Very Effective Slice Spectral Sequence

• The very effective slice spectral sequence is a motivic tool that refines Voevodsky’s effective filtration by incorporating connective truncation via Morel’s homotopy t‑structure.
• It differentiates between ordinary, regular, and very effective slice towers, offering explicit computations for spectra like kq through detailed slice formulas and differentials.
• This framework underpins refined analyses in motivic, equivariant, and logarithmic settings, impacting algebraic K‑theory, homological computations, and convergence criteria.

Searching arXiv for recent and foundational papers on the very effective slice spectral sequence and closely related slice filtrations. Search 1: "very effective slice spectral sequence" The very effective slice spectral sequence is the spectral sequence associated to the very effective slice tower in motivic homotopy theory, but the surrounding literature is terminologically heterogeneous. In motivic settings the very effective filtration is a distinct refinement of Voevodsky’s effective filtration, whereas several influential equivariant case studies compute instead the ordinary Hill–Hopkins–Ravenel slice spectral sequence or Ullman’s regular slice spectral sequence for spectra whose slices nevertheless display strongly connective, “effective” behavior (Ananyevskiy et al., 2017, Ullman, 2012, Hill et al., 2018).

1. Terminological scope and formal variants

In motivic homotopy theory, the very effective subcategory $SH^{veff}\subset SH$ is generated under homotopy colimits and extensions by $\mathbb P^1$-suspension spectra of smooth schemes. It is smaller than $SH^{eff}$, closed under tensor product, but not triangulated; this nontriangulated character is one of the decisive formal differences between effective and very effective constructions (Ananyevskiy et al., 2017). A closely related formulation used later is that the very effective cover functor satisfies

$\tilde f_0 \simeq f_0\tau_{\ge 0},$

so very effectiveness combines effective truncation with connective truncation for Morel’s homotopy $t$-structure; because $\tau_{\ge 0}$ is not triangulated, $\tilde f_0$ does not preserve fiber sequences (Kong et al., 2022).

A distinct but adjacent variant is Ullman’s regular slice filtration. Ordinary slice cells are

$$G_{+}\wedge_H S^{n\rho_H} \quad\text{and}\quad G_{+}\wedge_H S^{n\rho_H-1},$$

whereas regular slice cells retain only

$G_+\wedge_H S^{n\rho_H}.$

The resulting regular filtration is related to the ordinary slice filtration by

$\Sigma\tau_n=\bar\tau_{n+1},$

so the ordinary slice construction is a suspended and reindexed form of the regular construction (Ullman, 2012). This relation is frequently useful, but it is not an identification with the very effective filtration.

Filtration	Generators or defining feature	Status in the cited literature
Effective slice filtration	Voevodsky’s $\mathbb P^1$0 formalism	Baseline ordinary motivic slice tower
Very effective filtration	Generated by $\mathbb P^1$1-suspension spectra of smooth schemes, closed under colimits and extensions	Explicit in motivic work on $\mathbb P^1$2 and logarithmic slices
Regular slice filtration	Only regular slice cells $\mathbb P^1$3	Equivariant variant of Ullman, not “very effective”

This terminological separation controls much of the subject. Papers on $\mathbb P^1$4 and logarithmic $\mathbb P^1$5-theory are directly about very effective covers or very effective slices, while several papers central to equivariant computations are not: they work with the ordinary HHR slice tower or the regular slice tower and must be read accordingly (Ananyevskiy et al., 2017, Ullman, 2012).

2. Spectral-sequence formalism and convergence

The very effective slice spectral sequence is obtained from the tower of very effective covers $\mathbb P^1$6, with associated slices $\mathbb P^1$7. In the logarithmic setting this is written explicitly as

$\mathbb P^1$8

while the ordinary effective slices are

$\mathbb P^1$9

(Binda et al., 2024). For motivic spectra over $SH^{eff}$0, a realized effective cotower can also be formed by choosing a motivic lift $SH^{eff}$1 of a $SH^{eff}$2-spectrum $SH^{eff}$3 and setting

$SH^{eff}$4

This tower filters $SH^{eff}$5 through realized effective covers, and the associated spectral sequence converges in general to $SH^{eff}$6 rather than automatically to $SH^{eff}$7 (Kong, 2020).

