The local cohomology spectral sequence for topological modular forms

Abstract

We discuss proofs of local cohomology theorems for topological modular forms, based on Mahowald–Rezk duality and on Gorenstein duality, and then make the associated local cohomology spectral sequences explicit, including their differential patterns and hidden extensions.

1 Introduction

Several interesting ring spectra satisfy duality theorems relating local cohomology to Anderson or Brown–Comenetz duals. The algebraic precursor of these results is due to Grothendieck [29], and is a local analogue of Serre’s projective duality theorem. In each case there is a covariant local cohomology spectral sequence converging to the homotopy of the local cohomology spectrum, and a contravariant ${{\,\mathrm{Ext}\,}}$ spectral sequence computing the homotopy of a functionally dual spectrum. As a consequence of self-dualities intrinsic to the ring spectra in question, the results of the two calculations agree up to a shift in grading, in spite of their opposite variances. It is the purpose of this paper to make these self-dualities explicit for the connective topological modular forms spectrum. Figures 3 and 11 depict the 2- and 3-complete dualities, respectively. A reader wondering if this paper is of interest might glance at these figures; they do require explanation (given below), but the structural patterns are immediately and strikingly apparent in the pictures. A reader new to $tm\!f$ might prefer to start with the simpler charts for $p=3$, as preparation for the case of $p=2$. We also treat the much simpler case of the connective real K-theory spectrum.

We work at one prime p at a time, write $ko = ko^\wedge _p$ and $ku = ku^\wedge _p$ for the p-completed real and complex connective topological K-theory spectra, and write $tm\!f= tm\!f^\wedge _p$ for the p-completed connective topological modular forms spectrum. We also consider a 2-complete spectrum $tm\!f_1(3) = tm\!f_1(3)^\wedge _2$ and a 3-complete spectrum $tm\!f_0(2) = tm\!f_0(2)^\wedge _3$ related to elliptic curves with $\Gamma _1(3)$ and $\Gamma _0(2)$ level structures, respectively. These are all commutative $S_p$-algebras, where $S_p = S^\wedge _p$ denotes the p-completed sphere spectrum. See [20] and [15] for theoretical and computational background regarding topological modular forms. For $p=2$ there are Bott and Mahowald classes $B \in \pi _8(tm\!f)$ and $M \in \pi _{192}(tm\!f)$ detected by the modular forms $c_4$ and $\Delta ^8$, respectively. The homotopy groups $\pi _*(tm\!f)$ for $0 \le * \le 192$ are shown in Fig. 4. The red dots indicate B-power torsion classes, and the entire picture repeats M-periodically.

For any commutative $S_p$-algebra R and a choice of finitely generated ideal $J = (x_1, \dots , x_d) \subset \pi _*(R)$, the local cohomology spectrum $\Gamma _J R$ encapsulates the J-power torsion of $\pi _*(R)$, together with its right derived functors. There is a local cohomology spectral sequence

$$\begin{aligned} E_2^{s,t} = H^s_J(\pi _*(R))_t \Longrightarrow _s \pi _{t-s}(\Gamma _J R) \end{aligned}$$

(in Adams grading), which can be used to compute its homotopy. For $R = tm\!f$ at $p=2$ and $J = (B, M)$ the spectral sequence collapses to a short exact sequence

$$\begin{aligned} 0 \rightarrow H^2_{(B, M)}(\pi _*(tm\!f))_{n+2} \longrightarrow \pi _n(\Gamma _{(B, M)} tm\!f) \longrightarrow H^1_{(B, M)}(\pi _*(tm\!f))_{n+1} \rightarrow 0 \end{aligned}$$

in each topological degree n, cf. Figs. 5 and 6, while for $J = (2, B, M)$ its $E_2$-term is concentrated in filtration degrees $1 \le s \le 3$ and contains nonzero $d_2$-differentials, cf. Figs. 7, 8, 9 and 10.

