--- title: Very Effective Slice Spectral Sequence url: https://www.emergentmind.com/topics/very-effective-slice-spectral-sequence type: topic --- # Very Effective Slice Spectral Sequence 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 [1712.01349] [1206.0058] [1811.07960]. ## 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 [1712.01349]. 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 [2209.08603]. 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 [1206.0058]. 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 \(f_q,s_q\) formalism | Baseline ordinary motivic slice tower | | Very effective filtration | Generated by \(\mathbb P^1\)-suspension spectra of smooth schemes, closed under colimits and extensions | Explicit in motivic work on \(kq\) and logarithmic slices | | Regular slice filtration | Only regular slice cells \(G_+\wedge_H S^{n\rho_H}\) | Equivariant variant of Ullman, not “very effective” | This terminological separation controls much of the subject. Papers on \(kq\) and logarithmic \(K\)-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 [1712.01349] [1206.0058]. ## 2. Spectral-sequence formalism and convergence The very effective slice spectral sequence is obtained from the tower of very effective covers \(\tilde f_qE\), with associated slices \(\tilde s_qE\). In the logarithmic setting this is written explicitly as \[ \tilde s_i^\tau E:=\operatorname{cofib}(\tilde f_{i+1}^\tau E\to \tilde f_i^\tau E), \] while the ordinary effective slices are \[ s_i^\tau:=\operatorname{cofib}(f_{i+1}^\tau\to f_i^\tau) \] [2403.03056]. For motivic spectra over \(\mathbb R\), a realized effective cotower can also be formed by choosing a motivic lift \(\widetilde X\) of a \(C_2\)-spectrum \(X\) and setting \[ \mathrm{RT}(X)^*:=\mathrm{Re}(f^*\widetilde X),\qquad sc(X):=\lim_q \mathrm{RT}(X)^{q-1}. \] This tower filters \(X\) through realized effective covers, and the associated spectral sequence converges in general to \(\pi_*^{C_2}(sc(X))\) rather than automatically to \(\pi_*^{C_2}(X)\) [2004.00806]. A key convergence criterion in that realized setting is the lemma: if \(X\) is connective and \(X/\eta\) is slice complete, then \[ sc(X)\simeq X^\wedge_\eta. \] Applied to \(\ko\), the note argues that \(\ko/\eta\) is an extension of \(\kR\) and a slice, that \(\kR\) is slice complete, and therefore \[ sc(\ko)\simeq \ko^\wedge_\eta \] [2004.00806]. This places \(\eta\)-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: \[ \gamma_r^{p,q}:E_r^{p,q}(AH)\xrightarrow{\ \sim\ }E_{2r-1}^{\,3p+q,\;2p}(AN). \] 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 [1311.4179]. A homology-valued variant built directly from the very effective slice tower appears in the homological slice spectral sequence. For \(E\in\mathcal{SH}(k)\) and \(K=i_*H\mathbb F_2\), its \(E_2\)-page is \[ E_2^{s,w,t}(E;K)=K_{t-s,w}(P_t^tE), \] with Adams trigrading \((t-s,w,s)\), and it converges to the mod \(2\) homology of the global sections spectrum \(\Gamma(E)\). When \(E\) and \(K\) are slice \(\ge 0\), weight-zero terms satisfy the vanishing line \[ E_2^{s,0,t}(E;K)=0 \quad \text{if } s>t-s \text{ or } t<0 \] [2304.01960]. ## 3. The very effective cover \(kq\) and its slice spectral sequence The central motivic example is hermitian \(K\)-theory. The very effective cover of \(KQ\) is denoted \[ kq \to KQ, \qquad kq=f_0(KQ_{\ge 0}), \] and it is presented as the algebro-geometric analogue of connective real topological \(K\)-theory. If the base field admits a complex embedding, then its Betti realization is \[ \operatorname{Re}_B(kq)\simeq ko, \] and it fits into the connective Wood sequence \[ \Sigma^{1,1}kq \xrightarrow{\eta} kq \xrightarrow{f} kgl \xrightarrow{hyp} \Sigma^{2,1}kq \] [1712.01349]. The slices of \(kq\) are considerably smaller than those of periodic \(KQ\). When \(\operatorname{char}(F)\neq 2\), the nonnegative slices are \[ s_{q}kq = \begin{cases} \Sigma^{2n,2n}M/2 \vee \Sigma^{2n+2,2n}M/2\vee \dotsm \vee \Sigma^{4n-2,2n}M/2 \vee \Sigma^{4n,2n}M & q=2n, \\ \Sigma^{2n+1,2n+1}M/2 \vee \Sigma^{2n+3,2n+1}M/2 \vee \dotsm \vee \Sigma^{4n+1,2n+1}M/2 & q=2n+1, \end{cases} \] and the negative slices are zero [1712.01349]. The graded ring of slices is described multiplicatively by \[ s_\ast kq \cong M[\eta,\sqrt{\alpha}]/(2\eta=0,\eta^2\xrightarrow{\delta}\sqrt{\alpha}). \] These slice formulas make the ordinary slice spectral sequence for \(kq\) highly explicit: \[ \pi_{p,w}s_qkq \Rightarrow kq_{p,w}, \] with \(E_1\)-terms given by shifted motivic cohomology groups and first differential \(d_1\) expressed in motivic Steenrod operations involving \(Sq^1\), \(Sq^2\), \(Sq^3Sq^1\), and the classes \(\rho,\tau\) [1712.01349]. The same paper proves conditional convergence \[ \pi_{\star}s_{\ast}kq \Longrightarrow \pi_{\star}kq^{\wedge}_{\eta}, \] and