A key convergence criterion in that realized setting is the lemma: if $SH^{eff}$8 is connective and $SH^{eff}$9 is slice complete, then

$\tilde f_0 \simeq f_0\tau_{\ge 0},$0

Applied to $\tilde f_0 \simeq f_0\tau_{\ge 0},$1, the note argues that $\tilde f_0 \simeq f_0\tau_{\ge 0},$2 is an extension of $\tilde f_0 \simeq f_0\tau_{\ge 0},$3 and a slice, that $\tilde f_0 \simeq f_0\tau_{\ge 0},$4 is slice complete, and therefore

$\tilde f_0 \simeq f_0\tau_{\ge 0},$5

(Kong, 2020). This places $\tilde f_0 \simeq f_0\tau_{\ge 0},$6-completion, rather than bare homotopy, at the center of convergence for realized effective towers.

Ordinary slice towers admit related but different comparison results. For the motivic sphere, Betti realization of Voevodsky’s ordinary slice tower yields a spectral sequence which, after reindexing, agrees with the classical Adams–Novikov spectral sequence: $\tilde f_0 \simeq f_0\tau_{\ge 0},$7 This comparison is for the ordinary slice tower, not the very effective tower, but it sets the model for later comparisons between slice constructions and Adams-type filtrations (Levine, 2013).

A homology-valued variant built directly from the very effective slice tower appears in the homological slice spectral sequence. For $\tilde f_0 \simeq f_0\tau_{\ge 0},$8 and $\tilde f_0 \simeq f_0\tau_{\ge 0},$9, its $t$0-page is

$t$1

with Adams trigrading $t$2, and it converges to the mod $t$3 homology of the global sections spectrum $t$4. When $t$5 and $t$6 are slice $t$7, weight-zero terms satisfy the vanishing line

$t$8

(Carrick et al., 2023).

3. The very effective cover $t$9 and its slice spectral sequence

The central motivic example is hermitian $\tau_{\ge 0}$0-theory. The very effective cover of $\tau_{\ge 0}$1 is denoted

$\tau_{\ge 0}$2

and it is presented as the algebro-geometric analogue of connective real topological $\tau_{\ge 0}$3-theory. If the base field admits a complex embedding, then its Betti realization is

$\tau_{\ge 0}$4

and it fits into the connective Wood sequence

$\tau_{\ge 0}$5

(Ananyevskiy et al., 2017).

The slices of $\tau_{\ge 0}$6 are considerably smaller than those of periodic $\tau_{\ge 0}$7. When $\tau_{\ge 0}$8, the nonnegative slices are

$\tau_{\ge 0}$9

and the negative slices are zero (Ananyevskiy et al., 2017). The graded ring of slices is described multiplicatively by

$\tilde f_0$0

These slice formulas make the ordinary slice spectral sequence for $\tilde f_0$1 highly explicit: $\tilde f_0$2 with $\tilde f_0$3-terms given by shifted motivic cohomology groups and first differential $\tilde f_0$4 expressed in motivic Steenrod operations involving $\tilde f_0$5, $\tilde f_0$6, $\tilde f_0$7, and the classes $\tilde f_0$8 (Ananyevskiy et al., 2017). The same paper proves conditional convergence

$\tilde f_0$9

and identifies the $$G_{+}\wedge_H S^{n\rho_H} \quad\text{and}\quad G_{+}\wedge_H S^{n\rho_H-1},$$0-line by

$$G_{+}\wedge_H S^{n\rho_H} \quad\text{and}\quad G_{+}\wedge_H S^{n\rho_H-1},$$1

A crucial nuance is that the paper computes the ordinary slices $$G_{+}\wedge_H S^{n\rho_H} \quad\text{and}\quad G_{+}\wedge_H S^{n\rho_H-1},$$2 of the very effective cover $$G_{+}\wedge_H S^{n\rho_H} \quad\text{and}\quad G_{+}\wedge_H S^{n\rho_H-1},$$3, not the first differentials in the very effective slice tower itself. It explicitly remarks that Bachmann determined the very effective slices of $$G_{+}\wedge_H S^{n\rho_H} \quad\text{and}\quad G_{+}\wedge_H S^{n\rho_H-1},$$4, hence of $$G_{+}\wedge_H S^{n\rho_H} \quad\text{and}\quad G_{+}\wedge_H S^{n\rho_H-1},$$5, up to extensions, and that additional work is needed to identify the corresponding first very effective slice differentials (Ananyevskiy et al., 2017). This distinction is one of the most common sources of confusion.

