Two spectral sequences arising from a simplicial spectrum

Question

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.

Answer 1

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.