%~Mouliné par MaN_auto v.0.40.4 (9e9f6e09) 2026-07-21 16:09:21 \documentclass[AHL,Unicode]{cedram} \usepackage[all]{xy} \usepackage{spectralsequences} %%%%%%%%%%%%%%%%%%%%%% \newcommand{\Ascr}{\mathcal{A}} \newcommand{\Cscr}{\mathcal{C}} \newcommand{\Dscr}{\mathcal{D}} \newcommand{\Escr}{\mathcal{E}} \newcommand{\Fscr}{\mathcal{F}} \newcommand{\Gscr}{\mathcal{G}} \newcommand{\Hscr}{\mathcal{H}} \newcommand{\Mscr}{\mathcal{M}} \newcommand{\Oscr}{\mathcal{O}} \newcommand{\Pscr}{\mathcal{P}} \newcommand{\Sscr}{\mathcal{S}} \newcommand{\Tscr}{\mathcal{T}} \newcommand{\Cat}{\Cscr\mathrm{at}} \newcommand{\Ab}{\Ascr\mathrm{b}} \newcommand{\Sp}{\Sscr\mathrm{p}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%% \DeclareMathOperator*{\colim}{colim} \DeclareMathOperator*{\Tot}{Tot} \DeclareMathOperator{\Spec}{Spec} \DeclareMathOperator{\cofiber}{cofiber} \DeclareMathOperator{\coker}{coker} %%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\B}{\mathrm{B}} \renewcommand{\d}{\mathrm{d}} \newcommand{\D}{\mathrm{D}} 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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \graphicspath{{./figures/}} \newcommand*{\mk}{\mkern -1mu} \newcommand*{\Mk}{\mkern -2mu} \newcommand*{\mK}{\mkern 1mu} \newcommand*{\MK}{\mkern 2mu} \hypersetup{urlcolor=purple, linkcolor=blue, citecolor=red} \newcommand*{\relabel}{\renewcommand{\labelenumi}{(\theenumi)}} \newcommand*{\romanenumi}{\renewcommand*{\theenumi}{\roman{enumi}}\relabel} \newcommand*{\Romanenumi}{\renewcommand*{\theenumi}{\Roman{enumi}}\relabel} \newcommand*{\alphenumi}{\renewcommand*{\theenumi}{\alph{enumi}}\relabel} \newcommand*{\Alphenumi}{\renewcommand*{\theenumi}{\Alph{enumi}}\relabel} \newcommand*{\objectenumi}{\renewcommand*{\theenumi}{\textbf{\arabic{enumi}}}\relabel} \let\oldtilde\tilde \renewcommand*{\tilde}[1]{\mathchoice{\widetilde{#1}}{\widetilde{#1}}{\oldtilde{#1}}{\oldtilde{#1}}} \let\oldhat\hat \renewcommand*{\hat}[1]{\mathchoice{\widehat{#1}}{\widehat{#1}}{\oldhat{#1}}{\oldhat{#1}}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand*{\To}{\mathchoice{\Longrightarrow}{\Rightarrow}{\Rightarrow}{\Rightarrow}} \newcommand{\longhookrightarrow}{\lhook\joinrel\longrightarrow} \newcommand*{\hookto}{\mathchoice{\longhookrightarrow}{\hookrightarrow}{\hookrightarrow}{\hookrightarrow}} \SelectTips{cm}{11} \renewcommand*{\to}{\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}} \let\oldmapsto\mapsto \renewcommand*{\mapsto}{\mathchoice{\longmapsto}{\oldmapsto}{\oldmapsto}{\oldmapsto}} %~ This fixes spacing issues around \left( and \right) and other similar symbols \let\originalleft\left \let\originalright\right \renewcommand{\left}{\mathopen{}\mathclose\bgroup\originalleft} \renewcommand{\right}{\aftergroup\egroup\originalright} %%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \title{Spectral sequences, décalage, and the Beilinson $t$-structure} \alttitle{Suites spectrales, décalage, et la $t$-structure de Beilinson} \subjclass{18G40, 55T05, 14A30} \keywords{spectral sequences, $t$-structures} \author{\firstname{Benjamin} \lastname{Antieau}} \address{Department of Mathematics,\\ Northwestern University,\\ 2033 Sheridan Road, Evanston,\\ IL 60208 (USA)} \email{antieau@northwestern.edu} \thanks{The author was supported by NSF grants DMS-2120005, DMS-2102010, and DMS-2152235, by Simons Fellowships 666565 and 00005925, and by the Simons Collaboration on Perfection. This paper was completed while B.~Antieau was a guest at the MPIM} \CDRGrant[NSF]{DMS-2120005} \CDRGrant[NSF]{DMS-2102010} \CDRGrant[NSF]{DMS-2152235} \CDRGrant[Simons Fellowships]{666565} \CDRGrant[Simons Fellowships]{00005925} \begin{abstract} This paper explains the theory of spectral sequences via décalage and the Beilinson $t$-structure. \end{abstract} \begin{altabstract} Cet article explique la théorie des suites spectrales à travers le décalage et la $t$-structure de Beilinson. \end{altabstract} \datereceived{2024-11-15} \daterevised{2025-10-08} \dateaccepted{2026-01-23} \editors{V.~Colin and J.~Ayoub} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \maketitle %\break \begingroup \setcounter{figure}{0} \renewcommand{\thetable}{\Alph{table}} \begin{table}[!h] \caption{Spectral sequence conventions for decreasing filtrations $\F^\star$ and increasing filtrations $\F_\star$. We use $\To$ even when no convergence is implied.}\label{fig:cover_ssconventions} \centering %\makebox[\textwidth]{ \begin{tabular}{||l l l||} \hline $\pi_*\F^\star M$&$\E^1_{s,t}=\pi_{s+t}\gr^{-s}M\To\pi_{s+t}M$&$(-r,r-1)$\\ &$\F^s{\pi_nM}=\im(\pi_n\F^sM\to\pi_nM)$&$\gr^s\pi_nM\iso\E^\infty_{-s,s+n}$\\ \hline $\pi_*\F_\star M$&$\E^1_{s,t}=\pi_{s+t}\gr_sM\To\pi_{s+t}M$&$(-r,r-1)$\\ &$\F_s\pi_nM=\im(\pi_n\F_sM\to\pi_nM)$&$\gr_s\pi_nM\iso\E^\infty_{s,-s+n}$\\ \hline $\H^*\F^\star M$&$\E_1^{s,t}=\H^{s+t}\gr^sM\To\H^{s+t}M$&$(r,-r+1)$\\ &$\F^s\H^nM=\im(\H^n\F^sM\to\H^nM)$&$\gr^s\H^nM\iso\E_\infty^{s,-s+n}$\\ \hline $\H^*\F_\star M$&$\E_1^{s,t}=\H^{s+t}\gr_{-s}M\To\H^{s+t}M$&$(r,-r+1)$\\ &$\F_s\H^nM=\im(\H^n\F_sM\to\H^nM)$&$\gr_s\H^nM\iso\E_\infty^{-s,s+n}$\\ \hline \end{tabular} %} \end{table} \begin{table}[!h] \caption{Reindexing to $'\E^2$. The differentials have the same bidegree as in the $\E^1$-spectral sequences from Table~\ref{fig:cover_ssconventions}. The filtrations on the abutment are also defined in the same way as in the $\E^1$ spectral sequences.}\label{fig:cover_E2ssconventions} \centering %\makebox[\textwidth]{ \begin{tabular}{||l l l||} \hline $\pi_*\F^\star M$&$'\E^2_{s,t}=\E^1_{-t,s+2t}=\pi_{s+t}\gr^{t}M\To\pi_{s+t}M$&$\gr^s\pi_nM\iso\E^\infty_{-s+n,s}$\\ \hline $\pi_*\F_\star M$&$'\E^2_{s,t}=\E^1_{-t,s+2t}=\pi_{s+t}\gr_{-t}M\To\pi_{s+t}M$&$\gr_s\pi_nM\iso\E^\infty_{s+n,-s}$ \\ \hline $\H^*\F^\star M$&$'\E_2^{s,t}=\E_1^{-t,s+2t}=\H^{s+t}\gr^{-t}M\To\H^{s+t}M$&$\gr^s\H^nM\iso\E_\infty^{s+n,-s}$ \\ \hline $\H^*\F_\star M$&$'\E_2^{s,t}=\E_1^{-t,s+2t}=\H^{s+t}\gr_{t}M\To\H^{s+t}M$&$\gr_s\H^nM\iso\E_\infty^{-s+n,s}$ \\ \hline \end{tabular}% } \end{table} \begin{table}[!h] \caption{Adams indexing spectral sequence conventions.}\label{fig:cover_adamsssconventions} \centering %\makebox[\textwidth] \begin{tabular}{||l l l||} \hline $\pi_*\F^\star M$&$\E^1_{s,t}=\pi_{s}\gr^{t}M\To\pi_{s}M$&$(-1,r)$\\ &$\F^s{\pi_nM}=\im(\pi_n\F^sM\to\pi_nM)$&$\gr^s\pi_nM\iso\E^\infty_{n,s}$\\ \hline $\pi_*\F_\star M$&$\E^1_{s,t}=\pi_{s}\gr_{-t}M\To\pi_{s}M$&$(-1,r)$\\ &$\F_s\pi_nM=\im(\pi_n\F_sM\to\pi_nM)$&$\gr_s\pi_nM\iso\E^\infty_{n,-s}$\\ \hline $\H^*\F^\star M$&$\E_1^{s,t}=\H^{s}\gr^{-t}M\To\H^{s}M$&$(1,-r)$\\ &$\F^s\H^nM=\im(\H^n\F^sM\to\H^nM)$&$\gr^s\H^nM\iso\E_\infty^{n,-s}$\\ \hline $\H^*\F_\star M$&$\E_1^{s,t}=\H^{s}\gr_{t}M\To\H^{s}M$&$(1,-r)$\\ &$\F_s\H^nM=\im(\H^n\F_sM\to\H^nM)$&$\gr_s\H^nM\iso\E_\infty^{n,s}$\\ \hline \end{tabular}% } \end{table} \endgroup \goodbreak \section{Introduction} There has been a recent cultural movement best portrayed as: \[ \text{``there is a spectral sequence''}\To\text{``there is a filtration''}. \] This move is due to the fact that filtrations admit better functorial properties than spectral sequences. For example, Bhatt, Morrow, and Scholze use quasisyntomic descent in~\cite{bms2} to define new filtrations on topological Hochschild homology and related invariants of $p$-adic rings by reducing to particularly simple cases. This requires descending a filtration, where descending a spectral sequence would not classically be meaningful. Other factors in this evolution include the emergence of synthetic spectra~\cite{pstragowski-synthetic} and the even filtration~\cite{hrw} as foundational tools. The Beilinson $t$-structure plays an important role in~\cite{bms2} and one can also use the Beilinson $t$-structure to construct related filtrations on integral periodic cyclic homology, as in~\cite{antieau-derham}, or on $\TP$ of schemes in characteristic $p$, as in~\cite{an1}. Given a stable $\infty$-category $\Cscr$ which admits sequential limits and equipped with a $t$-structure, the Beilinson $t$-structure is a $t$-structure on $\F\Cscr$, the $\infty$-category of filtered objects in $\Cscr$, whose heart is the abelian category $\Ch^\bullet(\Cscr^\heart)$ of cochain complexes.\linebreak Given a filtration $\F^\star\in\F\Cscr$, the cochain complexes appearing as the homotopy objects with respect to the Beilinson $t$-structure are precisely the cochain complexes occurring on the $\E^1$-page of the associated spectral sequence. This $t$-structure was first defined and studied by Beilinson in~\cite{beilinson}; see~\cite{ariotta,bms2,raksit} for recent treatments. The goal of this paper is to explain the theory of spectral sequences as seen through the lens of the Beilinson $t$-structure. The main construction is the décalage $\Dec(\F)^\star$ of a filtration $\F^\star$. If $\Cscr$ additionally admits sequential colimits, then every filtered object $\F^\star$ of $\Cscr$ admits an underlying object $|\F^\star|$ obtained by forgetting the filtration, or taking the colimit. Each connective cover $\tau_{\geq n}^\B(\F^\star)$ in the Beilinson $t$-structure is, by definition, itself a filtered object in $\Cscr$. Thus, given a filtered object $\F^\star$ one can obtain a new filtered object $\Dec(\F)^\star$ by taking the underlying objects of the Whitehead tower \[ \dots\to\left|\tau_{\geq n+1}^\B(\F^\star)\right|\to\left|\tau_{\geq n}^\B(\F^\star)\right|\to\left|\tau_{\geq n-1}^\B(\F^\star)\right|\to\dots. \] This construction defines an endomorphism $\Dec\colon\F\Cscr\to\F\Cscr$, the décalage functor. The main result of the paper is that décalage turns the page. Specifically, if $\Cscr$ is a stable $\infty$-category equipped with a $t$-structure and $\F^\star\in\F\Cscr$, there is an associated spectral sequence defined, for example, in~\cite[Section~1.2.2]{ha}. Below, an isomorphism of pages between two spectral sequences includes compatibility of the isomorphism with the differentials. \begin{theol}[See Theorem~\ref{thm:decalage}]\label{thm:main} Let $\Cscr$ be a stable $\infty$-category which admits sequential limits and colimits and fix a $t$-structure on $\Cscr$. For $r\geq 1$, the $\E^r$-page of the spectral sequence for $\Dec(\F^\star)$ is naturally isomorphic to the $\E^{r+1}$-page of the spectral sequence for $\F^\star$. \end{theol} Let $\Dec^{(r-1)}(\F^\star)$ denote the $(r-1)$-fold composition of the décalage construction. \begin{coro} The $\E^r$-page of the spectral sequence for $\F^\star$ is naturally isomorphic to the $\E^1$-page of the spectral sequence for $\Dec^{(r-1)}(\F^\star)$. \end{coro} The corollary leads to a quick definition of the spectral sequence associated to a~filtered object $\F^\star\in\F\Cscr$. \begin{defi}\label{def:main} Let $\F^\star$ be a filtered object in a stable $\infty$-category $\Cscr$ with a $t$-structure and suppose that $\Cscr$ admits sequential limits and colimits. Write the associated coherent cochain complex \[ \dots\to\gr^{-1}[-1]\overset{\delta}\to\gr^0 \overset{\delta}\to\gr^1 [1]\to\dots \] obtained by pasting together the boundary maps arising in the cofiber sequences \[ \F^n /\F^{n+2}\to\gr^n\overset{\delta}\to\gr^{n+1}[1] \] and take homotopy objects with respect to the $t$-structure on $\Cscr$ to obtain cochain complexes \[ \dots\to\pi_{t+1}\gr^{-1} M\to\pi_t\gr^0 M\to\pi_{t-1}\gr^1 M\to\dots. \] These form the $\E^1$-page of the spectral sequence associated to $\F^\star$, where $\E^1_{s,t}=\pi_{s+t}\gr^{-s}$. To define the $\E^{r}$-page, for some $r\geq 1$, apply the décalage