$ \newcommand\nil{\mathrm{nil}} \newcommand\gfrak{\mathfrak{g}} \newcommand\A{\mathrm{A}} \newcommand\B{\mathrm{B}} \newcommand\C{\mathrm{C}} \newcommand\D{\mathrm{D}} \newcommand\E{\mathrm{E}} \newcommand\F{\mathrm{F}} \newcommand\G{\mathrm{G}} \newcommand\H{\mathrm{H}} \newcommand\h{\mathrm{h}} \newcommand\K{\mathrm{K}} \newcommand\L{\mathrm{L}} \newcommand\M{\mathrm{M}} \newcommand\N{\mathrm{N}} \newcommand\R{\mathrm{R}} \newcommand\t{\mathrm{t}} \newcommand{\bA}{\mathbf{A}} \newcommand{\bG}{\mathbf{G}} \newcommand{\bH}{\mathbf{H}} \newcommand{\bT}{\mathbf{T}} \newcommand{\bW}{\mathbf{W}} \newcommand{\Gm}{\bG_m} \newcommand\Ascr{\mathcal{A}} \newcommand\Cscr{\mathcal{C}} \newcommand\Dscr{\mathcal{D}} \newcommand\Escr{\mathcal{E}} \newcommand\Fscr{\mathcal{F}} \newcommand\Kscr{\mathcal{K}} \newcommand\Lscr{\mathcal{L}} \newcommand\Oscr{\mathcal{O}} \newcommand\Perf{\mathrm{Perf}} \newcommand\Perfscr{\mathcal{P}\mathrm{erf}} \newcommand\Acscr{\mathcal{A}\mathrm{c}} \newcommand\heart{\heartsuit} \newcommand\cn{\mathrm{cn}} \newcommand\op{\mathrm{op}} \newcommand\gr{\mathrm{gr}} \newcommand\Gr{\mathrm{Gr}} \newcommand\fil{\mathrm{fil}} \newcommand\Ho{\mathrm{Ho}} \newcommand\dR{\mathrm{dR}} \newcommand\dRhat{\widehat{\dR}} \newcommand\we{\simeq} \newcommand\Sym{\mathrm{Sym}} \newcommand\HH{\mathrm{HH}} \newcommand\HC{\mathrm{HC}} \newcommand\HP{\mathrm{HP}} \newcommand\TC{\mathrm{TC}} \newcommand{\bMap}{\mathbf{Map}} \newcommand{\End}{\mathrm{End}} \newcommand{\Mod}{\mathrm{Mod}} \newcommand{\coMod}{\mathrm{coMod}} \newcommand{\Fun}{\mathrm{Fun}} \newcommand{\bMap}{\mathbf{Map}} \newcommand\bE{\mathbf{E}} \newcommand\bZ{\mathbf{Z}} \newcommand\bQ{\mathbf{Q}} \newcommand\bC{\mathbf{C}} \newcommand\bAM{\mathbf{AM}} \newcommand\bLM{\mathbf{LM}} \newcommand\Spec{\mathrm{Spec}\,} \newcommand\CAlg{\mathrm{CAlg}} \newcommand\aCAlg{\mathfrak{a}\CAlg} \newcommand\dCAlg{\mathfrak{d}\CAlg} \newcommand{\Cat}{\mathrm{Cat}} \newcommand{\Sscr}{\mathcal{S}} \newcommand{\poly}{\mathrm{poly}} \newcommand{\perf}{\mathrm{perf}} \newcommand{\Dec}{\mathrm{Dec}} $

I am finally writing about a paper [1] I put on the arXiv last year about spectral sequences. This paper gives a new construction of all pages of the spectral sequence of a filtered object in a stable $\infty$-category equipped with a $t$-structure.

