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Let $X_\bullet$ be a simplicial spectrum, and let $X = |X_\bullet|$ be the geometric realization. Let's assume each $X_n$ is connective.

From this situation, we can form two filtrations on $X$: the skeletal filtration and the levelwise Postnikov filtration. You can probably guess what these will be, but let me spell it out for my own sake.

The skeletal filtration $$\mathrm{Fil}^\mathrm{sk}_*(X) = \{ 0 \to \mathrm{sk}^0(X_\bullet) \to \mathrm{sk}^1(X_\bullet) \to \mathrm{sk}^2(X_\bullet) \to \cdots \}$$ is an increasing filtration with graded pieces $\mathrm{gr}^\mathrm{sk}_s(X)= \mathrm{Fil}^\mathrm{sk}_s(X)/\mathrm{Fil}^\mathrm{sk}_{s-1}(X) = \Sigma^s X_s$. The resulting spectral sequence has signature $$E^1_{n,s}(\mathrm{sk}) = \pi_{n-s}(X_s) \implies \pi_n(X)$$ with differentials $d^r : E^r_{n,s} \to E^r_{n-1,s-r}$. The second page is quite nice: we find that $E^2_{n,s}(\mathrm{sk}) = \mathrm{H}_s(\pi_{n-s}(X_\bullet))$, where for each $j \in \mathbb{Z}$ we're viewing $\pi_j(X_\bullet)$ as a chain complex via Dold-Kan.

On the other hand, the levelwise Postnikov filtration $$\mathrm{Fil}^\mathrm{LP}_*(X) = \{ \cdots \to |\tau_{\geq 2} (X_\bullet)| \to |\tau_{\geq 1} (X_\bullet)| \to |\tau_{\geq 0} (X_\bullet)| = X\}$$ is a decreasing filtration with graded pieces $\mathrm{gr}^\mathrm{LP}_s(X) = \mathrm{Fil}^\mathrm{LP}_s(X)/\mathrm{Fil}^\mathrm{LP}_{s+1}(X) = \Sigma^s|\pi_s (X_\bullet)|$. The resulting spectral sequence has signature $$E^1_{n,s}(\mathrm{LP}) = \mathrm{H}_{n-s}(\pi_s(X_\bullet)) \implies \pi_n(X)$$ with differentials $d^r : E^r_{n,s} \to E^r_{n-1,s+r}$. Note in particular that the $E^1$ page here is the $E^2$ page coming from the skeletal filtration, up to a shift $s \mapsto n-s$.

Finally, the

Question. What exactly is the relationship between the skeletal filtration and the levelwise Postnikov filtration?

In particular, I'm looking for an explanation for the essentially matching spectral sequence pages.

I suspect the key is hidden in the fact that the two filtrations are two "edges" of a single diagram $\mathbb{N} \times \mathbb{N}^\mathrm{op} \to \mathrm{Spt}$, $(a,b) \mapsto \mathrm{sk}_a(\tau_{\geq b} (X_\bullet))$. Restricting to $b=0$ yields the skeletal filtration, while taking the colimit over $a$ yields the levelwise Postnikov filtration. I imagine this gives some way to interpolate between the two spectral sequences.

More generally, I imagine the various spectral sequences one can extract from a bilfiltered spectrum $M \in \mathrm{Fun}(\mathbb{Z},\mathrm{Fun}(\mathbb{Z},\mathrm{Spt}))$ have some sort of relationship, though maybe not as straightforward as shifting and shearing pages. In general, it seems our understanding of the various spectral sequences arising from a bifiltered spectrum is not quite at a satisfactory state. See e.g. this other question. I am very much interested in any developments on this front.

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The precise relation between the skeleton filtration and the levelwise Postnikov filtration is that the décalage of the first is isomorphic to the second. This is explained in [Ariotta: Coherent cochain complexes and Beilinson t-structures, §9]. As indicated in loc. cit., this should be explained in Hedenlund–Krause–Nikolaus, but this paper does not seem to appear, let me try to explain my understanding briefly, hopefully correct.

Let $F^\ast X$ be a filtered spectrum. Recall that there is a Beilinson Postnikov filtration $\tau_{\ge\star}^BF^\ast X$ on $F^\ast X$, which is a $(\ast,\star)$-bifiltered spectrum. The décalage of the filtered spectrum $F^\ast X$ is the underlying filtered spectrum obtained by taking colimit along $\ast\in\mathbb N$.

Under the Dold–Kan correspondence, the Beilinson t-structure on filtered spectra corresponds to the levelwise t-structure on simplicial spectra. Unraveling definitions, you see that, the décalage of the skeleton filtration becomes the levelwise Postnikov filtration.

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  • $\begingroup$ Ah, I see. The bifiltered spectrum $(a,b) \mapsto \mathrm{sk}^a(\tau_{\geq b}(X_\bullet))$ is exactly the Beilinson Postnikov filtration of the skeletal filtration. Neat, thanks! $\endgroup$
    – Brian Shin
    Dec 25, 2023 at 2:21
  • $\begingroup$ The original reference for this kind of result is probably: C.R.F. Maunder, "The spectral sequence of an extraordinary cohomology theory", Proc. Cambridge Philos. Soc. 59 (1963), 567–574, showing that the exact couple associated to the Postnikov tower is the first derived of the exact couple associated to the skeleton filtration. $\endgroup$ Dec 26, 2023 at 20:01
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    $\begingroup$ For the spectral systems arising from filtrations indexed by more-or-less arbitrary partially ordered sets, including $(\mathbb{Z}, \le)^n$ with $n=2$, see arxiv.org/abs/1308.3187 and arxiv.org/abs/2107.02130 by Benjamin Matschke. $\endgroup$ Dec 26, 2023 at 20:49
  • $\begingroup$ @JohnRognes I guess that the statement on exact couples only tells us something about homotopy groups, not on the level of filtered spectra. I heard this point of view from those who work with synthetic spectra. $\endgroup$
    – Z. M
    Dec 27, 2023 at 20:22
  • $\begingroup$ @Z.M Yes, Maunder's paper only answers the "I'm looking for an explanation for the essentially matching spectral sequence pages" part of the question. Of course, the exact couples know about all later differentials and terms of the spectral sequence, which is more than "something about homotopy groups". I wanted to mention Maunder's paper since it seemed to have been forgotten for a while, e.g. by Greenlees and May when they wrote Appendix B in their Generalized Tate Cohomology AMS Memoir. $\endgroup$ Dec 27, 2023 at 22:04

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