construction above $(r-1)$ times and take the $\E^1$-page of the resulting filtration, applying the transformation \[ \begin{pmatrix}-r+1&-r\\r&r+1 \end{pmatrix}= \begin{pmatrix}0&-1\\1&2 \end{pmatrix}^r \] to the indices. \end{defi} Definition~\ref{def:main} gives a sequence of pages for which it is not hard to prove that the cohomology groups of one page agree with the terms on the next page. (See Remark~\ref{rem:graded_decalage}.) In other words, Definition~\ref{def:main} produces a spectral sequence in the sense of any standard textbook, such as~\cite{mccleary}. The homotopy coherent nature of the décalage functor makes it possible to make $\infty$-categorical arguments about the nature of the spectral sequence attached to a filtered object via Definition~\ref{def:main}. For example, the multiplicative properties of these spectral sequences follow immediately from the fact that the décalage functor admits a lax symmetric monoidal structure. See Hedenlund's thesis~\cite{hedenlund-thesis} and also Section~\ref{sec:multiplicativity}. The content of Theorem~\ref{thm:main} is that Definition~\ref{def:main} yields $\E^{r+1}$-pages and $d^{r+1}$-dif\-ferentials which agree naturally for $r\geq 1$ with the usual definition of the spectral sequence of a filtration, for which our reference is~\cite[Construction~1.2.2.6]{ha}. The Beilinson $t$-structure implements in a homotopy coherent way Deligne's\linebreak décalage functor, which was used to construct canonical mixed Hodge structures on the singular cohomology of complex algebraic varieties in~\cite{deligne-hodge-2,deligne-hodge-3}. Suppose that $M^\cs$ is a cochain complex with a strict decreasing filtration $\F^\star M^\cs$, so that each $\F^nM^\cs$ is a subcomplex. Let $\Dec(\F)^\star M^\cs$ be Deligne's décalage construction (see Definition~\ref{def:delignedecalage}), which is a new strict filtration on $M^\bullet$. One can put on each $\Dec(\F)^s M^\cs$ a~secondary filtration $\G^\star\Dec(\F)^s M$ in such a way that $\G^\star\Dec(\F)^\star M$\linebreak becomes a strictly bifiltered cochain complex. \begin{theo}[See Theorem~\ref{thm:decalagecomparison}]\label{thm:main_comparison} The natural inclusion $\G^\star\Dec(\F)^s M_\cs\to\F^\star M_\cs$ induces an equivalence $\G^\star\Dec(\F)^s M\to\tau_{\geq s}^\B(\F)M$ of towers of filtered objects. \end{theo} \begin{coro} The décalage construction of Construction~\ref{const:decalage} agrees with Deligne's for strictly filtered chain complexes and there is a natural equivalence $\G^\star\Dec(\F)^\star M\we\tau_{\geq\star}^\B(\F)M$ of bifiltered objects of $\D(\bZ)$. \end{coro} An $\infty$-categorical form of décalage was constructed by Levine in~\cite{levine_anss} in the context of cosimplicial spectra. Levine does not use the language of the Beilinson $t$-structure, but does prove analogues of Theorems~\ref{thm:main} and~\ref{thm:main_comparison}; see~\cite[Proposition.~6.3]{levine_anss} and~\cite[Remark~6.4]{levine_anss}. Theorem~\ref{thm:main_comparison} was discovered independently by the author around 2019 and by Bhatt, Morrow, and Scholze~\cite{bms2}. The author discovered Theorem~\ref{thm:main} as stated shortly thereafter and communicated it to several people. See for example~\cite{ariotta,gikr,hedenlund-thesis,mor-profinite}, which contain expositions of some of the ideas of this paper. In particular, the paper~\cite{gikr} of Gheorghe--Isaksen--Krause--Ricka shows that if $\Tot^\star(\MU^\bullet)$ denotes the Adams tower associated to the Čech complex of $\bS\to\MU$, then $\Dec(\Tot^\star(\MU^\bullet))$ is what is now called the even filtration $\bS_\ev$ on the sphere spectrum in the language of~\cite{hrw}. Recently, Tyler Lawson~\cite{lawson-filtered} has given a generalized version of a décalage construction which applies more generally to filtrations in a broad class of $\infty$-categories, not just stable ones. Lawson gives a sketch of the agreement of $\E^r(\Dec(\F))$ and $\E^{r+1}(\F)$ in~\cite[Proposition~9.17]{lawson-filtered}. Besides the theoretical results mentioned above, this paper is intended as a brief manual on spectral sequences, so more is included than strictly necessary for the proofs of the main theorems. For example, we include discussions of indexing conventions and convergence, which do not in fact require the methods developed here. The convergence discussion is especially simplistic as we are awaiting forthcoming work of Hedenlund, Krause, and Nikolaus on a new approach to conditional convergence. Two applications of the décalage approach to spectral sequences are given, namely to multiplicativity of spectral sequences and to the proof that the two standard constructions of the Atiyah--Hirzebruch spectral sequence agree from the $\E_2$-page~on. \subsection*{Additional related work} Our perspective places filtered objects at the heart of the study of spectral sequences, including the higher pages. By contrast, refer to work of Livernet--White\-house \cite{livernet-whitehouse} or Cirici--Egas-Santander--Livernet~\cite{cirici-egas-santander-livernet} who study homotopy theories of extended spectral sequences and define décalage functors for these theories. In another direction, Cirici--Guillén~\cite{cirici-guillen} use Deligne's décalage to denote certain localizations of the homotopy theory of filtered complexes with applications to mixed Hodge theory. \subsection*{Acknowledgments} Peter Scholze suggested to me the problem of constructing a ``BMS'' filtration on periodic cyclic homology and related invariants without $p$-completeness assumptions. This led to~\cite{antieau-derham} and to an appreciation of the Beilinson $t$-structure. Akhil Mathew asked how the décalage used there relates to that defined by Deligne in~\cite{deligne-hodge-2}, leading to this paper. I thank both for their motivating questions. Additionally,\linebreak I thank Achim Krause and Thomas Nikolaus for explaining parts of their work on~convergence. This paper would never have been completed without the interest of Julie Creemers, Lennart Meier, and Itamar Mor. Additionally, Julie Creemers and Alice Hedenlund provided very helpful comments on a draft, for which I am very grateful. To the referee, thank you for the helpful suggestions. I would like to thank the NSF and the Simons Foundation. \section{\texorpdfstring{$t$}{t}-structures}\label{sec:t} Stable $\infty$-categories are the natural higher categorical models of triangulated categories. We include a brief overview of the theory. For proofs and more details, see~\cite[Chapter~1]{ha} or~\cite[Appendix~C]{sag}. \goodbreak \begin{defi}\leavevmode \begin{enumerate}\alphenumi \item\label{defi1.1.1} An $\infty$-category is pointed if it admits an object $0$ which is both initial and terminal. \item\label{defi1.1.2} If $\Cscr$ is pointed and admits finite colimits, then there is a suspension functor $\Sigma\colon\Cscr\to\Cscr$ obtained by taking the pushout of the functors $0\leftarrow\id\to 0$. The Spanier--Whitehead $\infty$-category of $\Cscr$ is $\SW(\Cscr)\we\colim(\Cscr\xrightarrow{\Sigma}\Cscr\xrightarrow{\Sigma}\dots)$. An $\infty$-category is prestable if it is pointed, admits finite colimits, the suspension functor $\Sigma$ is fully faithful, and the essential image of the fully faithful functor $\Cscr\to\SW(\Cscr)$ is closed under extensions. \item\label{defi1.1.3} An $\infty$-category $\Cscr$ is stable if it is pointed, admits finite colimits, and the suspension functor $\Sigma$ is an equivalence. If $\Cscr$ is prestable, then $\SW(\Cscr)$ is stable. \end{enumerate} \end{defi} \begin{defi} A functor $\Cscr\to\Dscr$ between stable $\infty$-categories is exact if it preserves finite colimits. \end{defi} \begin{enonce}{Fact} If $\Cscr$ is a stable $\infty$-category, then its homotopy category $\Ho(\Cscr)$ admits a natural triangulated structure. If $f\colon\Cscr\to\Dscr$ is an exact functor between stable $\infty$-categories, then $\Ho(f)\colon\Ho(\Cscr)\to\Ho(\Dscr)$ is a triangulated functor. \end{enonce} No non-trivial pointed $1$-category is stable or prestable. Indeed, in a pointed $1$-category the suspension functor is the constant functor on $0$. In particular, triangulated categories are neither prestable nor stable. \begin{exam}\leavevmode \begin{enumerate}\alphenumi \item\label{exam1.4.1} The $\infty$-category $\Sp$ is the stabilization (in $\Pr^\L$) of the $\infty$-category $\Sscr$ of anima. This is a symmetric monoidal stable $\infty$-category which is compactly generated by its unit, the sphere spectrum $\bS$. Thus, we will also write $\D(\bS)$ for $\Sp$ as $\Mod_{\bS}(\Sp)\we\Sp$. \item\label{exam1.4.2} If $A$ is an $\bA_\infty$-ring spectrum, then $\LMod_A(\D(\bS))=\D(A)$ is the $\infty$-category of left $A$-module spectra. If $A$ is a dg algebra, then the homotopy category of $\D(A)$ is equivalent to the usual triangulated category of left dg modules over $A$ up to quasi-isomorphism. In particular, this holds when $A$ is a discrete commutative ring. \item\label{exam1.4.3} If $X$ is a quasicompact and quasiseparated scheme, then $\D_\qc(X)=\lim_{\Spec R\to X}\linebreak\D(R)$ defines a stable $\infty$-category whose homotopy category is the familiar triangulated category of complexes of sheaves of $\Oscr_X$-modules with quasicoherent cohomology. \end{enumerate} \end{exam} The theory of $t$-structures plays a fundamental role in this paper. \begin{defi} A $t$-structure on a stable $\infty$-category $\Cscr$ consists of a pair $(\Cscr_{\geq 0},\Cscr_{\leq 0})$ of full subcategories of $\Cscr$ satisfying the following conditions: \begin{enumerate}\alphenumi \item\label{defi1.5.1} $\Cscr_{\geq 0}[1]\subseteq\Cscr_{\geq 0}$ and $\Cscr_{\leq 0}\subseteq\Cscr_{\leq 0}[1]$; \item\label{defi1.5.2} if $X\in\Cscr_{\geq 0}$ and $Y\in\Cscr_{\leq 0}$, then the mapping space $\Map_\Cscr(X,Y[-1])$ is contractible; \item\label{defi1.5.3} every object $X$ in $\Cscr$ fits into a fiber sequence $W\to X\to Z$ where $W\in\Cscr_{\geq 0}$ and $Z[1]\in\Cscr_{\leq 0}$. \end{enumerate} It is not hard to see that to give a $t$-structure on $\Cscr$ is equivalent to give one on its triangulated homotopy category $\Ho(\Cscr)$. \end{defi} \begin{rema} If $(\Cscr_{\geq 0},\Cscr_{\leq 0})$ is a $t$-structure, then the inclusions of $\Cscr_{\geq 0}$ and $\Cscr_{\leq 0}$ into $\Cscr$ admit right and left adjoints, respectively. This follows from a Bousfield localization argument using the orthogonality relation in~\eqref{defi1.5.2} to show that given $X\in\Cscr$ the sequence $W\to X\to Z$ is unique up to homotopy. The functor $\Cscr\to\Cscr_{\geq 0}$ is $\tau_{\geq 0}$, while $\Cscr\to\Cscr_{\leq 0}$ is $\tau_{\leq 0}$. No care will be taken to distinguish between these functors and the corresponding colocalization and localization functors $\Cscr\to\Cscr_{\geq 0}\hookto\Cscr$ and $\Cscr\to\Cscr_{\leq 0}\hookto\Cscr$. More generally, one defines subcategories $\Cscr_{\geq n}\we\Cscr_{\geq 0}[n]$ and $\Cscr_{\leq n}\we\Cscr_{\leq 0}[n]$ for any integer $n$. There are corresponding truncation functors $\tau_{\geq n}\colon\Cscr\to\Cscr_{\geq n}$ and $\tau_{\leq n}\colon\Cscr\to\Cscr_{\leq n}$ for any $n$. \end{rema} \begin{defi} The objects of $\Cscr_{\geq 0}$ are called connective while those of $\Cscr_{\leq 0}$ are coconnective or $0$-truncated. More generally, objects of $\Cscr_{\geq n}$ are $n$-connective and those of $\Cscr_{\leq n}$ are called $n$-truncated, or at times $n$-coconnective. Note: these notions depend on the chosen $t$-structure on $\Cscr$. Often, as is the case in spectra or $\D(\bZ)$, this $t$-structure is not specified, in which case the standard $t$-structure is always implicit. \end{defi} The following lemma guarantees that the order of application of truncation and connective cover functors is irrelevant. \begin{lemm} For any $m\leq n$, define $\Cscr_{[m,n]}=\Cscr_{\geq m}\cap\Cscr_{\leq n}$. There is a natural equivalence $\tau_{\geq m}\circ\tau_{\leq n}\we\tau_{\leq n}\circ\tau_{\geq m}$ of functors $\Cscr\to\Cscr_{[m,n]}$. \end{lemm} \begin{rema} Additionally, there are natural equivalences $\Sigma\circ\tau_{\geq n}\we\tau_{\geq n+1}\circ\Sigma$ and $\Sigma\circ\tau_{\leq n}\we\tau_{\leq n+1}\circ\Sigma$ and similarly for higher powers of $\Sigma$. \end{rema} \begin{defi} Given a $t$-structure on $\Cscr$, the heart of the $t$-structure is the intersection $\Cscr^\heart=\Cscr_{\geq 0}\cap\Cscr_{\leq 0}$. This turns out to be an abelian category, although it might be trivial. Given any integer $n$, one takes $\pi_n\colon\Cscr\to\Cscr^\heart$ to be the functor $\Sigma^{-n}\circ\tau_{\geq n}\circ\tau_{\leq n}\we\Sigma^{-n}\circ\tau_{\leq n}\circ\tau_{\geq n}\we\tau_{\leq 0}\circ\tau_{\geq 0}\circ\Sigma^{-n}\we\tau_{\geq 0}\circ\tau_{\leq 0}\circ\Sigma^{-n}$. \end{defi} \begin{rema} Given a fiber sequence $X\to Y\to Z$ in $\Cscr$, there is an induced long exact sequence \[ \dots\to\pi_nX\to\pi_nY\to\pi_nZ\to\pi_{n-1}X\to\dots \] in $\Cscr^\heart$. These sequences give a hands-on way for understanding stable $\infty$-categories using more familiar abelian categories. \end{rema} \begin{rema} If $(\Cscr_{\geq 0},\Cscr_{\leq 0})$ is a $t$-structure on a stable $\infty$-category $\Cscr$, then $\Cscr_{\geq 0}$ is a prestable $\infty$-category which is closed under extensions in $\Cscr$. The converse is true under the presence of enough colimits using a Bousfield localization argument. See~\cite[Proposition~1.4.4.11]{ha}. \end{rema} \begin{defi} Let $(\Cscr_{\geq 0},\Cscr_{\leq 0})$ be a $t$-structure on a stable $\infty$-category $\Cscr$. One says that the $t$-structure (or $\Cscr$) is right separated if the condition $X\in\Cscr_{\leq n}$ for all $n$ implies that $X$ is equivalent to zero. Similarly, $\Cscr$ is left separated if the condition $X\in\Cscr_{\geq n}$ for all $n$ implies that $X$ is equivalent to zero. Relatedly, one says that the $t$-structure is right complete if the natural map \[ \Cscr\to\lim\left(\dots\to\Cscr_{\geq -1}\xrightarrow{\tau_{\geq 0}}\Cscr_{\geq 0}\xrightarrow{\tau_{\geq 1}}\Cscr_{\geq 1}\to\dots\right) \] is an equivalence. And, $\Cscr$ is left complete if the natural map \[ \Cscr\to\lim\left(\dots\to\Cscr_{\leq 1}\xrightarrow{\tau_{\leq 0}}\Cscr_{\leq 0}\xrightarrow{\tau_{\leq -1}}\Cscr_{\leq -1}\to\dots\right) \] is an equivalence. In general, if $\Cscr$ is right or left complete, it is right or left separated. If $\Cscr_{\leq 0}$ is closed under countable coproducts in $\Cscr$, then $\Cscr$ is right separated if and only if it is right complete. If $\Cscr_{\geq 0}$ is closed under countable products in $\Cscr$, then $\Cscr$ is left separated if and only if it is left complete. \end{defi} \begin{exam}\leavevmode \begin{enumerate}\alphenumi \item\label{exam1.14.1} The $\infty$-category $\Sp\we\D(\bS)$ of spectra admits a $t$-structure where $\Sp_{\geq 0}$ is the full subcategory of spectra $X$ such that $\pi_nX=0$ for $n<0$ while $\Sp_{\leq 0}$ is the full subcategory of $X$ where $\pi_nX=0$ for $n>0$. The heart $\Sp^\heart$ is the abelian category $\Mod_{\bZ}$ of abelian groups. \item\label{exam1.14.2} More generally, if $A$ is any connective $\bE_1$-ring spectrum, then $\D(A)$ inherits\linebreak a $t$-structure where $\D(A)_{\geq 0}$ consists of the $A$-module spectra $X$ where\linebreak $\pi_nX=0$ for $n<0$ and $\D(A)_{\leq 0}$ consists of those where $\pi_nX=0$ for $n>0$. The connectivity assumption on $A$ guarantees that if $X$ is an $A$-module spectrum, then $\tau_{\geq 0}X$, where the truncation is taken in spectra, inherits a canonical $A$-module spectrum structure. The heart of this $t$-structure is naturally equivalent to the abelian category $\Mod_{\pi_0A}$ of left $\pi_0A$-modules. If $A$ is discrete, then this $t$-structure on $\D(A)$ recovers the usual $t$-structure on the derived $\infty$-category of $A$, which encodes the ``good truncations'' of complexes. These $t$-structures are right and left complete. \item\label{exam1.14.3} If $\Cscr$ is a stable $\infty$-category equipped with a $t$-structure $(\Cscr_{\geq 0},\Cscr_{\leq 0})$ and if $I$ is an $\infty$-category, then the pair $(\Fun(I,\Cscr_{\geq 0}),\Fun(I,\Cscr_{\leq 0}))$ forms a $t$-structure on $\Fun(I,\Cscr)$ with heart $\Fun(I,\Cscr)^\heart\we\Fun(I,\Cscr^\heart)\we\Fun(\Ho(I),\Cscr^\heart)$, where $\Ho(I)$ denotes the homotopy category of $I$. \end{enumerate} \end{exam} \section{Beilinson's \texorpdfstring{$t$}{t}-structure}\label{sec:beilinson} Filtered objects in a stable $\infty$-category have been studied in several places. See \cite{ariotta,gwilliam-pavlov,lurie-rotation,ha,moulinos-filtrations,raksit}. Let $\bZ$ denote the ordered set of integers viewed as a category so that there is\linebreak a (unique) morphism $m\to n$ if and only if $m\leq n$. Let $\bZ^\op$ be the opposite of $\bZ$. \begin{defi}\label{def:filtration} Let $\Cscr$ be an $\infty$-category. The $\infty$-category of decreasing filtrations in $\Cscr$ is \[ \F\Cscr=\Fun(\bZ^\op,\Cscr). \] The $\infty$-category of increasing filtrations in $\Cscr$ is $\Fun(\bZ,\Cscr)$. \end{defi} \begin{rema} There is an equivalence $\bZ\we\bZ^\op$ obtained by sending $m$ to $-m$. The theory of increasing filtrations and decreasing filtrations are thus equivalent, so we focus below on decreasing filtrations. \end{rema} A decreasing filtration in $\Cscr$ is represented as an infinite sequence \[ \F^\star \colon\cdots\to\F^{s+1}\to\F^s\to\F^{s-1}\to\cdots. \] Included in the data of a functor $\bZ^\op\to\Cscr$ is homotopy coherence data expressing for example a specific homotopy between $\F^{s+1}\to\F^{s-1}$ and the composition of $\F^{s+1}\to\F^s$ with $\F^s\to\F^{s-1}$. \begin{exam} If $\Cscr\we\D(\bZ)$ is the $\infty$-category of $\bZ$-module spectra, then $\FD(\bZ)$ is the $\infty$-category of filtered $\bZ$-module spectra. The homotopy category of $\FD(\bZ)$ is what is often called the derived category of filtered complexes. In fact, if $\F\Mod_{\bZ}$ is the Grothendieck abelian category of filtered abelian groups, then $\FD(\bZ)\we\D(\F\Mod_{\bZ})$ is its derived $\infty$-category. Indeed, one can check that the pointwise $t$-structure is $0$-complicial and separated, so the claim follows from~\cite[Proposition~C.5.3.2]{sag} and~\cite[Proposition~C.5.4.5]{sag}; see Proposition~\ref{prop:inheritance}$\MK$\eqref{prop2.31.6} below for more details. \end{exam} \begin{enonce}{Warning} By Definition~\ref{def:filtration}, a filtered object of $\F\Mod_{\bZ}$ is an arbitrary $\bZ^\op$-indexed sequence of maps of abelian groups. There is no requirement that the transition maps in the sequence be injective. When they are, the filtration is called strict. The category of strict filtrations on $\Mod_{\bZ}$ has kernels and cokernels, but is not abelian. \end{enonce} Often, one considers filtrations of specific objects. \begin{defi} A decreasing filtration on an object $X$ of $\Cscr$ is a filtration $\F^\star$ together with a map $\F^\star\to X$ of filtrations, where $X$ denotes the constant decreasing filtration on $X$. A filtration $\F^\star$ on $X$ is exhaustive if $X$ is a colimit of the diagram~$\F^\star$. In analogy with the ordinal $\omega+1$, one can consider the $\infty$-category of pairs $(\F^\star\to X)$ of filtrations on objects of $\Cscr$ as the functor category $\Fun(\bZ^\op+1,\Cscr)$, where $\bZ^\op+1$ is the union of $\bZ^\op$ and a new, terminal element $-\infty$. Of course, we have $\bZ^\op+1\we(\bZ^\op)^{\vartriangleright}\we(\bZ^\vartriangleleft)^\op$. \end{defi} The picture to have in mind here is a large commutative diagram \[ \xymatrix{ \cdots\ar[r]&\F^{s+1}\ar[dr]\ar[r]&\F^s\ar[d]\ar[r]&\F^{s+1}\ar[dl]\ar[r]& \cdots\\ &&X.&& } \] \begin{exam} If $\Cscr$ admits sequential colimits, then any filtration $\F^\star$ can be viewed as giving a filtration on $\F^{-\infty}=\colim_s\F^s$. This construction will appear frequently, so we denote it by $|\F^\star|=\F^{-\infty}$. If $\F^\star$ is a filtration on $X$, then there is a canonical map $|\F^\star|\to X$, which is an equivalence if the filtration is exhaustive. If $\Cscr$ has sequential colimits, the functor $|-|\colon\F\Cscr\to\Cscr$ is left adjoint to the constant filtration functor $\Cscr\to\F\Cscr$. \end{exam} \begin{nota} We use the notation $\F^\star$ for a generic filtration to emphasize the primacy of the filtration as opposed to a possible object being filtered. Sometimes, we will write $\F^\star X$ for a generic filtration as well, in which case, unless specified otherwise, $X=|\F^\star|$. \end{nota} \begin{defi} If $\Cscr$ has a final object $\ast$ and admits cofibers and if $\F^\star$ is a~decreasing filtration in $\Cscr$, the associated graded pieces are \[ \gr^s_\F=\cofiber\left(\F^{s+1}\to\F^s\right)=\frac{\F^{s+1}}{\F^s} \] for $s\in\bZ$. The collection of associated graded pieces forms a graded object $\gr^\star_\F$ of~$\Cscr$, i.e., an object of $\Fun(\bZ^\delta,\Cscr)$, where $\bZ^\delta$ denotes the set $\bZ$ viewed as a $1$-category with only identity morphisms. If $\F^\star$ is a filtration then its weights are the integers $s$ for which $\gr^s_\F\we\F^s/\F^{s+1}$ is non-zero. \end{defi} The underlying goal of spectral sequences, and this paper, is to explain how to understand $X$ from information about the graded pieces $\gr^\star_\F X$ of some filtration $\F^\star X$ on $X$. This understanding is what spectral sequences ``do''. For the remainder of this paper, filtrations are studied in the context of \emph{stable} $\infty$-categories. \begin{defi} A filtration $\F^\star$ in $\Cscr$ is complete if $\F^\infty=\lim_s\F^s\we 0$, the initial object of $\Cscr$. In particular, the limit exists. Let $\widehat{\F}\Cscr\subseteq\F\Cscr$ be the full subcategory of complete filtrations. If $\Cscr$ admits sequential limits, then the inclusion $\widehat{\F}\Cscr\subseteq\F\Cscr$ admits a left adjoint, called completion, given by $\widehat{\F}^\star=\F^\star/\F^\infty$. Let $\F^\star$ be a decreasing filtration on an object $X$ of $\Cscr$. The completion of $X$ with respect to the filtration is $\widehat{X}=\lim_s(X/\F^sX)\we X/(\lim_s\F^sX)\we X/(\F^{\infty}X)$. The completion $\widehat{\F}^\star$ is a complete filtration on $\widehat{X}$. \end{defi} \begin{exam} Let $p^\star\bZ$ denote the $p$-adic filtration of $\bZ$. Specifically, $p^s\bZ$ is the ideal $(p^s)\subseteq\bZ$ for $s\geq 0$ and $p^s\bZ=\bZ$ for $s\leq 0$. The graded pieces are $\gr^s\bZ\we (p^s)/(p^{s+1})\iso\bZ/p$ for $s\geq 0$ and $0$ for $s<0$. This filtration is exhaustive, but not complete: $\F^\infty=\lim_s(p^s)\we(\bZ_p/\bZ)[-1]$. The completion is the $p$-adic filtration $p^\star\bZ_p$ of $\bZ_p$. \end{exam} \begin{rema} There is a duality between complete exhaustive filtrations $\F^\star X$ on $X$ and non-complete filtrations on $0$ with limit $X$. Indeed, if $\F^\star X$ is complete and exhaustive, consider the cone of $\F^\star X\to X$, which is the filtration $\frac{X}{\F^\star X}$. The limit of this filtration is $\lim_s X/F^sX\we X$ and the colimit is $\colim_s X/F^sX\we 0$. Conversely, if $\F^\star$ is