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 ⇒ Hⁿ(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 → H¹(A) → E₂^0,1 → E₂^2,0 → H²(A).

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

References

• 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.