The final XR of 2021! This is about one of my favorite recent theorems, which is due to Ariotta [1]. I call it the $\E_1$-page theorem. There is a lot in the paper, especially about Toda brackets and related questions, which I will ignore.

The main idea is the following. Suppose that $\F^\star M$ is a decreasing filtered spectrum

$\cdots\F^{n+1}M\rightarrow\F^n M\rightarrow\F^{n-1}M\rightarrow\cdots.$

For integers $a<b$, let $\gr^{[a,b)}M$ be the cofiber of $\F^bM\rightarrow\F^aM$ and let $\gr^aM=\gr^{[a,a+1)}M$. Thus, $\gr^{[a,b)}$ is an iterated extension of $\gr^aM,\ldots,\gr^{b-1}M$.

When $b=a+2$, there is a cofiber sequence

$\gr^{a+1}M\rightarrow\gr^{[a,a+2)}M\rightarrow\gr^aM$

which gives rise to a boundary map $\delta\colon\gr^aM\rightarrow\gr^{a+1}M[1]$. It is not hard to check that there is a specified nullhomotopy of $\delta^2$ when this makes sense. In particular, the sequence

$\cdots\rightarrow\gr^{-1}M[-1]\xrightarrow{\delta}\gr^0M\xrightarrow{\delta}\gr^1M[1]\xrightarrow{\delta}\gr^2M[2]\rightarrow\cdots\quad(\color{red}{\ast})$

looks like a kind of cochain complex in spectra. Lurie uses this idea in Higher algebra to study spectral sequences and so forth.

Ariotta’s paper makes precise exactly what kind of structure this sequence inherits from the filtered spectrum $\F^\star M$ and the extent to which $\F^\star M$ can be recovered from the sequence.

Definition. Let $\Xi$ be the pointed category consisting of a point $\ast$ and an object $n$ for each integer $n$. Then, $\mathrm{Hom}_\Xi(n,n-1)=\{\delta,\ast\}$, $\mathrm{Hom}_\Xi(m,n)=\ast$ if $m\neq n+1$, and $\delta^2=\ast$, when this makes sense. The category $\Xi$ is the classifying category for cochain complexes in the following senese: if $\Ascr$ is an abelian category, then the category $\mathrm{Fun}_\ast(\Xi,\Ascr)$ of pointed functors $\Xi\rightarrow\Ascr$ is naturally equivalent to the category of chain complexes in $\Ascr$. Similarly, $\mathrm{Fun}_\ast(\Xi^\op,\Ascr)$ is naturally equivalent to the category of cochain complexes in $\Ascr$.

Definition. If $\Cscr$ is a stable $\infty$-category, then a coherent cochain complex in $\Cscr$ is a functor $\Xi^\op\rightarrow\Cscr$. The $\infty$-category of coherent cochain complexes in $\Cscr$ is $\mathrm{Fun}_\ast(\Xi^\op,\Cscr)$.

Note that a functor $\Xi^\op\rightarrow\Cscr$ specifies a sequence $\cdots\rightarrow X^{-1}\xrightarrow{\delta} X^0\xrightarrow{\delta} X^1\rightarrow\cdots$ with specified nullhomotopies $\delta^2\we 0$ and an infinite amount of coherent information relating the nullhomotopies.

Ariotta’s $\E_1$-page Theorem. If $\Cscr$ be a stable $\infty$-category, then there is a natural functor $\F\Cscr\rightarrow\mathrm{\Fun}_\ast(\Xi^\op,\Cscr)$ from the $\infty$-category of filtered objects in $\Cscr$ to the $\infty$-category of coherent cochain complexes in $\Cscr$ which realizes the construction $(\color{red}{\ast})$ above. If $\Cscr$ admits sequential limits, then this functor induces an equivalence $\widehat{\F\Cscr}\we\mathrm{\Fun}_\ast(\Xi^\op,\Cscr)$, where $\widehat{\F\Cscr}$ is the stable $\infty$-category of complete filtrations in $\Cscr$.

Why do I call this the $\E_1$-page theorem?

If $\Cscr$ admits a $t$-structure, then there is an induced pointwise $t$-structure on the functor category $\mathrm{Fun}_\ast(\Xi^\op,\Cscr)$. If $\Cscr$ admits sequential limits, then this $t$-structure yields the Beilinson $t$-structure studied in [2] on complete filtered objects $\widehat{\F\Cscr}$. Given a complete filtered object $\F^\star M$, the $\E_1$-page of the spectral sequence of $\F^\star M$, with respect to the fixed $t$-structure on $\Cscr$, is precisely the result of applying $\pi_*$ to the coherent cochain complex $(\color{red}{\ast})$. Ariotta’s theorem says that conversely, if one keeps track of enough homotopy coherence, then one can go backwards from the $\E_1$-page to the complete filtered object. This lends new strength to the idea that spectral sequences are filtrations…

It is possible that Ariotta’s theorem can be extracted from Raksit’s Koszul duality approach to filtered objects in [3], which in turn follows Lurie’s work in the rotation invariance paper. I have not checked the details.

Happy new year!

# References

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

[2] 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.

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