an exhaustive filtration on $0$ with limit $Z$, then taking the fiber of $Z\to\F^\star$ gives a complete exhaustive filtration on $Z$. As example, consider the two perspectives on the completeness of the $p$-adic filtration: $\lim_s p^s\bZ_p=0$ versus $\bZ_p\iso\lim_s \bZ_p/p^s$. \end{rema} \begin{exam}\label{ex:stupid} Let $M^\bullet\in\Ch^\bullet(\bZ)$ be a cochain complex. The stupid filtration $\sigma^\star M^\bullet$ on $M$ is the filtered cochain complex with $\sigma^s M^\bullet=M^{\bullet\geq s}$. The graded pieces are $\gr^sM^\bullet\iso M^s$ placed in cohomological degree $s$. The stupid filtration is a complete exhaustive filtration on $M^\bullet$ in $\Ch^\bullet(\bZ)$. Looking at underlying homotopy types produces a complete exhaustive filtration $\sigma^\star M$ on the image $M$ of $M^\bullet$ in the derived $\infty$-category $\D(\bZ)$. The graded pieces are $\gr^sM\we M^s[-s]$. We return to this example below in Construction~\ref{const:realization}. \end{exam} \begin{exam}\label{ex:whitehead} Let $X\in\D(\bS)$ be a spectrum and let $\F^\star X=\tau_{\geq\star}X$ be its Whitehead tower: \[ \cdots\to\tau_{\geq s+1}X\to\tau_{\geq s}X\to\tau_{\geq s-1}X\to\cdots. \] This is a complete and exhaustive filtration on $X$; the graded pieces are $\gr^sX\we\pi_sX[s]\in\D(\bS)$. \end{exam} \begin{enonce} {Warning} Whitehead towers are neither complete nor exhaustive in a general stable $\infty$-category with a $t$-structure. In the case of spectra, the usual, Postnikov $t$-structure is compatible with filtered colimits, so that the homotopy groups of a filtered colimit are the filtered colimit of the homotopy groups, and is left-complete, so that Postnikov towers converge. These properties are shared by many other $t$-structures which occur in algebra, geometry, and topology, but not all. For an extensive discussion, see~\cite[Section~2]{clausen-mathew}. \end{enonce} \begin{exam} Let $X$ be a CW complex, which we view as a space with a~certain increasing filtration $\F_\star X$ by cells. The filtration is complete and exhaustive by definition. Taking $\bZ$-cochains yields a filtered complex with \[ \F^s_{\CW}\R\Gamma(X,\bZ)=\fib\bigl(\R\Gamma(X,\bZ)\to\R\Gamma(X_{s-1},\bZ)\bigr), \] where $\R\Gamma(-,\bZ)$ denotes the singular cohomology functor (which is equivalent to sheaf cohomology for CW complexes). The choice of indexing here implies that $\F^0_{\CW}\R\Gamma(X,\bZ)\we\R\Gamma(X,\bZ)$ since the cohomology of the empty set vanishes. The CW filtration is complete and exhaustive on $\R\Gamma(X,\bZ)$ with associated graded pieces $\gr^s\R\Gamma(X,\bZ)\we(\R\Gamma(X_{s-1},\bZ)/\R\Gamma(X_{s},\bZ))[-1]\we \prod_{\text{$s$ cells}}\bZ[-s]$. This filtration is used below in Section~\ref{sec:mysterious}. \end{exam} \begin{nota} Let $\F^\star$ be a filtered object in a stable $\infty$-category. For $j\geq i$\linebreak finite, let $\gr^{[i,j)}_\F\Mk=\Mk\cofib(\F^{j}\to\F^i)\Mk=\Mk\frac{\F^i}{\F^j}$. For example, $\gr^{[i,i)}_\F\Mk\we\Mk 0$ and $\gr_\F^{[i,i+1)}\Mk\we\Mk\gr^i_\F$.\linebreak Additionally, let $\gr^{[i,\infty)}_\F=\cofib(\F^\infty\to\F^i)=\lim_{j\to\infty}\gr^{[i,j)}_\F$ and let $\gr^{(-\infty,j)}=\colim_{i\to-\infty}\gr^{[i,j)}_\F$. The graded pieces for other intervals are defined analogously. When unambiguous, $\gr^i$ is used for $\gr^i_\F$. \end{nota} \begin{enonce} {Construction}[Filtration on gradeds]\label{const:filtered_gradeds} It often convenient to view $\gr^{[i,j)}_\F$ as admitting a residual filtration \[ \cdots\to 0\to\frac{\F^{j-1}}{\F^j}\to\frac{\F^{j-2}}{\F^j}\to\cdots\to\frac{\F^i}{\F^j}\xrightarrow{=}\frac{\F^i}{\F^j}\to\cdots, \] where the term $\frac{\F^{j-1}}{\F^j}$ is in filtration weight $j-1$. This is a complete exhaustive filtration on $\gr^{[i,j)}_\F$ with natural identifications \[ \gr^a(\gr^{[i,j)}_\F)\we \begin{cases} \gr^a_\F&\text{if $i\leq a}[d]&\pi_{s+t}\gr^{[-s-r+1,-s+1)}\ar[d]\\ \pi_{s+t-1}\gr^{[-s+r,-s+2r)}\ar[r]&\E^r_{s-r,t+r-1}\ar[r]&\pi_{s+t-1}\gr^{[-s+1,-s+r+1)} } \] where the rows are epi-mono factorizations and the left and right vertical maps come from the boundary maps in homotopy associated to the fiber sequences in~\eqref{eq:ss}. The claim is that the outer square above commutes. This follows immediately from the fact that there is a canonical map from the top fiber sequence above to the bottom fiber sequence. By functoriality of epi-mono factorization, the dotted arrow exists. Denote it by $\d^r_{s,t}$. \end{enonce} \begin{defi}[Lurie's definition of the spectral sequence]\label{def:er} The bigraded collection $\E^r_{s,t}$ of objects of $\Cscr^\heart$ together with the differentials $\d^r_{s,t}\colon\E^r_{s,t}\to\E^r_{s-r,t+r-1}$ is the $\E^r$-page of the spectral sequence associated to $\F^\star$. \end{defi} Parts~\eqref{prop3.12.2}--\eqref{prop3.12.3} of the following proposition appears as \cite[Proposition~1.2.2.7]{ha}. \begin{prop}\label{prop:lurie} Let $\F^\star M$ be a filtered object in a stable $\infty$-category equipped with a $t$-structure. \begin{enumerate}\romanenumi \item\label{prop3.12.1} The definitions of the $\E^1$-page given in Definitions~\ref{def:e1} and~\ref{def:er} agree. \item\label{prop3.12.2} The composition $\d^r\circ\d^r$ vanishes for $r\geq 1$. \item\label{prop3.12.3} The homology of the chain complexes on the $\E^r$-page is naturally isomorphic to the terms of the $\E^{r+1}$-page for $r\geq 1$. \end{enumerate} In particular, the sequence of pages of Definition~\ref{def:er} defines a spectral sequence in the sense of Definition~\ref{def:ss}. \end{prop} \begin{proof}[{Proof of~\eqref{prop3.12.1}}] Part~\eqref{prop3.12.1} follows by observing that in the second definition $\E^1_{s,t}=\pi_{s+t}\gr^{-s}$, just as in the first definition. The check that the differentials agree follows from examining the boundary maps associated to the fiber sequences in~\eqref{eq:ss}. \end{proof} The proof of part~\eqref{prop3.12.3} of Proposition~\ref{prop:lurie} is nontrivial. It will follow from Theorem~\ref{thm:decalage} that Lurie's definition of the $\E^r$-pages agrees with the décalage definition of the $\E^r$-page given here. Inductively, this implies parts~\eqref{prop3.12.2} and~\eqref{prop3.12.3} of Proposition~\ref{prop:lurie}. In particular, the proof of Theorem~\ref{thm:decalage} does not use parts~\eqref{prop3.12.2} or~\eqref{prop3.12.3} of Proposition~\ref{prop:lurie}. Until the end of the section, unless specified otherwise, $\E^r_{s,t}(\F)$ refers to Lurie's definition. \begin{theo}[Comparison theorem]\label{thm:decalage} Let $\Cscr$ be a stable $\infty$-category with sequential limits and colimits and a $t$-structure. For $r\geq 1$, there are natural isomorphisms $\E^r_{-t,s+2t}(\Dec(\F))\iso\E^{r+1}_{s,t}(\F)$ compatible with the differentials on the $\E^r(\Dec(\F))$ and $\E^{r+1}(\F)$-pages. \end{theo} \begin{coro}\label{cor:decalage} Let $\Cscr$ be a stable $\infty$-category with sequential limits and colimits and a $t$-structure. For $r\geq 0$, there are natural isomorphisms $\E^1_{-(r-1)s-rt,rs+(r+1)t}\linebreak(\Dec^{(r)}(\F))\iso\E^{r+1}_{s,t}(\F)$ compatible with the differentials on the $\E^1(\Dec^{(r)}(\F))$ and $\E^{r+1}(\F)$-pages. \end{coro} \begin{rema} In particular, the décalage definition of the spectral sequence of a filtration agrees with Lurie's. \end{rema} \begin{rema}[Bounded décalage] The main situation we foresee where the hypotheses of Theorem~\ref{thm:decalage} are not met is when $\Cscr$ is a small abelian category admitting a bounded $t$-structure, such as $\D^b(X)$ for a regular noetherian scheme. However, as in Remark~\ref{rem:bounded}, we can embed $\Cscr$ inside $\Ind(\Cscr)$ and apply the theory of the Beilinson $t$-structure and décalage to $\F\Ind(\Cscr)$. As the $\Dec$ endofunctor preserves $\F^b\Cscr\subseteq\F\Ind(\Cscr)$ since the $t$-structure on $\Cscr$ is bounded, we see that Theorem~\ref{thm:decalage} also applies to $\F^b\Cscr$. \end{rema} The proof is given at the end of the section after some preliminaries. \begin{exam} If $\F^\star M$ is the constant filtration on $M$, then each $\tau_{\geq n}^\B(\F^\star M)$ is the constant filtration on $M$ and $\Dec(\F)^\star M$ is the constant filtration on $M$. \end{exam} \begin{lemm}\label{lem:truncationscomplete} If $\F^\star M$ is a complete filtration, then each $\tau_{\geq n}^\B(\F^\star M)$ is complete. \end{lemm} \begin{proof} There is a cofiber sequence \[ \tau_{\geq n}^\B(\F^\star M)\to\F^\star M\to\tau_{\leq n-1}^\B(\F^\star M) \] of filtered objects of $\Cscr$. As $\tau_{\leq n-1}^\B(\F^\star M)$ is necessarily complete (see Definition~\ref{def:noncompletebeilinson}), it follows that $\tau_{\geq n}^\B(\F^\star M)$ is complete. \end{proof} \begin{rema}\label{rem:no_short_exact} If $\F^\star M'\to\F^\star M\to\F^\star M''$ is a fiber sequence of fibrations, so that each $\F^sM'\to\F^s M\to\F^s M''$ is a fiber sequence, there is a long-exact sequence \[ \dots\to\pi_{n+1}^\B(\F^\star M'')\to\pi_{n}^\B(\F^\star M')\to\pi_{n}^\B(\F^\star M)\to\pi_{n}^\B(\F^\star M'')\to\dots \] of cochain complexes. It does not typically break up into short exact sequences, which explains why one does not often find ``short exact sequences of spectral sequences'' in nature. \end{rema} There is an important class of examples which run counter to the general situation of Remark~\ref{rem:no_short_exact} and which will be important in the proof of Theorem~\ref{thm:decalage}. First, we need a standard lemma. \begin{lemm}\label{lem:truncation_exact} Let $\Cscr$ be a stable $\infty$-category with a $t$-structure $(\Cscr_{\geq 0},\Cscr_{\leq 0})$. Suppose that $x\to y\to z$ is a cofiber sequence in $\Cscr$. \begin{enumerate}\romanenumi \item\label{lemm3.20.1} If $\pi_0y\to\pi_0z$ is an epimorphism in $\Cscr^\heart$, then $\tau_{\geq 0}x\to\tau_{\geq 0}y\to\tau_{\geq 0}z$ is a cofiber sequence in $\Cscr$. \item\label{lemm3.20.2} If $\pi_0x\to\pi_0y$ is a monomorphism in $\Cscr^\heart$, then $\tau_{\leq 0}x\to\tau_{\leq 0}y\to\tau_{\leq 0}z$ is a cofiber sequence in $\Cscr$. \end{enumerate} \end{lemm} \begin{proof} It suffices to prove part~\eqref{lemm3.20.1}. Consider the induced map $z\to x[1]$ and note that the composition $\tau_{\geq 0}z\to \tau_{\geq 0}x[1]\to\pi_{-1}x$ factors through the boundary map $\pi_0z\to\pi_{-1}x$, which is assumed to be zero. Thus, there is a lift $\tau_{\geq 0}z\to(\tau_{\geq 0}x)[1]$. As $\Cscr_{\geq 0}$ is prestable, there is by definition a fiber $w$ of this map in $\Cscr_{\geq 0}$ such that the square \[ \xymatrix{ w\ar[r]\ar[d]&\tau_{\geq 0}z\ar[d]\\ 0\ar[r]&(\tau_{\geq 0}x)[1] } \] is a fiber square and a cofiber square in $\Cscr_{\geq 0}$. As $\tau_{\geq 0}$ preserves fiber squares, we see that there is a map $w\to\tau_{\geq 0}y$ in $\Cscr_{\geq 0}$ with trivial fiber and which induces an isomorphism on $\pi_0$. If this map is an equivalence, then $\tau_{\geq 0}y\to\tau_{\geq 0}z\to(\tau_{\geq 0}x)[1]$ is a cofiber sequence in $\Cscr$, proving~\eqref{lemm3.20.1}. Now, suppose that $a\to b$ is a map in $\Cscr_{\geq 0}$ which has trivial fiber and induces a surjection on $\pi_0$. Let $f$ be the fiber of $a\to b$ in $\Cscr$, so that $\tau_{\geq 0}f\we 0$ is the fiber computed in $\Cscr_{\geq 0}$. We have a fiber sequence $a\to b\to f[1]$ and hence a~fiber sequence $a\to b\to\tau_{\geq 0}(f[1])$ in $\Cscr_{\geq 0}$. As $\tau_{\geq 0}(f[1])\iso\pi_{-1}f$ and as $\pi_0a\to\pi_0b$ is surjective, with have that $b\to\tau_{\geq 0}(f[1])$ is zero, so that $a\we b\oplus\Omega(\pi_{-1}f)\we b$,\linebreak as