11001

• Weibel, Charles A. (1994). An introduction to homological algebra. Cambridge Studies in Advanced Mathematics. Vol. 38. Cambridge University Press. ISBN 978-0-521-55987-4. MR 1269324. OCLC 36131259.