The $S_p$-module Anderson and Brown–Comenetz duals $I_{{\mathbb {Z}}_p} R$ and IR are defined as function spectra $F_{S_p}(R, I_{{\mathbb {Z}}_p})$ and $F_{S_p}(R, I)$, where $I_{{\mathbb {Z}}_p}$ and I are so designed that the associated homotopy spectral sequences collapse to a short exact sequence

$$\begin{aligned} 0 \rightarrow {{\,\mathrm{Ext}\,}}_{{\mathbb {Z}}_p}(\pi _{m-1}(R), {\mathbb {Z}}_p) \longrightarrow \pi _{-m}(I_{{\mathbb {Z}}_p} R) \longrightarrow {{\,\mathrm{Hom}\,}}_{{\mathbb {Z}}_p}(\pi _m(R), {\mathbb {Z}}_p) \rightarrow 0 \end{aligned}$$

and an isomorphism $\pi _{-m}(IR) \cong {{\,\mathrm{Hom}\,}}_{{\mathbb {Z}}_p}(\pi _m(R), {\mathbb {Q}}_p/{\mathbb {Z}}_p)$, respectively. The local cohomology duality theorems for $tm\!f$ at $p=2$ establish equivalences

$$\begin{aligned} \Gamma _{(B, M)} tm\!f\simeq & {} \Sigma ^{-22} I_{{\mathbb {Z}}_2}(tm\!f) \end{aligned}$$

(1.1)

$$\begin{aligned} \Gamma _{(2, B, M)} tm\!f\simeq & {} \Sigma ^{-23} I(tm\!f), \end{aligned}$$

(1.2)

which in particular imply that the covariantly defined $\pi _n(\Gamma _{(B, M)} tm\!f)$ and the contravariantly defined $\pi _{-m}(I_{{\mathbb {Z}}_2} tm\!f)$ are isomorphic for $n+m = -22$, and similarly that $\pi _n(\Gamma _{(2, B, M)} tm\!f)$ and ${{\,\mathrm{Hom}\,}}_{{\mathbb {Z}}_2}(\pi _m(tm\!f), {\mathbb {Q}}_2/{\mathbb {Z}}_2)$ are isomorphic for $n+m = -23$.

Figure 3 illustrates $\pi _*(tm\!f)$, $\pi _*(\Gamma _{(B, M)} tm\!f)$, $\pi _*(I_{{\mathbb {Z}}_2}(tm\!f))$ and the duality isomorphism, up to a degree shift, between the latter two graded abelian groups. More precisely, $\pi _*(tm\!f)$ is isomorphic to the ‘basic block’ $\pi _*(N)$ shown in the first part of the figure, tensored with ${\mathbb {Z}}[M]$. The local cohomology $\pi _*(\Gamma _{(B, M)} tm\!f)$ and the Anderson dual $\pi _*(I_{{\mathbb {Z}}_2} tm\!f)$ are isomorphic to $\pi _*(\Gamma _B N)$ and $\pi _*(I_{{\mathbb {Z}}_2} N)$ tensored with ${\mathbb {Z}}[M]/M^\infty$, respectively, up to appropriate degree shifts. The second part of the figure shows the covariantly defined $\pi _*(\Gamma _B N)$, while the third part shows the contravariantly defined $\pi _*(\Sigma ^{171} I_{{\mathbb {Z}}_2} N)$. The nearly mirror symmetric isomorphism between the latter two graded abelian groups thus exhibits the duality isomorphism (1.1), in its ‘basic block’ form $\Gamma _B N \simeq \Sigma ^{171} I_{{\mathbb {Z}}_2} N$. The same structure is presented in greater detail in Figs. 4, 5 and 6, where the additive generators are named and the module action by $\eta$, $\nu$ (and B) is shown by lines increasing the topological degree by 1, 3 (and 8), respectively. However, the distinctive symmetry implied by the duality theorem is most easily seen in the first figure.

Several different approaches lead to proofs of such local cohomology duality theorems. For fp-spectra X, i.e., bounded below and p-complete spectra whose mod p cohomology is finitely presented as a module over the Steenrod algebra, Mahowald and Rezk [38] determined the cohomology of the Brown–Comenetz dual of the finite E(n)-acyclisation $C^f_n X$. In many cases $C^f_n R$ is a local cohomology spectrum, and we show in Theorem 4.8 how this leads to duality theorems for $R = ko$ at all primes, and for $R = tm\!f$ at $p=2$ and $p=3$. This strategy ties nicely in with chromatic homotopy theory.

Next, Dwyer, Greenlees and Iyengar [21] showed that for augmented ring spectra $R \rightarrow k$ such that $\pi _*(R) \rightarrow k$ is algebraically Gorenstein, the k-cellularisation ${{\,\mathrm{Cell}\,}}_k R$ is often equivalent to a suspension of IR or $I_{{\mathbb {Z}}_p} R$, for $k = {\mathbb {F}}_p$ or $k = {\mathbb {Z}}_p$, respectively. We use descent methods to extend this to ring spectra with a good map to an augmented ring spectrum $T \rightarrow k$ satisfying the algebraic Gorenstein property, e.g., with $\pi _*(T) = k[x_1, \dots , x_d]$ polynomial over k. Moreover, ${{\,\mathrm{Cell}\,}}_k R$ is in many cases a local cohomology spectrum, and we show in Theorem 5.19 how this leads to duality theorems for $R = ko$ and $R = tm\!f$, at all primes p. This strategy emphasises commutative algebra inspired by algebraic geometry.