desired. \end{proof} \begin{exam}\label{ex:short_exact} Suppose that $\F^\star$ is a filtered object in $\Cscr$ and consider the fiber sequence $\gr^{[b,c)}\to\gr^{[a,c)}\to\gr^{[a,b)}$ of filtered objects using the filtrations of Construction~\ref{const:filtered_gradeds} for $a\leq b\leq c$ (including infinite values). Then, \[ 0\to\pi_0^\B\left(\gr^{[b,c)}\right)\to\pi_0^\B\left(\gr^{[a,c)}\right)\to\pi_0^\B\left(\gr^{[a,b)}\right)\to 0 \] is short exact. Indeed, the middle term is the cochain complex \[ \dots\to 0\to\pi_{-a}\gr^a\to\pi_{-a-1}\gr^{a+1}\to\dots\pi_{-c+1}\gr^{c-1}\to 0\to\dots, \] and the left-hand term is the inclusion of the stupid truncation $\sigma^{\geq b}$ with quotient the right-hand term the stupid truncation $\sigma^{\leq b-1}$. It follows by Lemma~\ref{lem:truncation_exact} that $\Dec(\gr^{[b,c)})\to\Dec(\gr^{[a,c)})\to\Dec(\gr^{[a,b)})$ is in fact a fiber sequence of filtered spectra. \end{exam} Recall that $\ins^sM$ denotes the filtration on $M$ obtained by left Kan extension of the constant functor on $M$ along the inclusion $\{s\}\hookto\bZ^\op$. Concretely, $\F^i\ins^s M\we 0$ for $i>s$ and $\F^i\ins^s M\we M$ for $i\leq s$, with all transition maps the identity. The filtration on $\gr^a$ of Construction~\ref{const:filtered_gradeds} is naturally equivalent to $\ins^a\gr^a$. \begin{lemm}\label{lem:ins_t_exact} Let $\Cscr$ be a stable $\infty$-category with sequential limits and with a $t$-structure $(\Cscr_{\geq 0},\Cscr_{\leq 0})$. The functor $\ins^s\colon\Cscr\to\F\Cscr$ is $t$-exact with respect to the $t$-structure $(\Cscr_{\geq -s},\Cscr_{\leq -s})$ on $\Cscr$ and the Beilinson $t$-structure associated to $(\Cscr_{\geq 0},\Cscr_{\leq 0})$ on $\F\Cscr$. \end{lemm} \begin{proof} The functor $\ins^s$ lands in $\widehat{\F}\Cscr$, so it is enough to see it is $t$-exact as a~functor to $\widehat{\F}\Cscr$ by Definition~\ref{def:noncompletebeilinson}. As a coherent cochain complex, $\ins^sM$ is simply $M[s]$ in degree $s$ and $0$ elsewhere. This is a $t$-exact assignment with the given $t$-structures. \end{proof} \begin{exam} The functor $\ins^0\colon\Cscr\to\F\Cscr$ is $t$-exact. \end{exam} \begin{lemm}\label{lem:ins_decalage} The filtration $\Dec^\star(\gr^sM)$ is equivalent to $\tau_{\geq -s+\star}\gr^s M$. \end{lemm} \begin{proof} It is enough to consider the case $s=0$, in which case we have that $\tau_{\geq\star}^\B(\gr^0M)=\tau_{\geq\star}^\B(\ins^0M)\we\ins^0(\tau_{\geq\star}M)$ by Lemma~\ref{lem:ins_t_exact}. Taking colimits gives the result. \end{proof} \begin{proof}[Proof of Theorem~\ref{thm:decalage}] Consider the following commutative diagram %regarder si on remet les mathbf{1} etc... \[ \resizebox{15cm}{!}{ \xymatrix{ \pi_0\gr^{[0,r+1)}_\F \ar[d]&\pi_0\gr^{[0,\infty)}_\Dec\gr^{[0,r+1)}_\F\ar[r]^-{\mathbf{2}}_-\twoheadrightarrow\ar[d]\ar[l]_-{\mathbf{1}}^-\iso&\pi_0\gr^{[0,r)}_\Dec\gr^{[0,r+1)}_\F\ar[d]&\pi_0\gr^{[0,r)}_\Dec\gr^{[0,\infty)}_\F\ar[l]^-\iso_-{\mathbf{3}}\ar[r]^-{\mathbf{4}}_-\twoheadrightarrow\ar[d]&\pi_0\gr^{[0,r)}_\Dec\ar[d]\\ \pi_0\gr^{[-r,1)}_\F&\pi_0\gr^{[-r+1,\infty)}_\Dec\gr^{[-r,1)}_\F\ar[l]_-{\mathbf{5}}^-\hookleftarrow\ar[r]^-{\mathbf{6}}_-\iso&\pi_0\gr^{[-r+1,1)}_\Dec\gr^{[-r,1)}_\F&\pi_0\gr^{[-r+1,1)}_\Dec\gr^{[-r,\infty)}_\F\ar[r]^-{\mathbf{8}}_-\iso\ar[l]_-{\mathbf{7}}^-\hookleftarrow&\pi_0\gr^{[-r+1,1)}_\Dec, }} \] referred to below as the $2\times 5$-diagram. To make precise what is meant here, for example $\gr^{[0,\infty)}_\Dec\gr^{[0,r+1)}_\F=\Dec^0(\gr^{[0,r+1)}_\F)$, where $\gr^{[0,r+1)}_\F$ is viewed as a filtration via Construction~\ref{const:filtered_gradeds}. Similarly, $\gr^{[0,r)}_\Dec$ is the $[0,r)$-associated graded of the décalage filtration associated to the original filtration $\F^\star$. The maps arise from functoriality of the décalage construction. Note that $\E^{r+1}_{0,0}(\F)$ is the epi-mono factorization of the left-hand map, while $\E^r_{0,0}(\Dec(\F))$ is the epi-mono factorization of the right-hand map. Justification of the displayed properties of maps $\mathbf{1},\dots,\mathbf{8}$ will show that there is a canonical isomorphism of these objects of $\Cscr^\heart$. \begin{enumerate}\objectenumi \item\label{object1} There is a fiber sequence of filtered objects \[\tau_{\geq 0}^\B\gr^{[0,r+1)}_\F\to\gr^{[0,r+1)}_\F\to\tau_{\leq -1}^\B\gr^{[0,r+1)}_\F \] whose colimit is $\gr^{[0,\infty)}_\Dec\gr^{[0,r+1)}_\F\to\gr^{[0,r+1)}_\F\to\gr^{(-\infty,-1]}_\Dec\gr^{[0,r+1)}_\F$. By Example~\ref{ex:short_exact}, $\gr^{(-\infty,-1]}_\Dec\gr^{[0,r+1)}_\F$ has a finite filtration (induced by $\F$) with associated graded pieces $\gr^{(-\infty,-1]}_\Dec\gr^a_\F$. By Lemma~\ref{lem:ins_decalage}, $\gr^{(-\infty,-1]}_\Dec\gr^a_\F\we\tau_{\leq -a-1}\gr^a_\F$ for $0\leq a\leq r$. Thus, $\gr^{(-\infty,-1]}_\Dec\gr^{[0,r+1)}_\F$ is an iterated extension of objects in $\Cscr_{\leq -1}$, so it is in $\Cscr_{\leq -1}$, which shows that $\mathbf{1}$ is an isomorphism. \item\label{object2} By Lemma~\ref{ex:short_exact}, $\gr^{[0,\infty)}_\Dec\gr^{[0,r+1)}_\F$ has a finite filtration with graded pieces $\gr^{[0,\infty)}_\Dec\gr^a_\F\we\tau_{\geq -a}\gr^a_F$ for $0\leq a\leq r$ and similarly $\gr^{[0,r)}_\Dec\gr^{[0,r+1)}_\F$ has a finite filtration with graded pieces $\gr^{[0,r)}_\Dec\gr^a_\F\we\tau_{[-a,-a+r)}\gr^a_\F$. The natural map is compatible with these filtrations, so on its fiber there is a finite filtration with graded pieces $\tau_{\geq -a+r}\gr^a_\F$ for $0\leq a\leq r$. It follows that the fiber is connective and hence the displayed map $\mathbf{2}$ on $\pi_0$ is surjective. \item\label{object3} The fiber of the morphism which induces $\mathbf{3}$ is $\gr^{[0,r)}_\Dec\gr^{[r+1,\infty)}_\F$, which has a~finite filtration (now using $\Dec$) whose associated graded pieces are given by $|\pi_a^\B\gr^{[r+1,\infty)}_\F[a]|$ for $0\leq a-r-1+a$. In particular, as $a-2$. Hence, $\mathbf{3}$ is an isomorphism. \item\label{object4} The map $\mathbf{4}$ is induced by a map $\gr^{[0,r)}_\Dec\gr^{[0,\infty)}_\F\to\gr^{[0,r)}_\Dec$, whose cofiber is $\gr^{[0,r)}_\Dec\gr^{(-\infty,-1]}_\F$, where $\gr^{(-\infty,-1]}_\F$ denotes the filtered object \[ \dots\to 0\to\frac{\F^{-1}}{\F^0}\to\frac{\F^{-2}}{\F^0}\to\dots. \] It is enough to show that $\pi_0$ of this cofiber vanishes. For this, one can again filter using the Beilinson Whitehead tower and reduce to considering $|\pi_a^\B\gr^{(-\infty,-1]}_\F[a]|$ for $0\leq a>}[ld]& \pi_0\gr^{[0,\infty)}_\Dec\gr^{[0,r+1)}_\F\ar[l]^-{\iso}\ar[r]_-{\twoheadrightarrow}\ar[dd]& \pi_0\gr^{[0,r)}_\Dec\gr^{[0,r+1)}_\F\ar[dd]& \pi_0\gr^{[0,r)}_\Dec\gr^{[0,\infty)}_\F\ar[l]^-\iso\ar[r]_-\twoheadrightarrow\ar[dd]& \pi_0\gr^{[0,r)}_\Dec\ar[dd]\ar@{->>}[dr]& \\ \E^{r+1}_{0,0}(\F)\ar[dd]_{\d^{r+1}_\F}&&&&&&\E^r_{0,0}(\Dec(\F))\ar[dd]^{\d^r_\Dec}\\ & \pi_{-1}\gr^{[r+1,2r+2)}_\F& \pi_{-1}\gr^{[0,\infty)}_\Dec\gr^{[r+1,2r+2)}_\F\ar[l]\ar[r]& \pi_{-1}P& \pi_{-1}\gr^{[r,2r)}_\Dec\gr^{[0,\infty)}_\F\ar[l]_-{\bB}^-\hookleftarrow\ar[r]& \pi_{-1}\gr^{[r,2r)}_\Dec& \\ \E^{r+1}_{-r-1,r}(\F)&&&&&&\E^r_{-r,r-1}(\Dec(\F)).\\ & \pi_{-1}\gr^{[r+1,2r+2)}_\F\ar@{=}[uu]\ar@{->>}[ul]& \pi_{-1}\gr^{[r,\infty)}_\Dec\gr^{[r+1,2r+2)}_\F\ar[l]^-\iso\ar[uu]^{\bA}_\iso\ar[r]_-\twoheadrightarrow& \pi_{-1}\gr^{[r,2r)}_\Dec\gr^{[r+1,2r+2)}_\F\ar[uu]& \pi_{-1}\gr^{[r,2r)}_\Dec\gr^{[r+1,\infty)}_\F\ar[l]^-\iso\ar[uu]\ar[r]_-\twoheadrightarrow& \pi_{-1}\gr^{[r,2r)}_\Dec\ar@{=}[uu]\ar@{->>}[ur]& } $} \] The middle $3\times 5$-diagram is commutative by commutativity of the $4\times 5$-diagram above. The outer trapezoids are commutative by definition of the $\d^{r+1}_\F$ and $\d^r_{\Dec}$-differentials. Now, we claim that the arrow marked $\bA$ is an isomorphism and that the arrow marked $\bB$ is injective. To check the claim for $\bA$, we can argue as in the case of $\mathbf{2}$ above to observe that the cofiber of $\gr^{[r,\infty)}_\Dec\gr^{[r+1,2r+2)}_\F\to\gr^{[0,\infty)}_\Dec\gr^{[r+1,2r+2)}_\F$ admits a finite filtration with graded pieces given by $|\pi_a^\B\gr^{[r+1,2r+2)}_\F[a]|$ for $0\leq aN$ for some fixed $N$. A left half-plane filtration is a filtration $\F^\star M$ such that $\gr^{-s}M\we 0$ for $s>N$. Lower and right half-plane filtrations are defined similarly. \item\label{defi5.2.2} By intersecting regions in the plane, one finds the notion of first, second, third, and fourth quadrant spectral sequences and filtrations. For example, a first quadrant filtration is one which is bounded below in the Beilinson $t$-structure and where $\gr^{-s}M\we 0$ for $-s$ sufficiently large. \item\label{defi5.2.3} Finally, a spectral sequence is column-bounded if it is left and right half-plane, so it is concentrated in a bounded vertical strip in the $(s,t)$-plane. A~filtration $\F^\star M$ is bounded if $\gr^sM\we 0$ for $s\notin[-N,N]$ for some integer~$N$. A filtration is row-bounded if it is bounded in the Beilinson $t$-structure. A~spectral sequence is row-bounded if $\E^1_{s,t}$ vanishes for $t\notin[-N,N]$, so the $\E^1$-page is concentrated in a bounded horizontal strip in the $(s,t)$-plane. \end{enumerate} \end{defi} \begin{lemm} If $\F^\star M$ is a filtration which is upper half-plane, then the associated spectral sequence is upper half-plane. Similarly for left half-plane, column-bounded, row-bounded, etc. \end{lemm} \begin{proof} Left to the reader. \end{proof} \begin{nota}[Boundaries and cycles]\label{not:boundaries} Each term $\E^r_{s,t}$ is a subquotient of $\pi_{s+t}\gr^{-s}M$, defining subobjects $\tilde{\B}^r_{s,t}\subseteq\tilde{\Z}^r_{s,t}$ of $\pi_{s+t}\gr^{-s}M$ consisting of boundaries and cycles with the properties that \[ \dots\subseteq\tilde{\B}^r_{s,t}\subseteq\tilde{\B}^{r+1}_{s,t}\subseteq\cdots \dots\subseteq\tilde{\Z}^{r+1}_{s,t}\subseteq\tilde{\Z}^{r}_{s,t}\subseteq\dots, \] and \[ \frac{\tilde{\Z}^r_{s,t}}{\tilde{\B}^r_{s,t}}\iso\E^{r+1}_{s,t}. \] Alternatively, the cycles may be defined as the terms $\E^{r+1}_{s,t}(\gr^{[-s,\infty)}_\F)$ (so there are no nonzero differentials hitting the $(s,t)$-term); the boundaries may be defined as the kernels of the natural maps $\E^{r+1}_{s,t}(\F^\star M)\to\E^{r+1}_{s,t}(\gr_\F^{(-\infty,-s]}M)$. \end{nota} \begin{defi}[The $\E^\infty$-page] Assume that $\Cscr$ admits sequential limits and colimits (which implies that $\Cscr^\heart$ admits countable unions and intersections). Let $\tilde{\B}^\infty_{s,t}=\colim_r\tilde{\B}^r_{s,t}$ and let $\tilde{\Z}^{\infty}_{s,t}=\lim_r\tilde{\Z}^r_{s,t}$, where the limits and colimits are computed in $\Cscr^\heart$.\footnote{Note that $\tilde{\Z}^\infty_{s,t}$ is necessarily a subobject of $\E^1_{s,t}\iso\pi_{s+t}\gr^{-s}$ since limits are left exact, but that the natural map $\tilde{\B}^\infty_{s,t}\to\E^1_{s,t}$ need not be injective in general. However, it will be injective as long as the $t$-structure on $\Cscr$ is compatible with sequential colimits.