The idea is quite simple. Let $\Cscr$ be a stable $\infty$-category which admits sequential limits and colimits. Suppose that $\Cscr$ moreover admits a $t$-structure. On the $\infty$-category $\F\Cscr=\Fun(\bZ^\op,\Cscr)$ of decreasing filtrations in $\Cscr$ there is a $t$-structure where an object $\F^\star$ is connective, i.e., in $\F\Cscr_{\geq 0}^\B$ if and only if $\gr^i\in\Cscr_{\geq -i}$, the latter with respect to the given $t$-structure above. This is called the Beilinson $t$-structure and has been considered in [2], [3], and [9]. The heart of this $t$-structure is equivalent to the abelian category of cochain complexes in $\Cscr^\heart$. It is not difficult to see that the homotopy objects $\pi_i^\B(\F^\star)$ are precisely the (co)chain complexes appearing on the $\E^1$-page of the spectral sequence associated to $\F^\star$.

The paper [1] provides an answer to the question of how to coherently construct the other pages. I use the $t$-structure to define an endofunctor of $\F\Cscr$, called $\Dec$. By definition, given an object $\F^\star\in\F\Cscr$ there is a Whitehead tower

\[\cdots\rightarrow\tau_{\geq n+1}^\B(\F^\star)\rightarrow\tau_{\geq n}^\B(\F^\star)\rightarrow\tau_{\geq n-1}^\B(\F^\star)\rightarrow\cdots.\]

Let $|-|\colon\F\Cscr\rightarrow\Cscr$ denote the colimit functor, which exists as we assume that $\Cscr$ admits sequential colimits. We can apply $|-|$ to the Whitehead tower to obtain a new decreasing filtration

\[\cdots\rightarrow|\tau_{\geq n+1}^\B(\F^\star)|\rightarrow|\tau_{\geq n}^\B(\F^\star)|\rightarrow|\tau_{\geq n-1}^\B(\F^\star)|\rightarrow\cdots.\]

This is $\Dec(\F^\star)$. More generally, we can iterate to obtain $\Dec^{(n)}(\F^\star)$.

This leads to a simple definition of all pages.

Definition. For $n\geq 1$, the $\E^n$-page of the spectral sequence attached to $\F^\star$ is the $\E^1$-page of the spectral sequence of $\Dec^{(n-1)}(\F^\star)$, with some reindexing.

The difficult part of the paper is the proof of the following result.

Theorem. This agrees with the definition of the higher pages given in [8].

One reason I find this interesting is that in the presence of a symmetric monoidal structure on $\Cscr$, and assuming that the $t$-structure on $\Cscr$ is compatible with the symmetric monoidal structure, one finds that $\Dec$ is lax symmetric monoidal. From this, one can easily prove multiplicativity results about all pages of a spectral sequence.

Another reason is that it makes it straightforward to compare, say, the two standard definitions of the Atiyah-Hirzebruch spectral sequence computing the $E$-cohomology of a space $X$, one using the Whitehead tower of $E$ and the other using a cellular filtration on $X$.

This work did not arise in a vacuum and some of our results are closely related to work of Deligne [4], Lawson [6], and Levine [7]. Some of the ideas also were explained in the PhD thesis [5] of Alice Hedenlund.

References

[1] Antieau, Spectral sequences, décalage, and the Beilinson t-structure, arXiv:2411.09115.

[2] Ariotta, Coherent cochain complexes and Beilinson $t$-structures, with an appendix by Achim Krause, arXiv:2109.01017.

[3] Bhargav Bhatt, Matthew Morrow, and Peter Scholze, Topological Hochschild homology and integral p-adic Hodge theory, Publ. Math. Inst. Hautes Études Sci. 129 (2019), 199–310.

[4] Deligne, Théorie de Hodge. II, Inst. Hautes Études Sci. Publ. Math. (1971), no. 40, 5–57.

[5] Hedenlund, Multiplicative Tate spectral sequences, PhD Thesis at Oslo.

[6] Lawson, Filtered spaces, filtered objects, arXiv:2410.08348.

[7] Levine, The Adams-Novikov spectral sequence and Voevodsky’s slice tower, Geom. Topol. 19 (2015), no. 5, 2691–2740.

[8] Lurie, Higher algebra, version dated 18 September 2017.

[9] Raksit, Hochschild homology and the derived de Rham complex revisited, arXiv:2007.025760.