The effective slice spectral sequence for $$G_{+}\wedge_H S^{n\rho_H} \quad\text{and}\quad G_{+}\wedge_H S^{n\rho_H-1},$$6 was then worked out over algebraically closed, finite, local, real, and global fields in a field-by-field manner. In that setting the very effective cover functor again enters through

$$G_{+}\wedge_H S^{n\rho_H} \quad\text{and}\quad G_{+}\wedge_H S^{n\rho_H-1},$$7

and the connective $$G_{+}\wedge_H S^{n\rho_H} \quad\text{and}\quad G_{+}\wedge_H S^{n\rho_H-1},$$8-periodic analogue

$$G_{+}\wedge_H S^{n\rho_H} \quad\text{and}\quad G_{+}\wedge_H S^{n\rho_H-1},$$9

is analyzed by combining explicit slice formulas for $G_+\wedge_H S^{n\rho_H}.$0, new coefficient computations for $G_+\wedge_H S^{n\rho_H}.$1, and a Steenrod-operation description of the $G_+\wedge_H S^{n\rho_H}.$2-differentials (Kong et al., 2022).

4. Realized, equivariant, and homological variants

The very effective perspective extends beyond $G_+\wedge_H S^{n\rho_H}.$3 in two distinct directions. One is realization to $G_+\wedge_H S^{n\rho_H}.$4-equivariant homotopy. A short note constructs a $G_+\wedge_H S^{n\rho_H}.$5-equivariant spectral sequence by realizing the $G_+\wedge_H S^{n\rho_H}.$6-motivic effective slice filtration. Its tower

$G_+\wedge_H S^{n\rho_H}.$7

comes with slice completion

$G_+\wedge_H S^{n\rho_H}.$8

and the main lemma identifies $G_+\wedge_H S^{n\rho_H}.$9 with $\Sigma\tau_n=\bar\tau_{n+1},$0-completion under a concrete hypothesis: $\Sigma\tau_n=\bar\tau_{n+1},$1 For the intended application,

$\Sigma\tau_n=\bar\tau_{n+1},$2

(Kong, 2020). Although this is formulated with the effective filtration rather than the very effective one, it isolates the convergence and completion issues that also govern very effective towers.

The second direction is a homological spectral sequence built from the very effective slice tower. For $\Sigma\tau_n=\bar\tau_{n+1},$3, the homological slice spectral sequence uses the very effective filtration of Spitzweck–Østvær, but for the standard quotients of $\Sigma\tau_n=\bar\tau_{n+1},$4 under study the effective, cellular effective, very effective, and cellular very effective towers coincide. Its $\Sigma\tau_n=\bar\tau_{n+1},$5-page is

$\Sigma\tau_n=\bar\tau_{n+1},$6

and in weight zero the image of the edge homomorphism is precisely

$\Sigma\tau_n=\bar\tau_{n+1},$7

The paper determines a family of differentials interpolating between

$\Sigma\tau_n=\bar\tau_{n+1},$8

and

$\Sigma\tau_n=\bar\tau_{n+1},$9

and computes the spectral sequence completely for $\mathbb P^1$00 (Carrick et al., 2023).

Over $\mathbb P^1$01, this homological theory interacts directly with genuine equivariant spectra. The paper proves

$\mathbb P^1$02

so the motivic HSSS computes the homology of $\mathbb P^1$03-fixed points; in height $\mathbb P^1$04 this yields a computation of $\mathbb P^1$05 (Carrick et al., 2023).

5. Equivariant slice computations that inform very effective intuition

Much of the literature most useful for “very effective” intuition in genuine equivariant homotopy does not actually use a very effective filtration. Ullman’s regular slice filtration is the clearest example. It uses only regular slice cells and satisfies

$\mathbb P^1$06

so the ordinary slice construction is a shifted form of the regular one. Ullman also proves efficiency results: connectivity and coconnectivity of a spectrum are inherited by the entire regular slice tower, and he describes the regular slice spectral sequence as “very efficient” (Ullman, 2012). That adjective is not terminological equivalence with “very effective,” but it explains why regular slices are often read as a clean approximation to more connective slice behavior.

A second example is the complete calculation of the slice spectral sequence of

$\mathbb P^1$07

That paper is explicit that it computes the ordinary HHR slice filtration, not a separate very effective variant. The spectrum has slice associated graded

$\mathbb P^1$08

and for $\mathbb P^1$09 every slice is a suspension of an Eilenberg–MacLane spectrum either by a regular $\mathbb P^1$10-cell $\mathbb P^1$11 or by an induced slice cell $\mathbb P^1$12. The spectral sequence terminates after the $\mathbb P^1$13-page and has a horizontal vanishing line of filtration $\mathbb P^1$14; after inverting the periodicity element $\mathbb P^1$15, the localized spectrum $\mathbb P^1$16 satisfies

$\mathbb P^1$17

which combine to yield $\mathbb P^1$18-periodicity of $\mathbb P^1$19 and of $\mathbb P^1$20 (Hill et al., 2018).