There is a growing list [8,9,10, 13, 14, 21, 23,24,25, 27, 28] of examples known to enjoy Gorenstein duality. Many of them are of equivariant origin, or have $R = C^*(X)$ for a manifold X, or arise from Serre duality in derived algebraic geometry. For instance, Stojanoska [43, 44] used Galois descent and homotopy fixed point spectral sequences to deduce Anderson self-duality for $Tm\!f$ from its covers $Tm\!f(2)$ (at $p=3$) and $Tm\!f(3)$ (at $p=2$). More recently, Bruner and Rognes [15] used a variant of the descent arguments above to directly deduce local cohomology duality theorems for $tm\!f$ at $p=2$ and $p=3$ from similar theorems for $tm\!f_1(3)$ and $tm\!f_0(2)$, respectively. We summarise these results in Theorems 6.1 and 6.2.

The main goal of this paper is to draw on the Hopkins–Mahowald calculation of $\pi _*(tm\!f)$, as presented in [15], to make the local cohomology spectral sequences for $R = tm\!f$ at $p=2$ and at $p=3$, with $J = (B, M)$ and $J = (p, B, M)$, completely explicit. In order to determine the differential patterns and some of the hidden (filtration-shifting) multiplicative extensions in these spectral sequences, we rely on the local cohomology duality theorems to identify the abutments with shifts of the Anderson and Brown–Comenetz duals of $tm\!f$. This is carried out in Sects. 8.1 and 8.2 for $p=2$, and in Sects. 8.3 and 8.4 for $p=3$. See also the explanations in Sect. 8.5 of the graphical conventions used in the charts. As a warm-up we first go through the corresponding, but far simpler, calculations for $R = ko$ at $p=2$ in Sect. 7.

2 Colocalisations

2.1 Small and proxy-small

The stable homotopy category of spectra and, more generally, the homotopy category of R-modules for any fixed S-algebra R, are prototypical triangulated categories. We keep the terminology from [21,  3.15, 4.6]: A full subcategory of a triangulated category is thick if it is closed under equivalences, integer suspensions, cofibres and retracts, and it is localising if it is furthermore closed under coproducts. An object A finitely builds an object X if X lies in the thick subcategory generated by A, and more generally A builds X if X lies in the localising subcategory generated by A. An R-module A is small if it is finitely built from R, and more generally it is proxy-small if there is a small R-module K that both builds A and is finitely built by A.

2.2 Acyclisation

We recall three related colocalisations. First, for any spectrum X and integer $n\ge 0$ let $C^f_n X \rightarrow X$ denote its finite E(n)-acycl(ic)isation, defined as the homotopy fiber of the finite E(n)-localization $X \rightarrow L^f_n X$ introduced by Miller [41,  §2]. Here E(n) denotes the n-th p-local Johnson–Wilson spectrum, with coefficient ring $\pi _* E(n) = {\mathbb {Z}}_{(p)}[v_1, \dots , v_{n-1}, v_n^{\pm 1}]$. The map $F(A, C^f_n X) \rightarrow F(A, X)$ is an equivalence for each finite E(n)-acyclic A, and $C^f_n X$ is built from finite E(n)-acyclic spectra. The finite localization is smashing, so there is a natural equivalence $C^f_n X \simeq X \wedge C^f_n S$, and for any R-module M the spectrum $C^f_n M$ admits a natural R-module structure. A p-local finite spectrum has type $n+1$ if it is E(n)-acyclic but not $E(n+1)$-acyclic. If X is p-local, then by Hopkins–Smith [33,  Thm. 7] any one choice of a finite spectrum A of type $n+1$ suffices to build $C^f_n X$. Inductively for each $n\ge 0$, Hovey and Strickland [35,  Prop. 4.22] build a cofinal tower of generalised Moore spectra S/I of type $n+1$, for suitable ideals $I = (p^{a_0}, v_1^{a_1}, \dots , v_{n-1}^{a_{n-1}}, v_n^{a_n})$, such that there are homotopy cofibre sequences

$$\begin{aligned} \Sigma ^{2(p^n-1)a_n} S/I' \overset{v_n^{a_n}}{\longrightarrow }S/I' \longrightarrow S/I \end{aligned}$$

with $I' = (p^{a_0}, v_1^{a_1}, \dots , v_{n-1}^{a_{n-1}})$. Here $S/() = S$ and $v_0 = p$. By [35,  Prop. 7.10(a)] there is a natural equivalence ${{\,\mathrm{hocolim}\,}}_I F(S/I, X) \simeq C^f_n X$, where S/I ranges over this tower.

