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Filtrations in $\mathbb{C}$-motivic stable homotopy theory

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Pith's one-line read This paper proves a uniform antidiagonal formula for the effective slices of C-motivic analogues of classical spectra, and applies it to compute the effective slices of the motivic modular forms spectrum mmf from the Adams-Novikov E2-page…

desk verdict Solid new slice computations for C-motivic analogues, but the proof of the key splitting theorem has a gap that needs to be closed before the central claim is fully established. read the letter →

arxiv 2608.04877 v1 pith:7PSOP2UM submitted 2026-08-05 math.AT math.AG

Konstantin Emming This is my paper · ORCID
classification math.ATmath.AG MSC 14F4255P4255T99
keywords C-motivichomotopytheoryslicefiltrationsfilteredspectraAdams-Novikovspectralsequencemotivicmodularformseffectivefiltrationconnectivevery
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to establish that the slice filtrations of $\mathbb{C}$-motivic spectra of the form $\Gamma_\star(X)$ can be computed entirely from classical data attached to $X$, using the filtered-spectrum model for the $\mathbb{C}$-motivic category. The main theorem gives a closed formula for the $q$-th effective slice of $\Gamma_\star(X)$ as a wedge of $\mathbb{C}$-motivic Eilenberg-MacLane spectra indexed by the antidiagonal $s+f=2q$ of the classical Adams-Novikov $E_2$-page, whenever $X$ is bounded below and has even $MU$-homology. This matters because it simultaneously reproves several slice conjectures and computes the previously unknown effective slices of the motivic modular forms spectrum $\mathrm{mmf}$. The paper also proves that the effective slice spectral sequence of $\Gamma_\star(X)$ is just a re-indexing of the classical Adams-Novikov spectral sequence of $X$.

What carries the argument

The carrying object is the $\mathbb{C}$-motivic analogue functor $\Gamma_\star\colon \mathrm{SH}^{\mathrm{cl}}\to \mathrm{SH}^{\mathrm{fil}}$, defined by $\Gamma_w(X)=\mathrm{Tot}(\tau_{\ge 2w}(X\wedge MU^{\wedge \bullet+1}))$, together with the equivalence $\mathrm{SH}_{\mathbb{C}}\simeq \mathrm{Mod}_{\Gamma_\star(S)}$ between the $2$-complete cellular $\mathbb{C}$-motivic category and modules over the filtered spectrum $\Gamma_\star(S)$. The paper modifies this construction in three ways—freezing weights below $q$ for the effective cover, applying classical connective covers $\tau_{\ge w+q}$ for the connective cover, and both for the very effective cover—to produce filtered-spectrum models for the covers. The homotopy groups of these models are computed by Bousfield-Kan spectral sequences that are truncations of the Adams-Novikov spectral sequence of $X$ below the antidiagonal $s+f=2w$. A $\mathbb{C}$-motivic analogue of Adams' theorem on modules over $M\mathbb{Z}$ (Proposition 7.8) then turns homotopy groups of the form $G[\tau]$ into wedges of Eilenberg-MacLane spectra.

What would settle it

Take $X=\mathrm{tmf}$ and let $q=1$. The theorem predicts that $s_1(\mathrm{mmf})$ is the wedge of $\Sigma^{s,1}M E^{s,f}_2$ over all $s+f=2$, so its homotopy groups should be free $\mathbb{Z}[\tau]$-modules on exactly the classes of the $s+f=2$ antidiagonal of the Adams-Novikov $E_2$-page of $\mathrm{tmf}$. A reader could compute $\pi_{*,*}(s_1(\mathrm{mmf}))$ from the filtered-spectrum description of the slice in Proposition 7.1 and the known $\mathrm{tmf}$ Adams-Novikov chart; any missing class, extra $\tau$-torsion, or non-split extension would falsify the wedge decomposition.

Watch

Extended reading notes

Core claim

The central claim is Corollary 7.11: for every bounded below classical spectrum $X$ with even $MU$-homology, the $q$-th effective slice of the $\mathbb{C}$-motivic analogue $\Gamma_\star(X)$ is equivalent to $$s_q(\Gamma_\star(X)) \simeq \bigvee_{s+f=2q} \$Sigma^{{s,q}}$ M $E^{{s,f}}$_2,$$ where $E^{s,f}_2$ is the $E_2$-page of the classical Adams-Novikov spectral sequence of $X$ in stem $s$ and filtration $f$, and $M$ is the functor taking a group to the associated $\mathbb{C}$-motivic Eilenberg-MacLane spectrum. The argument models each effective cover of $\Gamma_\star(X)$ by an explicit filtered spectrum, checks the cover is correct through homotopy-group characterizations, and then uses a $\mathbb{C}$-motivic analogue of Adams' splitting theorem for modules over $M\mathbb{Z}$ to assemble the slice from its homotopy groups. Applied to $X=\mathrm{tmf}$, this says the effective slices of $\mathrm{mmf}=\Gamma_\star(\mathrm{tmf})$ are wedges of Eilenberg-MacLane spectra indexed by the Adams-Novikov $E_2$-page of $\mathrm{tmf}$; applied to $S$, $MU$, $ku$, $ko$, and Eilenberg-MacLane spectra it recovers the earlier slice conjectures in the $2$-complete cellular setting.

