As we know, most of the spectral sequences are doubly graded. However, this "doubly graded" condition is not a part of the formal definition of spectral sequence. Is there any useful triply (quadruply, quintuply, etc.) graded spectral sequences? If not, is there a hope that some meaningful work can be done with this topic?

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Here be dragons! – Mariano Suárez-Alvarez Apr 10 '12 at 2:48
(It is a bit strange to have accepted an answer so quickly... I surely don't doubt Peter's spectral sequence is a great example, but if you wait a little more, other examples could come up —and it is more or less natural that not having an accepted answer will result in the question being more attractive to potential answerers) – Mariano Suárez-Alvarez Apr 10 '12 at 4:00
I am still new to this website and am not sure about many of its functions. Will wait more as you have suggested; thank you! – Zuriel Apr 10 '12 at 4:08

Well, there is an eponymous spectral sequence in my thesis (still never published in full, but there is an announcement, stuff about it in Ravenel's book, and papers by Tangora and others). Quite generally, take a connected graded algebra $A$ over a field $k$, filter it for example by the powers of its augmentation ideal $IA$ (the elements of positive degree). Then the associated graded algebra $E^0A$ is bigraded. The filtration leads to a spectral sequence that converges from $Ext_{E^0A}(k,k)$ to $Ext_A(k,k)$. There is a homological grading and a bigrading from $E^0A$ that give a trigrading.

Incidentally, in the opposite direction, Bockstein spectral sequences of spaces are monograded.

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Of course this has been studied. You need but google "trigraded complex", and much wisdom will be found. OK, some wisdom, notably Ravenel's Complex Cobordism and Stable Homotopy Groups of Spheres, and a very cute presentation by Noah Forman on

Graham Denham. The combinatorial laplacian of the tutte complex. J. Algebra, 242(1):160-175, 2001.

and related matters.

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As others have said, there are plenty of examples of spectral sequences that have a third grading such that each $d_r$ preserves the grading up to a shift depending only on $r$. This is all fairly straightforward.

However, there are some more interesting questions along the same lines. When Ravenel was trying to disprove the Telescope Conjecture he had certain spectra with a doubly-indexed filtration, say $X_{ij}$. Given a slope $m>0$ one can define a singly-indexed filtration by $F_kX=\bigcup_{i+mj\geq k}X_{ij}$, and using this we obtain a spectral sequence converging to $\pi_*(X)$ (the "localised parametrised Adams spectral sequence"). Ravenel's approach was to analyse how this changes when we vary $m$. For a long time he was asking whether there was some kind of spectral-sequency gadget that incorporated all values of $m$ at the same time. I don't think anyone ever had a satisfactory answer to that.

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Hi there! Funny enough, I was going to answer a similar question. My background is mostly differential geometric, and I'm interested in spectral sequences arising in the context of lagrangian formalism and variational calculus. Back in 1957, Dedecker mentioned a spectral system associated to a "bifiltration" (Calcul des variations et topology algébrique, page 82), which may be seen as a "trigraded spectral sequence". I'm aware that by googling "trigraded spectral sequence" one gets a lot of results, but it's hard to find a place where main definitions and basic results of the theory of multi-graded spectral sequences (if any) are concisely collected. So, I humbly post my own concerns:

• If, in the standard construction of a spectral sequence out of a filtered complex, we replace the filtered complex by a more general object, do we still get interesting results? By a "more general object" I mean an analogous of a filtered complex, where the indexing set is not necessarily totally ordered (e.g., a bifiltered complex), and by "interesting results" I mean analogues to the converging theorem and/or the geometrical interpretation in the context of fibered spaces.
• If such a theory is known and developed, where I can find some reference? I would like to get to know the state of the art in this matter.
• In the particular case when the main complex is a differential algebra (say, the de Rham complex of a smooth manifold) and such a complex is bi-filtered by the powers of two distinct differential ideals, is it known how to construct a trigraded spectral sequence out of these data and, possibly, how to interpret it geometrically?

Any sort of hint will be warmly welcome.

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J. L. Verdier dissertation covers multiple-graded complexes.

Des catégories dérivées des catégories abéliennes. (French. French summary) [On derived categories of abelian categories] With a preface by Luc Illusie. Edited and with a note by Georges Maltsiniotis. Astérisque No. 239 (1996), xii+253 pp. (1997).

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