2.3 Cellularisation

Second, let k and M be R-modules. The k-cellularisation of M is the R-module map ${{\,\mathrm{Cell}\,}}_k M \rightarrow M$ such that $F_R(k, {{\,\mathrm{Cell}\,}}_k M) \rightarrow F_R(k, M)$ is an equivalence, and such that ${{\,\mathrm{Cell}\,}}_k M$ is built from k in R-modules. It can be realised as the cofibrant replacement in a right Bousfield localisation of the stable model structure on R-modules in symmetric spectra, cf. [31,  §5.1, §4.1], hence always exists.

Lemma 2.1

If two R-modules k and $\ell$ mutually build one another, then ${{\,\mathrm{Cell}\,}}_k M \simeq {{\,\mathrm{Cell}\,}}_{\ell } M$ for all R-modules M. Conversely, if ${{\,\mathrm{Cell}\,}}_k M \simeq {{\,\mathrm{Cell}\,}}_{\ell } M$ for all M, then k and $\ell$ mutually build one another.

Proof

If k builds $\ell$, then $F_R(\ell , {{\,\mathrm{Cell}\,}}_k M) \rightarrow F_R(\ell , M)$ is an equivalence. If $\ell$ builds k, then ${{\,\mathrm{Cell}\,}}_k M$ is built from $\ell$. If both conditions hold, then $M \rightarrow {{\,\mathrm{Cell}\,}}_k M$ is the $\ell$-cellularisation of M.

If ${{\,\mathrm{Cell}\,}}_k \ell \simeq {{\,\mathrm{Cell}\,}}_{\ell } \ell = \ell$, then k builds $\ell$, and if $k = {{\,\mathrm{Cell}\,}}_k k \simeq {{\,\mathrm{Cell}\,}}_\ell k$ then $\ell$ builds k, so if both hold then k and $\ell$ build one another. $\square$

Lemma 2.2

If A is a p-local finite spectrum of type $n+1$, and M is a p-local R-module, then $C^f_n M \simeq {{\,\mathrm{Cell}\,}}_{R \wedge A} M$ as R-modules.

Proof

We know that R builds M in R-modules, and A builds $C^f_n S_{(p)}$ in S-modules, so $R \wedge A$ builds $M \wedge C^f_n S_{(p)} \simeq C^f_n M$ in R-modules. Moreover, $F_R(R \wedge A, C^f_n M) \rightarrow F_R(R \wedge A, M)$ is an equivalence, since this the same map as $F(A, C^f_n M) \rightarrow F(A, M)$. Hence $C^f_n M$ is the $R \wedge A$-cellularisation of M. $\square$

Let ${\mathcal {E}}= F_R(k, k)$ be the endomorphism S-algebra of the R-module k. An R-module M is effectively constructible from k if the natural map

$$\begin{aligned} F_R(k, M) \wedge _{{\mathcal {E}}} k \longrightarrow M \end{aligned}$$

is an equivalence. It is proved in [21,  Thm. 4.10] that, if k is proxy-small, then this map always realises the k-cellularisation of M. Hence ${{\,\mathrm{Cell}\,}}_k M$ is determined by the right ${\mathcal {E}}$-module structure on $F_R(k, M)$, for proxy-small k.