} The sequential limits and colimits can be obtained by computing them in $\Cscr$ and then applying $\pi_0$. The $\E^\infty$-page of the spectral sequence is the lattice of objects with \[ \E^\infty_{s,t}=\coker\left(\tilde{\B}^\infty_{s,t}\to\tilde{\Z}^\infty_{s,t}\right). \] \end{defi} \begin{rema} If $\F^\star M$ is upper half-plane or left half-plane, then the $\E^\infty$-page can be accessed in a slightly different way. For example, if $\F^\star M$ is $(-N)$-connective in the Beilinson $t$-structure (so it is upper half-plane), then for $n\geq N+t+1$, no differential can hit $\E^n_{s,t}$ so one obtains a decreasing sequence \[ \dots\subseteq\E^{N+t+3}_{s,t}\subseteq\E^{N+t+2}_{s,t}\subseteq\E^{N+t+1}_{s,t} \] of objects of $\Cscr^\heart$. Since $\F^\star M$ is upper half-plane or left half-plane, for any pair $(s,t)$, the boundary sequence $\tilde{\B}^r_{s,t}$ stabilizes, which implies that the two definitions of the $\E^\infty$-page agree. It follows that $\E^\infty_{s,t}$ is the intersection of these objects. \end{rema} Recall that a decreasing system $\G^\star N$ of objects of $\Cscr$ (and in particular $\Cscr^\heart$) is pro-zero if for each $n\in\bZ$ there is an $m\geq n$ such that $\G^mN\to\G^nN$ is nullhomotopic (zero if in $\Cscr^\heart$). \begin{lemm}\label{lem:prozero} Let $\Cscr$ be a stable $\infty$-category with sequential limits. If $\G^\star N$ is\linebreak a pro-zero filtered object of $\Cscr$, then it is a complete filtration, i.e., $\lim_s\G^s N\we 0$. \end{lemm} \begin{proof} Let $M\in\Cscr$. Then, $\F^\star X=\bMap_\Cscr(M,\F^\star N)$ is a pro-zero filtered spectrum and $\lim_s\F^s X=\bMap_\Cscr(M,\lim_s\F^s N)$. Since $M$ was arbitrary, it follows that it is enough to show that $\lim_s\F^s X\we 0$ if $\F^\star X$ is a pro-zero filtered spectrum. However, for each $n\in\bZ$, $\pi_n\F^\star X$ is a pro-zero filtered abelian group, so that ${\lim_s}^1\pi_n\F^s X=0=\lim_s\pi_n\F^s X$ by the Mittag--Leffler condition. By the Milnor sequence \[ 0\to{\lim_s}^1\pi_{n+1}\F^s X\to\pi_n\lim_s\F^sX\to\lim_s\pi_n\F^sX\to 0, \] which follows from the fact that the $t$-structure on spectra is compatible with countable products, it follows that $\pi_n\lim_s\F^sX\we 0$ for all $n$. In other words, $\lim_s\F^s X\we 0$. \end{proof} If $\Cscr$ is a stable $\infty$-category with sequential limits, derived limits of objects of $\Cscr^\heart$ are always computed in $\Cscr$. This notion depends on the ambient $\infty$-category $\Cscr$ and not just on the abelian category $\Cscr^\heart$. \begin{theo}[Convergence Theorem]\label{thm:convergence} Suppose that $\Cscr$ admits sequential limits and colimits and that the fixed $t$-structure on $\Cscr$ is compatible with sequential colimits. Let $\F^\star M$ be a filtration on an object $M$ of $\Cscr$. \begin{enumerate}\romanenumi \item\label{theo5.8.1} If $\F^\star M$ is an exhaustive filtration of $M$, then for each $n$ the induced filtration $\F^\star\pi_nM$ is exhaustive. \item\label{theo5.8.2} If for some integer $n$ the pro-object $\{\pi_n\F^sM\}_s$ is pro-zero, then the filtration $\F^\star\pi_nM$ is pro-zero and derived complete. \item\label{theo5.8.3} There are natural inclusions \[ \gr^s\pi_nM\subseteq\E^\infty_{-s,n+s} \] for each $s$ and $n$. The inclusions are equalities if $\F^\star M$ is column-bounded, row-bounded, first quadrant, or third quadrant. \end{enumerate} \end{theo} \begin{proof}[{Proof of Theorem~\ref{thm:convergence}}] The fact that $\F^\star\pi_nM$ is exhaustive follows from the compatibility of the $t$-structure on $\Cscr$ with sequential colimits and the fact that $\F^\star M$ is exhaustive, so $\colim_s\F^sM\we M$. This proves~\eqref{theo5.8.1}. For~\eqref{theo5.8.2}, since $\pi_n\F^\star M\to\F^\star\pi_nM$ is a surjection of pro-objects of $\Cscr^\heart$ and since $\pi_n\F^\star M$ is pro-zero, it follows that $\F^\star\pi_nM$ is pro-zero and in particular derived complete by Lemma~\ref{lem:prozero}. Now, consider the commutative diagram \[ \xymatrix{ &&&0\ar[d]\\ 0\ar[r]&\ker(\pi_n\F^{s+1}M\to\pi_nM)\ar[r]\ar[d]&\pi_n\F^{s+1}M\ar[r]\ar[d]&\F^{s+1}\pi_nM\ar[r]\ar[d]&0\\ 0\ar[r]&\ker(\pi_n\F^sM\to\pi_nM)\ar[r]\ar[d]&\pi_n\F^sM\ar[r]\ar[d]&\F^s\pi_nM\ar[r]\ar[d]&0\\ 0\ar[r]&B\ar[r]\ar[d]&Z\ar[d]\ar[r]&\frac{\F^s\pi_nM}{\F^{s+1}\pi_nM}\ar[d]\ar[r]&0\\ &0&0&0 } \] of exact sequences, where $B$ and $Z$ are defined to be the vertical cokernels of the upper left square. This implies that $B$ and $Z$ are subobjects of $\pi_n\gr^sM$ and in fact they are subobjects of $\tilde{\Z}^\infty_{-s,s+n}$. Indeed, $Z\subseteq\tilde{\Z}^\infty_{-s,s+n}$ is the subobject of permanent cycles that actually lift to $\pi_n\F^sM$. The kernel $\ker(\pi_n\F^sM\to\pi_nM)$ is the union of the kernels of $\pi_n\F^sM\to\pi_n\F^{s-N}M$ since the $t$-structure is compatible with sequential colimits. The commutative diagram \[ \xymatrix{ \pi_n\F^{s+1}M\ar[r]\ar[d]&\pi_n\F^{s+1}M\ar[d]\\ \pi_n\F^sM\ar[r]\ar[d]&\pi_n\F^{s-N}M\ar[d]\\ \pi_n\gr^sM\ar[r]&\pi_n\gr^{[s-N,s+1)}M, } \] in which the columns are exact, implies that $\ker(\pi_n\F^sM\to\pi_nM)$ maps to \[ \ker\left(\pi_n\gr^sM\to\pi_n\gr^{[s-N,s+1)}M\right), \] which is $\tilde{\B}^n_{-s,s+n}$. It follows there is a commutative diagram of exact sequences \[ \xymatrix{ &0\ar[d]&0\ar[d]\\ 0\ar[r]&B\ar[r]\ar[d]&Z\ar[d]\\ 0\ar[r]&\tilde{\B}^\infty_{-s,s+n}\ar[r]&\tilde{\Z}^{\infty}_{-s,s+n}. } \] This diagram presents $B$ as a pullback. To prove this, write $B^N=\ker(\pi_n\F^sM\to\pi_n\F^{s-N}M)$. Consider the intersection $\B^N_{-s,s+n}\cap Z\subseteq\pi_n\gr^sM$ and let $C$ be the inverse image of the intersection in $\pi_n\F^sM$. By construction, the map $C\hookto\pi_n\F^sM\to\pi_n\F^{s-N}M\to\gr^{[s-N,s+1)}M$ maps to zero, so $C$ maps to $\im(\pi_n\F^{s+1}M\to\pi_n\F^{s-N}M)$ in $\pi_n\F^{s-N}M$. Let $\tilde{C}$ be the pullback in \[ \xymatrix{ \tilde{C}\ar[r]\ar[d]&C\ar[d]\\ \pi_n\F^{s+1}M\ar[r]&\im\left(\pi_n\F^{s+1}M\to\pi_n\F^{s-N}M\right). } \] Since the bottom map is an epimorphism, so is $\tilde{C}\to C$. Let $f$ be the composition $\tilde{C}\to C\hookto\pi_n\F^sM$ and let $g$ be the composition $\tilde{C}\to\pi_n\F^{s+1}M\to\pi_n\F^sM$. Consider the map $\tilde{C}\xrightarrow{g-f}\pi_n\F^sM$. By construction, $g-f$ maps $\tilde{C}$ into $B^N$. The kernel of $\tilde{C}\to C$ is isomorphic to $\ker(\pi_n\F^{s+1}M\to\pi_n\F^{s-N}M)$, which maps to zero in $\pi_n\gr^sM$ (using any of the maps $f$, $g$, or $g-f$). Thus, there is a commutative square \[ \xymatrix{ \tilde{C}\ar[r]\ar[d]^{g-f}&C\ar[d]\\ B^N\ar[r]&\B^N_{-s,s+n}\cap Z. } \] The top and right arrows are surjective and hence the bottom arrow is surjective as well. Since the $t$-structure is compatible with sequential colimits, $B\iso\cup\B^N\to\cup(\B^N_{-s,s+n}\cap Z)\iso(\cup\B^n_{-s,s+n})\cap Z$ is a surjection. It follows that the induced map $Z/B\to\tilde{\Z}^\infty_{-s,s+n}/\tilde{\B}^\infty_{-s,s+n}\iso\E^\infty_{-s,s+n}$ is injective. It is an isomorphism if $Z\to\tilde{\Z}^{\infty}_{-s,s+n}$ is surjective. This is the case if $\F^\star M$ is column-bounded, row-bounded, first quadrant, or third quadrant since in those cases $\pi_n\F^sM\to\pi_n\F^{[s,s+N+1)}M$ is an isomorphism for large $N$. This proves Theorem~\ref{thm:convergence}$\MK$\eqref{theo5.8.3}. \end{proof} \section{Indexing conventions}\label{sec:reindexing} It should be understood that the choice of $\E^1_{s,t}=\pi_{s+t}\gr^{-s}M$ is entirely a convention. One could make many other choices. This one is the Serre grading convention. \begin{table}[!h] \caption{Spectral sequence conventions for decreasing filtrations $\F^\star$ and increasing filtrations $\F_\star$. We use $\To$ even when no convergence is implied.}\label{fig:ssconventions} \centering %\makebox[\textwidth]{ \begin{tabular}{||l l l||} \hline $\pi_*\F^\star M$&$\E^1_{s,t}=\pi_{s+t}\gr^{-s}M\To\pi_{s+t}M$&$(-r,r-1)$\\ &$\F^s{\pi_nM}=\im(\pi_n\F^sM\to\pi_nM)$&$\gr^s\pi_nM\iso\E^\infty_{-s,s+n}$\\ \hline $\pi_*\F_\star M$&$\E^1_{s,t}=\pi_{s+t}\gr_sM\To\pi_{s+t}M$&$(-r,r-1)$ \\ &$\F_s\pi_nM=\im(\pi_n\F_sM\to\pi_nM)$&$\gr_s\pi_nM\iso\E^\infty_{s,-s+n}$\\ \hline $\H^*\F^\star M$&$\E_1^{s,t}=\H^{s+t}\gr^sM\To\H^{s+t}M$&$(r,-r+1)$\\ &$\F^s\H^nM=\im(\H^n\F^sM\to\H^nM)$&$\gr^s\H^nM\iso\E_\infty^{s,-s+n}$\\ \hline $\H^*\F_\star M$&$\E_1^{s,t}=\H^{s+t}\gr_{-s}M\To\H^{s+t}M$&$(r,-r+1)$\\ &$\F_s\H^nM=\im(\H^n\F_sM\to\H^nM)$&$\gr_s\H^nM\iso\E_\infty^{-s,s+n}$\\ \hline \end{tabular} %} \end{table} Sometimes it is useful to reindex a spectral sequence to start at the second page instead of the first, in which case we write the resulting spectral sequence as $'\E^2$. This is a purely psychological move, but it leads to some useful comparisons as one can see in Section~\ref{sec:mysterious}. The matrices to go back and forth between the $\E^1$ and $\E^2$-conventions are \[ \begin{pmatrix}0&-1\\1&2 \end{pmatrix}\colon\text{from $\E^2$ to $\E^1$}\quad\text{and}\quad \begin{pmatrix}2&1\\-1&0 \end{pmatrix}\colon\text{from $\E^1$ to $\E^2$}. \] \begin{table}[!h] \caption{Reindexing to $'\E^2$. The differentials have the same bidegree as in the $\E^1$-spectral sequences from Table~\ref{fig:ssconventions}. The filtrations on the abutment are also defined in the same way as in the $\E^1$ spectral sequences.}\label{fig:E2ssconventions} \centering %\makebox[\textwidth]{ \begin{tabular}{||l l l||} \hline $\pi_*\F^\star M$&$'\E^2_{s,t}=\E^1_{-t,s+2t}=\pi_{s+t}\gr^{t}M\To\pi_{s+t}M$&$\gr^s\pi_nM\iso\E^\infty_{-s+n,s}$\\ \hline $\pi_*\F_\star M$&$'\E^2_{s,t}=\E^1_{-t,s+2t}=\pi_{s+t}\gr_{-t}M\To\pi_{s+t}M$&$\gr_s\pi_nM\iso\E^\infty_{s+n,-s}$\\ \hline $\H^*\F^\star M$&$'\E_2^{s,t}=\E_1^{-t,s+2t}=\H^{s+t}\gr^{-t}M\To\H^{s+t}M$&$\gr^s\H^nM\iso\E_\infty^{s+n,-s}$\\ \hline $\H^*\F_\star M$&$'\E_2^{s,t}=\E_1^{-t,s+2t}=\H^{s+t}\gr_{t}M\To\H^{s+t}M$&$\gr_s\H^nM\iso\E_\infty^{-s+n,s}$\\ \hline \end{tabular} %} \end{table} \begin{table}[!h] \caption{Adams indexing spectral sequence conventions.}\label{fig:adamsssconventions} \centering %\makebox[\textwidth]{ \begin{tabular}{||l l l||} \hline $\pi_*\F^\star M$&$\E^1_{s,t}=\pi_{s}\gr^{t}M\To\pi_{s}M$&$(-1,r)$\\ &$\F^s{\pi_nM}=\im(\pi_n\F^sM\to\pi_nM)$&$\gr^s\pi_nM\iso\E^\infty_{n,s}$\\ \hline $\pi_*\F_\star M$&$\E^1_{s,t}=\pi_{s}\gr_{-t}M\To\pi_{s}M$&$(-1,r)$\\ &$\F_s\pi_nM=\im(\pi_n\F_sM\to\pi_nM)$&$\gr_s\pi_nM\iso\E^\infty_{n,-s}$\\ \hline $\H^*\F^\star M$&$\E_1^{s,t}=\H^{s}\gr^{-t}M\To\H^{s}M$&$(1,-r)$\\ &$\F^s\H^nM=\im(\H^n\F^sM\to\H^nM)$&$\gr^s\H^nM\iso\E_\infty^{n,-s}$\\ \hline $\H^*\F_\star M$&$\E_1^{s,t}=\H^{s}\gr_{t}M\To\H^{s}M$&$(1,-r)$\\ &$\F_s\H^nM=\im(\H^n\F_sM\to\H^nM)$&$\gr_s\H^nM\iso\E_\infty^{n,s}$\\ \hline \end{tabular}%} \end{table} There is also a common reindexing in homotopy theory called the Adams grading convention. The beautiful charts one sees of the Adams spectral sequence are all given in this convention. The matrices to go back and forth between the Serre and Adams grading conventions are \[ \begin{pmatrix}0&-1\\1&1 \end{pmatrix}\colon\text{from Adams to Serre}\quad\text{and}\quad \begin{pmatrix}1&1\\-1&0 \end{pmatrix}\colon\text{from Serre to Adams}. \] Note that Adams spectral sequences are not spectral sequences if one defines a spectral sequence to have differentials of a certain bidegree. The main advantage of Adams indexing is that all contributions to $\pi_n$ of the abutment are in the $n^{\rm th}$ column and all differentials leaving or entering this column relate to the $(n+1)^{\rm st}$ or $(n-1)^{\rm st}$ column. This makes displaying spectral sequences more compact in some situations. \goodbreak \section{Multiplicativity}\label{sec:multiplicativity} Compare the material in this section to related results in Hedenlund's thesis~\cite{hedenlund-thesis}. \begin{defi}\label{def:graded_lines} The $\E^r$-page construction defines a functor \[ \E^r\colon\F\Cscr\to\Gr\left(\Ch^\bullet(\Cscr^\heart)\right) \] from the $\infty$-category of filtrations in $\Cscr$ to the $1$-category of graded objects in cochain complexes in $\Cscr^\heart$. The weight $n$ piece of the $\E^r$-page is the weight $n$ piece of the $\E^1$-page of $\Dec^{(r-1)}(\F)$, which is \[ \E^1_{-\bullet,n}\left(\Dec^{(r-1)}(\F)\right). \] Under the