MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Given a double complex in the first quadrant, one can derive from it a (homological or cohomological) spectral sequence converging to the (co)homology of the total complex of the double complex.

My question is: When is a (homological or cohomological) spectral sequence coming from a double complex?

share|cite|improve this question
up vote 15 down vote accepted

There are two different ways to understand the question:

  1. If I see an abstract spectral seqeunce, is there a double complex such that its spectral sequence is isomorphic to the given spectral sequence? I do not have an answer to that question and, to be honest, do not believe it is an interesting question.

  2. For wich set of names ''$XY$''; the $XY$-spectral sequence can be derived from a double complex?

The answer is that, as a general rule (it might have exceptions), all $XY$-spectral sequences whose $E_2$-terms and $E_{\infty}$ terms are purely homological can be derived from filtered complexes; and most of them in fact from double complexes.


  1. The spectral sequence of a simplicial space (Segal; ''Classfying spaces and spectral sequences'') can be reformulated using a double complex (a simplicial space $X_{\bullet}$ gives rise to a simplicial chain complex $C_{\ast} X_{\bullet}$ and thus a double complex.

  2. The Serre spectral sequence is a special case of the above; a direct construction using a double complex was given by A. Dress, ''Zur Spectralsequenz von Faserungen''.

  3. Special cases of 2. include the Lyndon-Hochschild-Serre spectral sequ. for group extensions; special cases of 1. include the Bousfield-Kan spectral sequ. of a homotopy colimit and some others.

  4. The Eilenberg-Moore spectral sequence comes from a double complex.

  5. Purely algebraic versions: Grothendieck-spectral sequence. Probably the spectral sequence of a Lie algebra extension fits into here. The Van Est spectral sequence for Lie algebra cohomology also comes from a double complex.

The Bockstein spectral sequence is a purely homological construction, it can be derived from a filtered complex; but it does not seem to stem from a double complex. Other counterexamples are the typical spectral sequence of stable homotopy theory (Atiyah-Hirzebruch, Adams spectral sequence): they cannot be derived from filtered complexes. In fact, if $E$ is a generalized homology theory, you cannot write $E_{\ast} (X)$ of a space in a sensible way as the homology of a chain complex functorially associated with $X$.

share|cite|improve this answer
Thanks a lot! The couterexamples are interesting. – Hiro Nov 2 '12 at 16:59

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.