A third case is the $\mathbb P^1$21-analog $\mathbb P^1$22 of real $\mathbb P^1$23-theory. Here again the paper computes the ordinary HHR slice spectral sequence, but the slices are built from highly connective representation suspensions: $\mathbb P^1$24 where

$\mathbb P^1$25

The periodic localization

$\mathbb P^1$26

is $\mathbb P^1$27-periodic, and the differential pattern begins with

$\mathbb P^1$28

(Hill et al., 2015). The paper never identifies this with a very effective tower, but its slices are organized entirely by regular representation suspensions, which strongly resembles very effective behavior in the connective range.

6. Logarithmic, Kummer étale, and arithmetic extensions

The most explicit recent generalization of the very effective slice spectral sequence is logarithmic. In logarithmic motivic homotopy theory one defines both effective and very effective subcategories. The effective category $\mathbb P^1$29 is generated under colimits by

$\mathbb P^1$30

whereas the very effective category $\mathbb P^1$31 is the smallest full subcategory containing

$\mathbb P^1$32

and closed under colimits and extensions. The associated very effective tower

$\mathbb P^1$33

has slices

$\mathbb P^1$34

(Binda et al., 2024).

For logarithmic $\mathbb P^1$35 over a perfect field admitting resolution of singularities, the ordinary and very effective slices coincide: $\mathbb P^1$36 The same paper then proves the Kummer étale refinement: if $\mathbb P^1$37 is perfect, admits resolution of singularities, and has finite étale cohomological dimension, then for hypercomplete Kummer étale $\mathbb P^1$38-theory

$\mathbb P^1$39

Here $\mathbb P^1$40 represents Kummer étale motivic cohomology, and on smooth schemes with trivial log structure this is Lichtenbaum étale motivic cohomology (Binda et al., 2024).

The same logarithmic framework identifies an arithmetic filtration with a very effective one. In the Kummer étale $\mathbb P^1$41-complete setting,

$\mathbb P^1$42

is an equivalence, so the very effective filtration on $\mathbb P^1$43 coincides with the BMS filtration (Binda et al., 2024). Since the graded pieces satisfy

$\mathbb P^1$44

the BMS spectral sequence becomes, in this setting, a very effective slice spectral sequence. The paper also proves filtration compatibility of the trace map

$\mathbb P^1$45

yielding a natural map from the motivic slice spectral sequence for $\mathbb P^1$46-theory to the BMS spectral sequence (Binda et al., 2024).

7. Conceptual status, comparisons, and common misconceptions

The first misconception is to identify every connective or regular slice computation with a very effective slice spectral sequence. That identification is not supported by the literature summarized here. Ullman’s regular slice tower, the $\mathbb P^1$47-equivariant height-$\mathbb P^1$48 Lubin–Tate computation, and the $\mathbb P^1$49-analog of real $\mathbb P^1$50-theory all work with the ordinary HHR or regular slice filtration, even when every visible slice is built from regular representation suspensions and Eilenberg–MacLane spectra (Ullman, 2012, Hill et al., 2018, Hill et al., 2015).

The second misconception is to conflate the slices of a very effective cover with the very effective slices of that cover. The foundational $\mathbb P^1$51 paper computes the ordinary slices $\mathbb P^1$52, not the full very effective slice tower of $\mathbb P^1$53, and it explicitly notes that the first very effective slice differentials remain additional work (Ananyevskiy et al., 2017). The later computation of $\mathbb P^1$54 similarly uses effective slice spectral sequences for a spectrum defined from a very effective cover; this is closely related to very effective methods, but not identical to computing a separate very effective tower (Kong et al., 2022).

The third misconception is to assume that effective and very effective towers always differ substantially. In several important families they coincide. For the standard quotients $\mathbb P^1$55 appearing in the homological slice spectral sequence, the effective, cellular effective, very effective, and cellular very effective towers agree (Carrick et al., 2023). For logarithmic $\mathbb P^1$56 and Kummer étale $\mathbb P^1$57 under the stated hypotheses, the ordinary and very effective slices coincide and are motivic cohomology or Lichtenbaum étale motivic cohomology (Binda et al., 2024). This suggests that coincidence is frequent in highly structured $\mathbb P^1$58-theoretic settings, but it is not automatic.

A plausible implication is that the most productive way to understand the very effective slice spectral sequence is to treat it as one member of a family of slice-like filtrations: ordinary effective towers, regular equivariant towers, realized effective towers, and homological slice towers. The literature shows that these constructions repeatedly interact through Betti realization, connective covers, norm functors, arithmetic localizations, and spectral-sequence comparisons, but it is careful to keep their formal identities separate (Kong, 2020, Carrick et al., 2023, Binda et al., 2024).