ρ-Bockstein Spectral Sequence in Motivic Homotopy

• ρ‑Bockstein Spectral Sequence is a computational tool in algebraic topology and motivic homotopy theory that refines Ext-group calculations by filtering with the parameter ρ.
• It leverages a cobar complex filtered by powers of ρ to connect ℂ-motivic and ℝ-motivic Ext computations, thereby enhancing Adams spectral sequence analyses.
• The method resolves hidden extensions and facilitates effective slice spectral sequence computations, linking motivic results to classical stable homotopy theory.

The $\rho$-Bockstein spectral sequence is a computational tool in algebraic topology and motivic homotopy theory that refines Ext-group calculations by incorporating a deformation parameter $\rho$, typically arising from motivic cohomology over fields such as $\mathbb{R}$. Its main usage is in resolving extensions and differentials in algebraic spectral sequences associated to spectra whose coefficient rings contain $\rho$ as a non-nilpotent class, yielding results directly relevant to Adams-type spectral sequences and stable homotopy computations.

1. Algebraic Structure and Definition

The $\rho$-Bockstein spectral sequence is constructed by filtering a cobar complex (or similar algebraic structure) by powers of $\rho$, where $\rho$ has bidegree $(-1,-1)$ in $\mathbb{R}$-motivic cohomology. Given a coefficient ring $M_2 = \mathbb{F}_2[\tau,\rho]$ and dual Steenrod algebra $A$, this filtration results in a graded object

$$Gr_\rho(M_2, A) \cong (\tilde{M}_2, A_\mathbb{C}) \otimes \mathbb{F}_2[\rho]$$

where $(\tilde{M}_2, A_\mathbb{C})$ refers to the corresponding structures over $\mathbb{C}$, which lack the $\rho$-term. The resulting spectral sequence has $E_1$-page: $$E_1 = \mathrm{Ext}_{A_\mathbb{C}}(\tilde{M}_2, \tilde{M}_2)[\rho] \Longrightarrow \mathrm{Ext}_A(M_2, M_2)$$ The differentials $d_r$ detect the failure of classes to be cycles after multiplication by powers of $\rho$, with $d_r$ acting as

$d_r: E_r^{s,f,w} \to E_r^{s+1, f+r, w}$

where $(s,f,w)$ index stem, Adams filtration, and weight, and the differentials preserve coweight defined as $s + f - w$ (Dugger et al., 2015, Belmont et al., 2020). This structure allows one to "lift" Ext computations from $\mathbb{C}$-motivic to $\mathbb{R}$-motivic settings by successively recovering the impact of $\rho$.

2. Computational Workflow and Differential Patterns

The construction begins with explicit knowledge of $\mathbb{C}$-motivic Ext groups, which serve as input (modulo $\rho$). The cobar complex is filtered: $C \supset \rho C \supset \rho^2 C \supset \cdots$ and the associated graded object corresponds to the easier Ext computation over $\mathbb{C}$. The differentials record how multiplication by $\rho$ interacts with these classes, often yielding relations such as

$d_1(\tau) = \rho \cdot h_0$

and more generally,

$d_k(\tau^{2^k}) = \rho^{2^k} h_\ell$

for appropriate $k$ and $h_\ell$ (Dugger et al., 2015, Culver et al., 2020, Belmont et al., 2020).

A table organizing basic input/output structure:

Step	Description	Reference Example
1	Compute $$\mathrm{Ext}_{A_\mathbb{C}}(\tilde{M}_2,\tilde{M}_2)$$	C-motivic Ext chart
2	Filter by powers of $\rho$	$C \supset \rho C \supset \ldots$
3	Identify $E_1$-page as above	$E_1$ mod-$\rho$ with free $\rho$-action
4	Compute differentials via Leibniz rule and explicit formulas	$d_1(\tau) = \rho h_0$
5	Assemble $E_\infty$-page for $\mathbb{R}$-motivic Ext	Output is Ext$_A(M_2,M_2)$

For truncated motivic Brown-Peterson spectra and motivic Morava $K$-theories, there are explicit formulas such as

$d_{2^{n+1}-1}(\tau^{2^n}) = \rho^{2^{n+1}-1} v_n$

which are shown to govern differentials in associated effective slice spectral sequences (Culver et al., 2020).