regrading, this corresponds to \[ \E^r_{(r-1)n-r\cs,-(r-2)n+(r-1)\cs}, \] which is the line of slope $\frac{1-r}{r}=-1+\frac{1}{r}$ through the point of height $n-\frac{n}{r}$ on the $t$-axis. This cochain complex is indexed so that the cohomological degree~$0$ term is $\E^r_{(r-1)n,-(r-2)n}$ and the cohomological degree~$1$ term is $\E^r_{(r-1)n-r,-(r-2)n+(r-1)}$. Note that the cohomological degree $n$ term in weight $n$ is $\E^r_{-n,n}$. See Figure~\ref{fig:weight1generalss} for an illustration of the weight $1$ term on the first three pages of a general spectral sequence. \end{defi} The choice of indexing for the $\E^r$-page is not arbitrary but is enforced by the décalage approach to spectral sequences; see Theorem~\ref{thm:decalage}. We note the following lemma for the readers convenience. \begin{lemm}\label{lem:lookup} The term $\E^r_{s,t}$ is the cohomological degree $(r-2)s+(r-1)t$ term in weight $(r-1)s+rt$. \end{lemm} \begin{sseqdata}[name = generalss, homological Serre grading, classes = {draw = none }, draw orphan edges, run off = -] \foreach \x in {-7,...,3} { \foreach \y in {-2,...,6} { \class["\bullet"](\x,\y) } } \foreach \x in {-4,...,0} { \foreach \y in {0,...,4} { \d1(\x,\y) \replacetarget["\bullet"] } } \foreach \y in {0,...,4} { \d1(1,\y) \replacetarget["\bullet"] \replacesource["\bullet"] } \foreach \x in {-4,...,0} { \foreach \y in {0,...,4} { \d2(\x,\y) \replacetarget["\bullet"] } } \foreach \x in {-4,...,0} { \d2(\x,-1) \replacetarget["\bullet"] \replacesource["\bullet"] } \foreach \x in {1,2} { \foreach \y in {-1,...,4} { \d2(\x,\y) \replacetarget["\bullet"] \replacesource["\bullet"] } } \foreach \x in {-4,...,3} { \foreach \y in {0,...,4} { \d3(\x,\y) } } \foreach \x in {-1,0,1,2,3} { \d3(\x,-2) \d3(\x,-1) } \d3(-2,-2) \d3(-2,-1) \d3(-3,-1) \end{sseqdata} \begin{figure}[!h] \centering \printpage[name = generalss, page = 1,xscale=0.8,yscale=0.8,x range = {-4}{0},y range = {0}{4}] \quad \printpage[name = generalss, page = 2,xscale=0.8,yscale=0.8,x range = {-4}{0},y range = {0}{4}] \quad \printpage[name = generalss, page = 3,xscale=0.8,yscale=0.8,x range = {-4}{0},y range = {0}{4}] \caption{The $\E_1$, $\E_2$, and $\E_3$-pages of a general spectral sequence.} \label{fig:generalss} \end{figure} \begin{sseqdata}[name = weight1page1, homological Serre grading, classes = {draw = none } ] \class["\bullet"](-3,1) \class["\bullet"](-2,1) \class["\bullet"](-1,1) \class["\circ"](0,1) \class["\bullet"](1,1) \class["\bullet"](2,1) \class["\bullet"](3,1) \foreach \x in {-2,...,3} { \d1(\x,1) } \end{sseqdata} \begin{sseqdata}[name = weight1page2, homological Serre grading, classes = {draw = none } ] \class["\circ"](1,0) \class["\bullet"](-1,1) \class["\bullet"](-3,2) \class["\bullet"](-5,3) \class["\bullet"](3,-1) \class["\bullet"](5,-2) \class["\bullet"](7,-3) \d2(7,-3) \d2(5,-2) \d2(3,-1) \d2(1,0) \d2(-1,1) \d2(-3,2) \end{sseqdata} \begin{sseqdata}[name = weight1page3, homological Serre grading, classes = {draw = none } ] \class["\circ"](2,-1) \class["\bullet"](-1,1) \class["\bullet"](-4,3) \class["\bullet"](-7,5) \class["\bullet"](5,-3) \class["\bullet"](8,-5) \class["\bullet"](11,-7) \d3(11,-7) \d3(8,-5) \d3(5,-3) \d3(2,-1) \d3(-1,1) \d3(-4,3) \end{sseqdata} \begin{figure}[h!] \centering \printpage[name = weight1page1, page = 1,xscale=0.8,yscale=0.8,x range = {-2}{2},y range = {0}{4}] \quad \printpage[name = weight1page2, page = 2,xscale=0.8,yscale=0.8,x range = {-3}{5},y range = {-2}{2}] \quad \printpage[name = weight1page3, page = 3,xscale=0.5,yscale=0.5,x range = {-4}{8},y range = {-5}{3}] \caption{The weight $1$ parts of the $\E_1$, $\E_2$, and $\E_3$-pages of a general spectral sequence. The $\circ$ class is degree $0$ of the cochain complex.} \label{fig:weight1generalss} \end{figure} The reader is referred to~\cite[Chapter~2]{ha} for all conventions about symmetric monoidal structures and the related theory of $\infty$-operads. For a symmetric monoidal structure $\Cscr^\otimes$ on an $\infty$-category $\Cscr$, the tensor product of two objects $X$ and $Y$ is written $X\otimes_\Cscr Y$. In the case where $\Cscr=\Perf(k)^\otimes$ or $\Cscr=\D(k)^\otimes$ for an $\bE_\infty$-ring $k$, this notation will be simplified to $X\otimes_kY$. \goodbreak Let $\Cscr$ be a stable $\infty$-category. If $\Cscr$ admits both a symmetric monoidal structure $\Cscr^\otimes$ such that the tensor product is exact in each variable and a $t$-structure $(\Cscr_{\geq 0},\Cscr_{\leq 0})$, the $t$-structure is said to be compatible with the symmetric monoidal structure if $\Cscr_{\geq 0}$ contains the unit object $\1_\Cscr$ and is closed under tensor products. In this case, the prestable $\infty$-category $\Cscr_{\geq 0}$ inherits a symmetric monoidal structure $\Cscr_{\geq 0}^\otimes$ and the inclusion $\Cscr_{\geq 0}\subseteq\Cscr$ is symmetric monoidal. Suppose that $\Cscr$ is a stable $\infty$-category with a symmetric monoidal structure $\Cscr^\otimes$ which is exact in each variable and suppose also that $\Cscr$ admits sequential colimits and that the tensor product preserves sequential colimits in each variable. Using that $\bZ^\op$ naturally admits a symmetric monoidal structure given by addition of integers, there is an induced Day convolution symmetric monoidal structure $\F\Cscr^\otimes$ on the $\infty$-category $\Fun(\bZ^\op,\Cscr)$ of decreasing filtrations in $\Cscr$. This is constructed in general in the $\infty$-categorical context by Glasman~\cite{glasman}; see also~\cite[Example~2.2.6.17]{ha}. To describe the tensor product of any given pair of decreasing filtrations $\F^\star$ and $\G^\star$, one considers the left Kan extension \[ \xymatrix{ \bZ^\op\times\bZ^\op\ar[rrr]^{(s,s')\mapsto\F^s\otimes_\Cscr\F^{s'}}\ar[d]^+&&&\Cscr\\ \bZ^\op.\ar@{.>}[urrr] } \] The left Kan extension exists by our assumption that $\Cscr$ admits sequential colimits and hence all countable colimits since $\Cscr$ is stable. The constant filtrations $\Cscr\subseteq\F\Cscr$ form a $\otimes$-ideal in $\F\Cscr$ since if $\F^\star$ is the constant filtration on $M$, then $\F^\star\otimes_{\F\Cscr}\G^\star$ is the constant filtration on $M\otimes_\Cscr\G^{-\infty}$, as one sees from the left Kan extension definition above. It follows that if $\Cscr$ admits sequential limits, then $\widehat{\F}\Cscr$ admits a symmetric monoidal structure $\widehat{\F}\Cscr^\otimes$. The tensor product here is the completed tensor product $\widehat{\otimes}$ obtained by computing the tensor product in $\F\Cscr$ and then completing. \begin{enonce}{Warning} The category $\Xi^\op$ which classifies cochain complexes is also symmetric monoidal. However, the induced Day convolution symmetric monoidal structure on $\widehat{\F}\Cscr\we\Fun_*(\Xi^\op,\Cscr)$ does not agree with the symmetric monoidal structure on $\widehat{\F}\Cscr$ induced by completing the Day convolution symmetric monoidal structure on $\Fun(\bZ^\op,\Cscr)$. Indeed, as pointed out by Thomas Nikolaus, this does not even work in the $1$-categorical situation of ordinary cochain complexes of abelian groups. The unit object of $\Fun_*(\Xi^\op,\Ab)$ is the cochain complex $\dots\to0\to\bZ\xrightarrow{\id}\bZ\to 0\to \dots$, where the non-trivial entries are in cohomological degrees $0$ and $1$. \end{enonce} Left Kan extension along $\bZ^\delta\to\bZ^\op$ induces a symmetric monoidal functor\linebreak $\ins^\star: \Gr\Cscr^\otimes\to\F\Cscr^\otimes$. \begin{prop}\label{prop:grtensor} The associated graded functor $\F\Cscr\to\Gr\Cscr$ naturally upgrades to a symmetric monoidal functor $\F\Cscr^\otimes\to\Gr\Cscr^\otimes$ and the induced composition $\Gr\Cscr^\otimes\to\F\Cscr^\otimes\to\Gr\Cscr^\otimes$ is equivalent to the identity as a symmetric monoidal functor. \end{prop} \begin{proof} This is not a formal result and is proved in~\cite[Proposition~3.2.1]{lurie-rotation} when $\Cscr\we\Sp$. That proof works in general. \end{proof} The proposition gives one of the main ways to understand the tensor product on filtered objects. Indeed, the tensor product on $\Gr\Cscr^\otimes$ is straightforward. If $X(\star)$ and $Y(\star)$ are graded objects in $\Cscr$, then \[ (X\otimes_{\Gr\Cscr} Y)(s)\we\bigoplus_{i+j=s} X(i)\otimes_\Cscr Y(j). \] For example, this is used in the next lemma to understand the compatibility of the Day convolution symmetric monoidal structure on $\F\Cscr$ with Beilinson $t$-structures. \begin{lemm} Suppose that $\Cscr$ is a stable $\infty$-category admitting sequential limits and colimits and equipped with a symmetric monoidal structure $\Cscr^\otimes$ whose tensor product is exact and preserves sequential colimits in each variable. If $\Cscr$ is additionally equipped with a $t$-structure $(\Cscr_{\geq 0},\Cscr_{\leq 0})$ compatible with $\Cscr^{\otimes}$, then the induced Beilinson $t$-structure on $\F\Cscr$ is compatible with the Day convolution symmetric monoidal structure on $\F\Cscr^\otimes$. \end{lemm} \begin{proof} The unit object of $\F\Cscr^\otimes$ is $\ins^0\1_\Cscr$ which has $\gr^0\ins^0\1_\Cscr\we\1_\Cscr$ and\linebreak $\gr^s\ins^0\1_\Cscr\we 0$ for $s\neq 0$. This object is connective in the Beilinson $t$-structure. The tensor product formula $\gr^s(\F^\star\otimes_{\F\Cscr}\G^\star)\we\oplus_{i+j=s}\gr^i_\F\otimes_\Cscr\gr^j_\G$ shows that if $\F^\star$ and $\G^\star$ are connective in the Beilinson $t$-structure, then so is $\F^\star\otimes_{\F\Cscr}\G^\star$. \end{proof} If $\Cscr$ is a stable $\infty$-category equipped with a symmetric monoidal structure $\Cscr^\otimes$ whose tensor product is exact in each variable and a compatible $t$-structure $(\Cscr_{\geq 0},\Cscr_{\leq 0})$, then $\Cscr^\heart$ inherits a symmetric monoidal structure $\Cscr^{\heart,\otimes}$. Indeed, by definition, the $\infty$-category $\Cscr_{\geq 0}$ of connective objects inherits a symmetric monoidal structure $\Cscr_{\geq 0}^\otimes$ by compatibility. The unit object of $\Cscr^{\heart,\otimes}$ is $\pi_0\1_\Cscr$ and the tensor product is formed by forming the tensor product in $\Cscr_{\geq 0}$ and applying $\pi_0$. \begin{exam}[Koszul symmetric monoidal structure] Let $\Cscr$ be a symmetric monoidal stable $\infty$-category with sequential colimits whose tensor product is exact and preserves sequential colimits in each variable with a compatible $t$-structure. Then, $\Gr\Cscr$ admits a symmetric monoidal structure via Day convolution and a compatible $t$-structure where a graded object $X(\star)$ is connective if $X(i)\in\Cscr_{\geq i}$ for all $i\in\bZ$. The heart $(\Gr\Cscr)^\heart$ is equivalent to the abelian category $\Gr(\Cscr^\heart)$ of graded objects in $\Cscr^\heart$ and the induced symmetric monoidal structure agrees with the usual monoidal structure on graded objects, with a symmetric braiding given by the Koszul sign rule. We will call this the Koszul symmetric monoidal structure on $\Gr(\Cscr^\heart)$. \end{exam} \begin{lemm}\label{lem:homotopylaxmonoidal} Suppose that $\Cscr$ is a stable $\infty$-category with sequential colimits equipped with a symmetric monoidal structure $\Cscr^\otimes$ whose tensor product is exact and preserves sequential colimits in each variable and a compatible $t$-structure $(\Cscr_{\geq 0},\Cscr_{\leq 0})$. Then, the functors \[ \tau_{\geq\star}\colon\Cscr\to\F\Cscr \] and \[ \pi_*\colon\Cscr\to\Gr(\Cscr^\heart) \] admit lax symmetric monoidal structures, where $\F\Cscr$ is equipped with the Day convolution symmetric monoidal structure and $\Gr(\Cscr^\heart)$ is equipped with the Koszul symmetric monoidal