Load-bearing premise

The results rest on identifying the filtered-spectrum model with the $2$-complete cellular $\mathbb{C}$-motivic category and on assuming the connective and very effective filtrations computed in that model are the genuine motivic filtrations; the paper proves this agreement for the effective filtration only.

Editorial extensions

If this is right

  • For $X=\mathrm{tmf}$, the formula gives all effective slices of $\mathrm{mmf}$ in closed form, and the same formula reproduces the previously known slices of the spheres, $MGL$, $kq$, and $kgl$ in the $2$-complete cellular $\mathbb{C}$-motivic category.
  • The effective slice spectral sequence of $\Gamma_\star(X)$ and the classical Adams-Novikov spectral sequence of $X$ carry the same information: $d_r(x)=\tau^r y$ in the slice spectral sequence exactly when $d_{2r+1}(x)=y$ in the Adams-Novikov spectral sequence.
  • The connective and very effective slices of $\Gamma_\star(X)$ also have explicit filtered-spectrum descriptions; in particular the Betti realization of the $q$-th very effective cover is the classical $2q$-connective cover of $X$.
  • For Landweber exact classical spectra $X$, the effective and very effective covers of $\Gamma_\star(X)$ coincide, extending a known agreement result to all objects of this form.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Our inference: if odd-primary versions of the filtered-spectrum model are constructed, the same antidiagonal formula would give odd-primary slices of $\mathbb{C}$-motivic analogues, including $\mathrm{mmf}$ at odd primes.
  • Our inference: since $\Gamma_\star(X)\simeq \nu X$ for bounded below $X$ with even $MU$-homology, these slice statements can be read as statements about synthetic spectra, and a synthetic proof might bypass the d\'ecalage comparison used here.
  • Our inference: the weight-by-weight identification of the slice spectral sequence with the Adams-Novikov spectral sequence suggests that slice filtrations of $\Gamma_\star(X)$ could be used as a bookkeeping device for classical stable homotopy computations, turning motivic weight into a new grading in large stem calculations.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper studies the effective, connective, and very effective filtrations in the 2-complete, cellular C-motivic stable homotopy category, using the filtered-spectrum model of Gheorghe–Isaksen–Krause–Ricka and the motivic analogue functor Γ⋆. The main construction associates to a bounded-below classical spectrum X the C-motivic analogue Γ⋆(X), gives filtered-spectrum models for its covers, and then computes the slices of these filtrations. The central result, Theorem 1.2 (Corollary 7.11), identifies the q-th effective slice of Γ⋆(X) with a wedge of C-motivic Eilenberg–MacLane spectra indexed by the classical Adams–Novikov E2-page of X, provided X has even MU-homology. This recovers earlier slice computations for S, MU, ku, ko, and MGL and yields a new computation for the motivic modular forms spectrum mmf. The paper also proves a comparison between the effective slice spectral sequence of Γ⋆(X) and the classical Adams–Novikov spectral sequence of X, stated as Theorem 1.3 (Theorem 8.1).

Significance. If the main results are correct, the paper gives a unified and computationally explicit framework for slice filtrations of C-motivic analogues, and the computation of the slices of mmf is a genuinely new contribution. The filtered-spectrum models for the effective, connective, and very effective covers are concrete and potentially useful for further calculations. The recovery of known results for MGL, the sphere, kq, and other spectra is a useful consistency check, and the comparison with the Adams–Novikov spectral sequence in Section 8 is conceptually attractive. The paper is not machine-checked, but many of the constructions are quite explicit, which is an asset.