3. Relationship to Adams and Effective Slice Spectral Sequences

Once the $\rho$-Bockstein spectral sequence computes $\mathrm{Ext}_A(M_2,M_2)$, these Ext-groups serve as input for the $\mathbb{R}$-motivic Adams spectral sequence: $$E_2 = \mathrm{Ext}_A^{s,f,w}(M_2, M_2) \implies \hat{\pi}_{s,w}^R(S^0)$$ with Adams differentials $d_r$ further refining the calculation. In practice, the Adams spectral sequence often collapses at a low page (e.g., $E_2$) in computations for low Milnor-Witt stems (Dugger et al., 2015). The $\rho$-Bockstein spectral sequence also completely determines the effective slice spectral sequence for certain motivic spectra (e.g., $kgl$, $k(n)$, $BPGL\langle m \rangle$), with a rigorous one-to-one correspondence between differentials in the $\rho$-Bockstein and slice spectral sequences (Culver et al., 2020).

Comparison square diagram (see Theorem 3.1 in (Culver et al., 2020)): $\begin{tikzcd} & \bigoplus_{q} \mathrm{Ext}(H_{**}(s_q^n E)) \arrow[dl,Rightarrow,"n\text{-aESSS}"'] \arrow[dr,Rightarrow,"\bigoplus mASS"] & \bigoplus_{q}\pi_{**}(s_q^nE) \arrow[rr,Rightarrow,"n\text{-ESSS}"'] && \pi_{**}(E) \end{tikzcd}$ where the algebraic effective slice spectral sequence ($n$-aESSS) is fully governed by $\rho$-Bockstein differentials.

4. Analysis of Hidden Extensions and Mahowald Invariants

The $\rho$-Bockstein spectral sequence computation makes possible the detection and resolution of "hidden extensions"—multiplications by $\rho$, $2$, or $\eta$ not visible on the $E_2$-page but present in the actual stable homotopy groups. These hidden multiplicative structures are resolved using combinatorial, multiplicative, and Toda bracket techniques (e.g., Moss Convergence Theorem) (Belmont et al., 2020). A paramount example is the calculation and verification of Mahowald invariants, where the representation

$$a = \rho^k \cdot \beta,\quad \mathrm{MI}(a) = \mathrm{image}(\beta)$$

connects classical stable homotopy elements to their motivic analogues. Agreement in observed ranges signals the reliability of the $\rho$-Bockstein method for classical comparison.

5. Implications for Motivic and Equivariant Homotopy Theory

The usage of the $\rho$-Bockstein spectral sequence catalyzes progress on several fronts:

• Enables translation from $\mathbb{C}$-motivic to $\mathbb{R}$-motivic Ext-group computations (bridge via $\rho$ filtration) (Dugger et al., 2015, Belmont et al., 2020).
• Provides computational control over Adams and slice spectral sequences for connective motivic and equivariant spectra (formal determination of differentials, e.g., $d_3(\tau^2) = \rho^3 v_1$) (Culver et al., 2020).
• Supports calculation of low-dimensional Milnor-Witt stems and reveals deviations from classical stable stems (e.g., altered order of Hopf maps) (Dugger et al., 2015).
• Yields explicit charts of stable stems with confirmed hidden extensions and Mahowald invariants (Belmont et al., 2020).
• Transfers computational results under Betti realization to $C_2$-equivariant setting (Hill-Hopkins-Ravenel slice spectral sequence for $kR$, $kR(n)$, $BPR\langle n\rangle$).

6. Limitations and Future Directions

While the $\rho$-Bockstein spectral sequence delivers precise algebraic information in a wide range, its limitations include:

• Dependence on explicit $\mathbb{C}$-motivic Ext calculations, which may be difficult in higher dimensions or for complicated spectra.
• Management of extensions and multiplicative structures at $E_\infty$-page requires auxiliary analysis (Massey products, Toda brackets).
• Analogy with other Bockstein-type spectral sequences (e.g., $u$-Bockstein, quantum-categorified Bockstein sequences) reveals both conceptual unity and contextual disparity: the periodicity generator, grading conventions, and application range differ (Kato et al., 2012, Putyra et al., 2021).
• Continued investigation of the cohomology of quotients of the dual Steenrod algebra and its implications for motivic modular forms, Adams spectral sequences, and ultimately the understanding of classical stable stems (Emming, 14 Sep 2025).

7. Historical and Conceptual Placement

The $\rho$-Bockstein spectral sequence represents an evolution of Bockstein-type methods for tracking torsion changes induced by additional algebraic parameters (here, $\rho$ from motivic cohomology over $\mathbb{R}$), refining classical approaches by providing an algorithmic pathway from base field computations to full motivic or equivariant results. Its integration with modern slice and Adams spectral sequence technology aligns it with some of the most active computational strategies in motivic and equivariant homotopy theory.

In summary, the $\rho$-Bockstein spectral sequence is a central tool for translating and resolving algebraic and cohomological data from “mod-$\rho$” to full motivic contexts, underpinning contemporary progress across Milnor-Witt stem computations, stable homotopy theory, and the determination of Adams-type invariants.