structure. \end{lemm} \begin{proof}[{Proof of Lemma~\ref{lem:homotopylaxmonoidal}}] Consider the $t$-structure on $\F\Cscr$ with connective objects those filtrations $\F^\star$ such that $\F^s$ is in $\Cscr_{\geq s}$ for all $s\in\bZ$. This is not a Beilinson $t$-structure, but it is still compatible with the Day convolution symmetric monoidal structure on $\Cscr$. It is called the positive $t$-structure by Raksit in~\cite{raksit}. We will call it the Postnikov $t$-structure and denote its connective objects by $(\F\Cscr)_{\geq 0}^\P$. In particular, since the inclusion $\F\Cscr_{\geq 0}^\P\subseteq\F\Cscr$ is symmetric monoidal, the right adjoint $\tau_{\geq 0}^\P$ is lax symmetric monoidal. The heart of the Postnikov $t$-structure is $\Gr(\Cscr^\heart)$ with the Koszul sign rule. Now, $\tau_{\geq\star}$ is the composition of the lax symmetric monoidal functors \[ \Cscr\xrightarrow{\text{constant}}\F\Cscr\xrightarrow{\tau_{\geq 0}^\P}\F\Cscr_{\geq 0}^\P \] and $\pi_*$ is the composition of the lax symmetric monoidal functors \[ \Cscr\xrightarrow{\text{constant}}\F\Cscr\xrightarrow{\tau_{\geq 0}^\P}\F\Cscr_{\geq 0}^\P\xrightarrow{\pi_0^\P}\Gr(\Cscr^\heart). \] This completes the proof. \end{proof} \begin{enonce} {Construction}[The $\E^r$-monoidal structure] Let $\Cscr^\otimes$ be a stable symmetric monoidal $\infty$-category which admits sequential limits and colimits, whose tensor product is exact and preserves sequential colimits in each variable, and which is equipped with a compatible $t$-structure. We view the $\E^r$-page functor as a functor from $\F\Cscr$ to $\Gr(\Ch^\bullet(\Cscr^\heart))$. We want to view the latter category as being symmetric monoidal. We do this by observing that it is the heart of $\F(\F\Cscr)$ with respect to the $t$-structure on bifiltrations $\F^{\star,\star}$ where the connective objects are those where $\F^{i,\star}$ is $i$-connective in the Beilinson $t$-structure. This $t$-structure is ``half-Postnikov, half-Beilinson.'' Call this the $\E^1$-symmetric monoidal structure $\Gr(\Ch^\bullet(\Cscr^\heart))^{\otimes_{\E^1}}$. We picture the objects of $\Gr(\Ch^\bullet(\Cscr^\heart))$ as a collection of objects in the plane (a bigraded) with differentials of bidegree $(-1,0)$. To a matrix $M$ in $\GL_2(\bZ)$, we can apply $M$ to this plane to obtain a new bigraded object with differentials of bidegree $M (\begin{smallmatrix} -1\\0 \end{smallmatrix})$. We use the matrix \[ \begin{pmatrix} 0&-1\\1&2 \end{pmatrix}^{-r+1}= \begin{pmatrix} r&r-1\\-r+1&-r+2 \end{pmatrix}. \] Using the conventions from Definition~\ref{def:graded_lines}, we see that the category of bigraded objects with differentials of bidegree $(-r,r-1)$ is equivalent again to $\Gr(\Ch^\bullet(\Cscr)^\heart)$. We can thus view the application of this matrix as given an autoequivalence of $\Gr(\Ch^\bullet(\Cscr)^\heart)$. We use this equivalence to transport the symmetric monoidal structure to obtain a new symmetric monoidal structure $\Gr(\Ch^\bullet(\Cscr^\heart))^{\otimes_{\E^r}}$. \end{enonce} \begin{theo}\label{thm:multiplicativity} Let $\Cscr^\otimes$ be a stable symmetric monoidal $\infty$-category which admits sequential limits and colimits, whose tensor product is exact and preserves sequential colimits in each variable, and which is equipped with a compatible $t$-structure. For each $r\geq 1$, the $\E^r$-page functor upgrades to a lax symmetric monoidal functor $\F\Cscr^\otimes\to\Gr(\Ch^\bullet(\Cscr^\heart))^{\otimes_{\E^r}}$. \end{theo} \begin{proof} For $r=1$, the result follows by applying Lemma~\ref{lem:homotopylaxmonoidal} with respect to the Beilinson $t$-structure. Now, we claim that the functor $\F^\star\mapsto\tau_{\geq\star}^\B(\F)$ from $\F\Cscr$ to $\F\F\Cscr$ is lax symmetric monoidal. This is another application of Lemma~\ref{lem:homotopylaxmonoidal}. Thus, $\Dec^{(r-1)}$ is lax symmetric monoidal by composition. It follows that $\F^\star\mapsto\E^1(\Dec^{(r-1)}(\F))$ is lax symmetric monoidal with target the $\E^1$-symmetric monoidal structure on $\Gr(\Ch^\bullet(\Cscr^\heart))$. The theorem follows by reindexing. \end{proof} \begin{coro}\label{cor:multiplicativity} Let $\Cscr^\otimes$ be a stable symmetric monoidal $\infty$-category which admits sequential limits and colimits, whose tensor product is exact and preserves sequential colimits in each variable, and which is equipped with a compatible $t$-structure. Suppose that $\Oscr$ is an $\infty$-operad. For each $r\geq 1$ the $\E^r$-page functor upgrades to a functor \[ \E^r\colon\Alg_\Oscr(\F\Cscr^\otimes)\to\Alg_\Oscr\left(\Gr(\Ch^\bullet(\Cscr^\heart))^{\otimes_{\E^r}}\right). \] \end{coro} \begin{exam}[Rings] If $\F^\star R$ is a filtered $\bE_1$-ring spectrum, then each page $\E^r$ of the spectral sequence for $\F^\star R$ is an associative algebra object in $\Gr(\Ch^\bullet(\Cscr^\heart))$. If $\F^\star R$ is an $\bE_\infty$-ring spectrum, then each page $\E^r$ of the spectral sequence for $\F^\star R$ is a commutative algebra object in $\Gr(\Ch^\bullet(\Cscr^\heart))$. These cases follow from Theorem~\ref{thm:multiplicativity} applied to the $\bE_1$ and $\bE_\infty$ operads. To make this more explicit, if $\F^\star R$ is an $\bE_\infty$-ring, then $\E^r_{-r\cs,(r-1)\cs}$ is a cdga in $\Cscr^\heart$ and the graded collection \[ \left\{\E^r_{(r-1)t-r\cs,-(r-2)t+(r-1)\cs}\right\}_{t\in\bZ} \] forms a graded-commutative ring in graded cochain complexes, meaning that there are morphisms $\E^r_{s,t}\otimes\E^r_{s',t'}\to\E^r_{s+s',t+t'}$ satisfying the bigraded Koszul sign rule $xy=(-1)^{(s+t)(s'+t')}yx$ and satisfying the following Leibniz rule for the horizontal differentials: $\d^1(xy)=\d^1(x)y+(-1)^{s+t}x\d^1(y)$ if $x\in\E^r_{s,t}$. These sign rules become the ordinary sign rules on the total complex associated to this graded cochain complex. These are justified by the fact $\E^r_{s,t}$ has weight $(n-1)s+nt$ and cohomological degree $(n-2)s+(n-1)t$ by Lemma~\ref{lem:lookup}, so the total degree $s+t$ has the same parity as $(n-1)s+nt+(n-2)s+(n-1)t=(2n-3)s+(2n-1)t$. \end{exam} \section{The Atiyah--Hirzebruch spectral sequence}\label{sec:mysterious} Let $X$ be a CW complex, which we view as an anima $X$ with an exhaustive increasing filtration $\F_\star X$ where $\F_sX=\emptyset$ for $s<0$ and each $\F_sX/\F_{s-1}$ is equivalent to a bouquet of $s$-spheres, called the $s$-cells of $X$, for $s\geq 0$. Taking singular cochains $\R\Gamma(X,\bZ)$ we obtain an object of $\D(\bZ)$ which we equip with the decreasing filtration \[ \F^s\R\Gamma(X,\bZ)=\fib\left(\R\Gamma(X,\bZ)\to\R\Gamma(X_{s-1},\bZ)\right). \] This filtration is complete and exhaustive with graded pieces given by $\gr^s\R\Gamma(X,\bZ)\we\prod_{\text{$s$-cells}}\bZ[-s]$. The associated (cohomological) spectral sequences is \[ \E_1^{s,t}=\H^{s+t}\gr^s\R\Gamma(X,\bZ)\iso\H^{s+t}\prod_{\text{$s$-cells}}\bZ[-s]\iso \begin{cases} \prod_{\text{$s$-cells}}\bZ&\text{if $t=0$ and}\\ 0&\text{otherwise.} \end{cases} \] The spectral sequence is thus concentrated on the ray from $(0,0)$ to $(\infty,0)$, giving a~cochain complex \[ 0\to\prod_{\text{$0$-cells}}\bZ\to\prod_{\text{$1$-cells}}\bZ\to\prod_{\text{$2$-cells}}\bZ\to\dots. \] The differentials in this cochain complex are precisely the differentials in the cochain complex computing the CW cohomology of $X$. In particular, $\pi_0^\B(\F^\star\R\Gamma(X,\bZ))$ is the CW cohomology cochain complex. Thus, $\E^2_{s,0}\iso\E^\infty_{s,0}\iso\H^s_\CW(X,\bZ)$ and the spectral sequence collapses at the $\E^2$-page. One finds that \[ \Dec(\F)^s\R\Gamma(X,\bZ)\we \begin{cases} 0&\text{if $s>0$ and}\\ \R\Gamma(X,\bZ)&\text{if $s\leq 0$.} \end{cases} \] \begin{rema}[Multiplicative structures and CW cohomology] It is well-known that the $\bE_\infty$-algebra $\R\Gamma(X,\bZ)$ cannot generally be computed by a cdga. This implies that for such $X$ no CW filtration $\F^\star\R\Gamma(X,\bZ)$ can be given the structure of an $\bE_\infty$-algebra object in $\FD(\bZ)$. If it could be, then $\pi_0^\B$ would admit the structure of a commutative algebra object by the results of Section~\ref{sec:multiplicativity}; this would be a cdga computing $\R\Gamma(X,\bZ)$. \end{rema} Suppose that $X$ is a CW complex and fix a spectrum $M$. We will write $\H^i(M)=\pi_{-i}M$ below for convenience. There are two classical ways to construct a spectral sequence for computing $\pi_*\R\Gamma(X,M)$, either by filtering $M$ or by filtering $X$. First, consider the cellular filtration given as in the previous section by \[ \F^s\R\Gamma(X,M)=\fib\left(\R\Gamma(X,M)\to\R\Gamma(X_{s-1},M)\right). \] The associated graded pieces are \[ \gr^s_\F\R\Gamma(X,M)\we(\R\Gamma(X_{s-1},M)/\R\Gamma(X_{s},M))[-1]\we\prod_{\text{$s$-cells}}M[-s]. \] The associated spectral sequence has \[ \E_1^{s,t}(\F)=\H^{s+t}\left(\prod_{\text{$s$-cells}}M[-s]\right)\iso\prod_{\text{$s$-cells}}\H^t(M)\To\H^{s+t}(X,M). \] By dévissage one can reduce to the case when $M\we\bZ$ to compute the differential on the $\E_1$-page as the CW cohomology differential. It follows that \[ \E_2^{s,t}(\F)=\H^s(X,\H^t(M)). \] Second, consider the Whitehead tower \[ \G^\star M=\tau_{\geq\star}M \] of $M$ and take cochains to obtain a new filtration $\G^\star\R\Gamma(X,M)=\R\Gamma(X,\tau_{\geq\star}M)$ with graded pieces $\gr^s_\G\R\Gamma(X,M)\we\R\Gamma(X,\gr^sM)\we\R\Gamma(X,(\H^{-s}M)[s])$. The $\E_1$-page of the associated spectral sequence is \[ \E_1^{s,t}(\G)=\H^{s+t}(X,(\H^{-s}M)[s])\iso\H^{2s+t}(X,\H^{-s}M)\To\H^{s+t}\R\Gamma(X,M). \] Via the standard reindexing of an $\E_1$-spectral sequence to an $'\E_2$-spectral sequence discussed in Section~\ref{sec:reindexing}, one obtains an $'\E_2$-spectral sequence \[ '\E_2^{s,t}(\G)=\H^s(X,\H^t(M))\To\H^{s+t}(X,M). \] It is natural to ask for the relationship between these two spectral sequences. \begin{prop} There is an equivalence $\Dec(\F)^\star\R\Gamma(X,M)\we\G^\star\R\Gamma(X,M)$ of filtered spectra, functorial in $M$ and the CW complex structure on $X$. \end{prop} \begin{proof} Consider the bifiltration $\H^{a,b}=\fib\left(\R\Gamma(X,\tau_{\geq a}M)\to\R\Gamma(X_{s-1},\tau_{\geq a}M)\right)$. If $a$ is fixed, then $\frac{\H^{a,s}}{\H^{a,s+1}}\we\prod_{\text{$s$-cells}}(\tau_{\geq a} M)[-s]$, which is $(a-s)$-connective. In particular, the natural map $\H^{a,\star}\to\F^\star$ factors through $\tau_{\geq a}^\B(\F)^\star$. It is in fact an equivalence. Indeed, both $\H^{a,\star}$ and $\tau_{\geq a}^\B(\F)^\star$ are complete filtrations (the latter because $\F^\star$ is, see Lemma~\ref{lem:truncationscomplete}), so it suffices to check this on associated graded pieces. However, \[ \frac{\H^{a,s}}{\H^{a,s+1}}\we\prod_{\text{$s$-cells}}(\tau_{\geq a}M)[-s]\we\tau_{\geq a-s}\left(\prod_{\text{$s$-cells}}M[-s]\right)\we\tau_{\geq a-s}\gr^s_\F\we\gr^s\tau_{\geq a}^\B(\F), \] where the second equivalence follows from the compatibility of the $t$-structure on $\Sp$ with arbitrary products. It follows that $\H^{a,\star}\we\tau_{\geq a}^\B(\F)$ and hence that $|\H^{a,\star}|\we\Dec(\F)^a$. However, $|\H^{a,\star}|\we\G^a$. This completes the proof. \end{proof} The following consequence was proved by Maunder in~\cite[Theorem~3.3]{maunder} with different methods. \begin{coro} There is an isomorphism of spectral sequences $\E^r_{s,t}(\F)\iso{'\E^r_{s,t}(\G)}$ starting at the second page. \end{coro} One often says that $\E^r(\F)$ and $'\E^r(\G)$ agree from the $\E^2$-page on. \addcontentsline{toc}{section}{References} \bibliography{20260123_antieau} \end{document}