When reading various literature on spectral sequences one always comes across two setups:

  • A chain complex with an increasing filtration
  • A cochain complex with a decreasing filtration

My question is why the other two options are never mentioned:

  • A chain complex with an decreasing filtration
  • A cochain complex with a increasing filtration

I worked through the construction of a spectral sequence associated to an increasing filtration on a cochain complex, and it seems to work just fine.

Is the reason for this convention in the literature just that the treated cases show up more frequently in practice or is there some deeper reason?

  • 2
    $\begingroup$ There is no reason but force of habit. Also if you change indices $i\to -i$ you can change between chain/cochain complexes and increasing/decreasing filtrations. $\endgroup$ – Donu Arapura Aug 15 '18 at 16:54
  • 2
    $\begingroup$ If you work with bi-infinite filtrations that subsumes both cases. In the case of decreasing your filtration you need to make sure your chain complex is complete with respect to this filtration or the spectral sequence computes something other than its homology. This is all well-explained in Boardman's spectral sequence paper. $\endgroup$ – Mike Miller Aug 15 '18 at 16:54

There is not a conceptual reason why increasing filtrations cannot happen on cochain complexes, or vice versa. A prominent example of this type of spectral sequence is the Eilenberg-Moore spectral sequence $$ Tor^{H^*(Z)}_{**}(H^* Y, H^* X) \Rightarrow H^*(Y \times_Z X) $$ for the cohomology groups of a (homotopy) pullback. Another one of this type, though it doesn't really arise from filtering a complex, is the Adams spectral sequence $$ Ext_{A^*}^{**}(H^* X, \Bbb Z/p) \Rightarrow \pi_*^s(X)^\wedge_p $$ computing the stable homotopy groups of $X$ using $Ext$ over the Steenrod algebra.

The most common spectral sequences can be derived from an expression of a space $X$ as the colimit of a sequence of subspaces, which naturally lends itself to an increasing chain filtration or a decreasing cochain filtration. On the other hand, many spectral sequences like the ones I just mentioned are second-quadrant or fourth-quadrant spectral sequences. They have attendant convergence issues that make them a little less easy to use (discussed at length in Boardman's paper that Mike Miller mentioned), and hence they are not as well known to nonspecialists.

  • $\begingroup$ Thanks for the answer. So is it true that if I have a first quadrant double (co)chain complex, I can in either case (chain/cochain) consider 4 filtrations: horizontal/vertical and increasing/decreasing, and these will all give spectral sequences converging to the (co)homology of the total complex? Here by convergence I just mean that after finitely many steps the terms stabilise to an object isomorphic to some $gr^i H^j(Tot)$. $\endgroup$ – user2520938 Aug 15 '18 at 21:36
  • $\begingroup$ You can certainly consider both the horizontal and vertical filtrations. If it is a double chain complex, you can make an increasing filtration, and if it is a cochain complex, you can make a decreasing filtration, simply generated by certain rows/columns. However, because of the constraint that your subcomplexes must be closed under the differential, if you want to construct a filtration in the other direction you must do it in a very different way. $\endgroup$ – Tyler Lawson Aug 15 '18 at 21:42
  • $\begingroup$ Of course, that was dumb. That actually also sort of justifies the focus on the two cases cochain+descreasing, chain+increasing for me, since in my application I will usually be dealing with this first quadrant double complex setup. Either way thanks for your time. $\endgroup$ – user2520938 Aug 15 '18 at 21:51

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