major comments (3)
  1. [Section 7.1, Lemma 7.5] The proof of Lemma 7.5 is not valid as written. The claim is that if α:C→Y induces an isomorphism on π_{s,∗}, then id∧α:MZ∧C→MZ∧Y also induces an isomorphism on π_{s,∗}. The argument uses a commutative diagram involving the Hurewicz map h:S→MZ and the module multiplication μ:MZ∧Y→Y, and asserts that all maps in the diagram induce isomorphisms on π_{s,∗}. This is not generally true for h∧id:S∧C→MZ∧C. For C=S^{s,w}, the map π_{s,∗}(S^{s,w})→π_{s,∗}(MZ∧S^{s,w}) is the Hurewicz map into an Eilenberg–MacLane spectrum, and it is not an isomorphism in positive stems where the sphere has torsion; the statement that cellular approximation of MZ 'keeps π_{0,∗} unchanged' does not imply isomorphisms on the relevant bigraded homotopy groups. Since Proposition 7.8 uses Lemma 7.5 to conclude that the maps Σ^{s,w}MG_s→Y induce isomorphisms on homotopy, the wedge decomposition in Corollary 7.11 is not established by the present proof.
  2. [Section 7.1, Proposition 7.8] Proposition 7.8 is load-bearing for Theorem 1.2, and its proof has a second gap beyond Lemma 7.5. Lemma 7.3 constructs C as the cofiber of a map between wedges of spheres. Such a compact cofiber generally has homotopy in adjacent stems, e.g. the cofiber of multiplication by 2 on S^{s,w} has nontrivial π_{s+1,∗}. Therefore an isomorphism on π_{s,∗}(C)→π_{s,∗}(Y) does not by itself control the behavior of the map α in neighboring stems, and the conclusion that the wedge map ⋁_s Σ^{s,w}MG_s→Y induces isomorphisms on all homotopy groups is not justified. A repair would need either a stronger splitting theorem for MZ-modules or a careful verification that the maps in the wedge decomposition have no spurious higher-stem contributions. As it stands, the central effective-slice formula rests on an unproven assertion.
  3. [Sections 1.5, 6.2, and 6.3] The paper explicitly states in Section 1.5 that Heard's theorem identifies the cellular effective filtration with the full C-motivic effective filtration, but that no analogous statement has been proven for the connective and very effective filtrations. Nevertheless, Propositions 6.7 and 6.11 present the filtered spectra Γ≥q⋆(X) and Γ̃q⋆(X) as models for the q-th connective and q-th very effective covers, and Sections 7.2 and 7.3 use them to compute slices. As written, these computations are only established in the cellular subcategory, and unless a comparison theorem for the connective and very effective filtrations is supplied, the claims should be explicitly restricted to the cellular category. This is a substantial limitation of the paper's scope as stated.
minor comments (6)
  1. [Section 5.4, Lemma 5.13] In Lemma 5.13(1), the connective cover is written as f_q(Y), but the cover in question is Y_{\ge q}; the notation is inconsistent with the surrounding text.
  2. [Proposition 7.1 and Section 7.1] The notation τ^{<2q+2}_{\ge 2q} is not defined in Section 2 or at its first use in Proposition 7.1; a brief definition would improve readability.
  3. [Section 7.1, proof of Lemma 7.5] The diagram in the proof of Lemma 7.5 is not fully labeled; in particular the maps h∧id and μ are not marked, which makes the commutativity claim harder to check.
  4. [Section 8, Remark 8.2] The weight formula (s+f)/2 in Remark 8.2 silently assumes that s+f is even; since the even-MU-homology hypothesis makes odd antidiagonals vanish, this is consistent, but the assumption should be stated explicitly.
  5. [Section 1, Corollary 7.13] The sentence 'tmf is bounded below since it is by definition a connective cover' is slightly imprecise: tmf is a connective ring spectrum by construction, but it is not literally defined as a connective cover of another fixed spectrum; the phrasing could be clarified.
  6. [Section 1.5] The statement that the odd-primary analogues 'are likely to hold' is presented as a remark, but it would be helpful to specify which parts of the argument actually depend on 2-adic completion beyond the existence of τ.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the ANSS/slice comparison follows from the external GIKR filtered-spectrum model and Levine's decalage theorem, not from the paper's own conclusion.

