New paper: spectral sequences, décalage, and the Beilinson t-structure
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.