identifies the \(0\)-line by \[ K^{MW}_{*}(F) \overset{\cong}{\longrightarrow} \bigoplus_{n\in \mathbb{Z}}\pi_{n,n}kq. \] A crucial nuance is that the paper computes the ordinary slices \(s_q(kq)\) of the very effective cover \(kq\), not the first differentials in the very effective slice tower itself. It explicitly remarks that Bachmann determined the very effective slices of \(KQ\), hence of \(kq\), up to extensions, and that additional work is needed to identify the corresponding first very effective slice differentials [1712.01349]. This distinction is one of the most common sources of confusion. The effective slice spectral sequence for \(kq\) 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 \[ \tilde f_0 \simeq f_0\tau_{\ge 0}, \] and the connective \(v_1\)-periodic analogue \[ L:=fib(\psi^3-1:kq\to kq) \] is analyzed by combining explicit slice formulas for \(kq\), new coefficient computations for \(HZ/2^n\), and a Steenrod-operation description of the \(d_1\)-differentials [2209.08603]. ## 4. Realized, equivariant, and homological variants The very effective perspective extends beyond \(kq\) in two distinct directions. One is realization to \(C_2\)-equivariant homotopy. A short note constructs a \(C_2\)-equivariant spectral sequence by realizing the \(\mathbb R\)-motivic effective slice filtration. Its tower \[ \mathrm{RT}(X)^* = \mathrm{Re}(f^*\widetilde X) \] comes with slice completion \[ sc(X):=\lim_q \mathrm{RT}(X)^{q-1}, \] and the main lemma identifies \(sc(X)\) with \(\eta\)-completion under a concrete hypothesis: \[ X\text{ connective and }X/\eta\text{ slice complete} \quad\Longrightarrow\quad sc(X)\simeq X^\wedge_\eta. \] For the intended application, \[ sc(\ko)\simeq \ko^\wedge_\eta \] [2004.00806]. 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 \(BPGL\langle m\rangle\), the homological slice spectral sequence uses the very effective filtration of Spitzweck–Østvær, but for the standard quotients of \(MGL\) under study the effective, cellular effective, very effective, and cellular very effective towers coincide. Its \(E_2\)-page is \[ E_2^{*,*,*}(BPGL\langle m\rangle;i_*H\mathbb F_2) \cong (\mathcal A_*\square_{\mathcal A(0)_*}\mathbb F_2)[\rho,x_1,\bar v_1,\ldots,\bar v_m], \] and in weight zero the image of the edge homomorphism is precisely \[ \mathcal A_*\square_{\mathcal A(m)_*}\mathbb F_2 \cong (\mathcal A//\mathcal A(m))^*. \] The paper determines a family of differentials interpolating between \[ \mathcal A_*\square_{\mathcal A(0)_*}\mathbb F_2 = \mathbb F_2[\zeta_1^2,\zeta_2,\ldots] \] and \[ \mathcal A_*\square_{\mathcal A(m)_*}\mathbb F_2 = \mathbb F_2[\zeta_1^{2^{m+1}},\zeta_2^{2^m},\ldots,\zeta_{m+1}^2,\zeta_{m+2},\ldots], \] and computes the spectral sequence completely for \(m\le 3\) [2304.01960]. Over \(k=\mathbb R\), this homological theory interacts directly with genuine equivariant spectra. The paper proves \[ \Gamma(BPGL\langle m\rangle)\xrightarrow{\simeq} BP_{\mathbb R}\langle m\rangle^{C_2}, \] so the motivic HSSS computes the homology of \(C_2\)-fixed points; in height \(2\) this yields a computation of \(H_*\mathrm{tmf}_0(3)\) [2304.01960]. ## 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 \[ \Sigma \tau_n = \bar\tau_{n+1}, \] 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” [1206.0058]. 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 \[ BP^{((C_4))}\langle 2\rangle. \] That paper is explicit that it computes the ordinary HHR slice filtration, not a separate very effective variant. The spectrum has slice associated graded \[ S^0[G\cdot \bar r_1,\dots,G\cdot \bar r_{2^m-1}]\wedge H, \] and for \(BP^{((C_4))}\langle 2\rangle\) every slice is a suspension of an Eilenberg–MacLane spectrum either by a regular \(C_4\)-cell \(S^{n\rho_4}\wedge H\) or by an induced slice cell \({C_4}_+\wedge_{C_2}S^{m\rho_2}\wedge H\). The spectral sequence terminates after the \(E_{61}\)-page and has a horizontal vanishing line of filtration \(61\); after inverting the periodicity element \(D_2\), the localized spectrum \(\Psi\) satisfies \[ S^{3\rho_4}\wedge \Psi \simeq \Psi,\qquad S^{8-8\sigma}\wedge \Psi \simeq \Psi,\qquad S^{32+32\sigma-32\lambda}\wedge \Psi \simeq \Psi, \] which combine to yield \(384\)-periodicity of \(\Psi^{C_4}\) and of \(E_4^{hC_{12}}\) [1811.07960]. A third case is the \(C_4\)-analog \(K_{[2]}\) of real \(K\)-theory. Here again the paper computes the ordinary HHR slice spectral sequence, but the slices are built from highly connective representation suspensions: \[ P_t^t k_{[2]}= \begin{cases} \displaystyle \bigvee_{0\le m\le t/4} X_{m,t/2-m}, & t\ge 0 \text{ even},\\ *, & \text{otherwise}, \end{cases} \] where \[ X_{m,n}= \begin{cases} \Sigma^{m\rho_4}H\underline{\mathbb Z}, & m=n,\\ G_+\wedge_{G'} \Sigma^{(m+n)\rho_2}H\underline{\mathbb Z}, & m