full rationale

The derivation is not circular. Section 3 defines Gamma_w(X) = Tot(tau_{≥2w}(X ∧ MU^{∧•+1})), so the BKSS of Gamma_⋆(X) is visibly the truncated classical Adams–Novikov spectral sequence; however, the effective, connective, and very effective filtrations are defined in Section 4 from subcategories generated by motivic spheres, not from that BKSS. Propositions 6.4, 6.7, and 6.11 prove that the filtered models are the respective covers by checking the homotopy characterizations of Section 5, which are proven rather than assumed. The differential comparison in Theorem 8.1 is obtained by applying Levine's decalage theorem ([Lev15, Proposition 6.3] via [vN25, Theorem 2.80]), an external result, to the model. Theorem 1.2 combines the homotopy computation of Proposition 7.2 with the MZ-module splitting Proposition 7.8; nothing is fitted and no parameter is renamed as a prediction. The sole author has no self-citations carrying the argument; the reliance on [GIKR22, Theorem 6.12] is an external equivalence, not a self-citation. Section 1.5 explicitly admits that the connective and very effective filtrations are only known in the cellular model, which is a scope limitation, not a circularity. The skeptical objection to Lemma 7.5 identifies an unsupported inference: the proof's assertion, 'for µ it follows because it is true for h ∧ id,' is a non-sequitur, but that is a correctness gap in the proof as written, not a reduction of the theorem to its own input. Accordingly, the circularity score is 0 and no circular steps are recorded.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claims are pure mathematics with no fitted constants. They rest on the filtered-spectrum equivalence, ANSS convergence and even-MU vanishing, Levine's decalage comparison, the MZ-module property of slices, and the cellular detection theorem. The most fragile entry is the unproven agreement of the cellular connective and very effective filtrations with the full motivic ones.

assumptions (7)
  • domain assumption The equivalence SH_C equivalent to Mod_{Gamma_star(S)} of GIKR22 Theorem 6.12, and the identification of bigraded homotopy groups with the Bousfield-Kan data of Gamma_star(X).
    The paper computes C-motivic covers and slices through filtered spectra and transfers the results along this equivalence. If the equivalence is wrong, Sections 6 through 8 describe filtered spectra rather than C-motivic spectra.
  • domain assumption For bounded below X, the MU-based Adams-Novikov spectral sequence converges to the 2-completed homotopy groups of X, and the BKSS of Gamma_w(X) is the ANSS truncated below the 2w-th antidiagonal.
    This is the foundation for reading homotopy groups from the spectral sequence pictures in Sections 3, 6, and 8.
  • domain assumption Even MU homology of X implies the ANSS has no d2-differentials and all odd antidiagonals are trivial, per Ravenel 4.4.2.
    This assumption is the reason the effective slices have the clean direct-sum form in Theorem 1.2 and is used in Proposition 7.2 and Lemma 8.6.
  • domain assumption Levine's decalage comparison, as reformulated in van Nigtevecht Theorem 2.80, applies to the 2-complete filtered-spectrum objects Gamma_w(X).
    Theorem 8.1 uses this as the black box to identify the decalage spectral sequence with the truncated ANSS.
  • domain assumption Effective slices of the spectra considered are modules over the motivic Eilenberg-MacLane spectrum MZ.
    Proposition 7.8 needs the MZ-module structure to split slices into wedges of Eilenberg-MacLane spectra; this is cited to Pelaez and GRSØ12.
  • domain assumption Homotopy groups detect equivalences in the cellular C-motivic category.
    Used in Proposition 5.5 and Proposition 7.8 to conclude that maps inducing isomorphisms on all bigraded homotopy groups are equivalences; standard in the cellular subcategory, cited to Dugger-Isaksen.
  • domain assumption The effective filtration of a cellular object agrees with its filtration in the full C-motivic category, by Heard Theorem 3.16; no analogous statement is proven for the connective or very effective filtrations.
    Section 1.5 explicitly states this asymmetry. The conditional verdict is based on the unproven connective and very effective part.

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Pith. "Pith review of Filtrations in $\mathbb{C}$-motivic stable homotopy theory." pith.science (2026). https://pith.science/paper/7PSOP2UM

@misc{pith2026260804877,
  author       = {Pith},
  title        = {Pith review of: Filtrations in $\mathbbC$-motivic stable homotopy theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7PSOP2UM}},
  note         = {Machine review of arXiv:2608.04877}
}
abstract

We study the effective, connective, and very effective filtrations in the $\mathbb{C}$-motivic, $2$-complete, cellular, stable homotopy category. We do so by using the filtered spectrum model for this category due to Gheorghe-Isaksen-Krause-Ricka, and in particular the motivic analogue functor $\Gamma_\star$. Then we can express the covers making up the respective filtrations of a nice motivic analogue $\Gamma_\star(X)$ via filtered spectra, and use these to compute the slices. Applying this in the case of $X$ being the sphere spectrum, $\text{MU}$, $\text{ku}$, or an Eilenberg-MacLane spectrum recovers a number of conjectures due to Voevodsky. Applying it to $\text{ko}$ recovers a computation of Ananyevskiy-R\"ondigs-{\O}stv{\ae}r. We can also apply it to $\text{tmf}$ and compute the effective slices of the motivic modular forms spectrum $\text{mmf}$. We also study the effective slice spectral sequence for $\Gamma_\star(X)$, which turns out to contain the same information as the classical Adams-Novikov spectral sequence for $X$.

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