2.4 Local cohomology

Third, suppose that R is a commutative S-algebra, and let $J = (x_1, \dots , x_d)$ be a finitely generated ideal in the graded ring $\pi _*(R)$. For each $x \in \pi _*(R)$ define the x-power torsion spectrum $\Gamma _x R$ by the homotopy (co-)fibre sequence

$$\begin{aligned} \Sigma ^{-1} R\left[ \frac{1}{x}\right] \overset{\alpha }{\longrightarrow }\Gamma _x R \overset{\beta }{\longrightarrow }R \overset{\gamma }{\longrightarrow }R\left[ \frac{1}{x}\right] . \end{aligned}$$

For any R-module M let

$$\begin{aligned} \Gamma _J M = \Gamma _{x_1} R \wedge _R \dots \wedge _R \Gamma _{x_d} R \wedge _R M \end{aligned}$$

be the local cohomology spectrum. By [26,  §1, §3], this R-module only depends on the radical $\sqrt{J}$ of the ideal J. The convolution product of the short filtrations $\alpha : \Sigma ^{-1} R[1/x_i] \rightarrow \Gamma _{x_i} R$ for $1 \le i \le d$ leads to a length d decreasing filtration of $\Gamma _J M$, with subquotients

$$\begin{aligned} F^s/F^{s+1} \simeq \bigvee _{1 \le i_1< \dots < i_s \le d} \Sigma ^{-1} R\left[ \frac{1}{x_{i_1}}\right] \wedge _R \dots \wedge _R \Sigma ^{-1} R[\frac{1}{x_{i_s}}] \wedge _R M. \end{aligned}$$

In Adams indexing, the associated spectral sequence has $E_1$-term

$$\begin{aligned} E_1^{s,t} = \pi _{t-s}(F^s/F^{s+1}) \cong \bigoplus _{1 \le i_1< \dots < i_s \le d} \pi _t\left( M\left[ \frac{1}{x_{i_1} \cdot \ldots \cdot x_{i_s}}\right] \right) \end{aligned}$$

for $0 \le s \le d$, with differentials $d_r : E_r^{s,t} \rightarrow E^{s+r,t+r-1}$. The $d_1$-differentials are induced by the various localisation maps $\gamma : R \rightarrow R[1/x_i]$, and the cohomology of $(E_1, d_1)$ defines the local cohomology groups of the $\pi _*(R)$-module $\pi _*(M)$, in the sense of Grothendieck [29]. This construction defines the local cohomology spectral sequence

$$\begin{aligned} E_2^{s,t} = H^s_J(\pi _*(M))_t \Longrightarrow _s \pi _{t-s}(\Gamma _J M), \end{aligned}$$

(2.1)

which is a strongly convergent $\pi _*(R)$-module spectral sequence, cf. [26,  (3.2)]. As in the topological case, the local cohomology groups $H^*_J(\pi _*(M))$ only depend on J through its radical in $\pi _*(R)$, not on the explicit generators $x_1, \dots , x_d$. We emphasise that the local cohomology spectrum $\Gamma _J M$, and the associated spectral sequence, are covariantly functorial in M.

Definition 2.3

Given a finite sequence $J = (x_1, \dots , x_d)$ of elements in $\pi _*(R)$, we let

$$\begin{aligned} R/J = R/x_1 \wedge _R \dots \wedge _R R/x_d, \end{aligned}$$

where each R/x is defined by a homotopy cofibre sequence

$$\begin{aligned} \Sigma ^{|x|} R \overset{x}{\longrightarrow }R \longrightarrow R/x \longrightarrow \Sigma ^{|x|+1} R. \end{aligned}$$

We shall also write R/J for this R-module in contexts where J is interpreted as the ideal in $\pi _*(R)$ generated by the given sequence of elements. This is, however, an abuse of notation, since R/J depends upon the chosen generators for the ideal, not just on the ideal J itself. We may refer to R/J and $\Gamma _J R$ as the Koszul complex and the stable Koszul complex, respectively.

Lemma 2.4

If $J = (x_1, \dots , x_d)$ is a finitely generated ideal in $\pi _*(R)$, and M is any R-module, then ${{\,\mathrm{Cell}\,}}_{R/J} M \simeq \Gamma _J M$ as R-modules.

Proof

We show that $\Gamma _J M$ is the R/J-cellularisation of M. An inductive argument, as in the proof of [21,  Prop. 9.3], shows that R/J finitely builds $R/x_1^m \wedge _R \dots \wedge _R R/x_d^m$ for each $m\ge 1$. Passing to the colimit over m, it follows that R/J builds $\Gamma _J R$. Since R builds M, it also follows that R/J builds $\Gamma _J M$. Finally, $F_R(R/J, \Gamma _J M) \rightarrow F_R(R/J, M)$ is an equivalence, because $F_R(R/J, N[1/x_i]) \simeq *$ for each R-module N and any $1 \le i \le d$. $\square$

With notation as above, if $R \wedge A$ and R/J mutually build one another, then $C^f_n M \simeq {{\,\mathrm{Cell}\,}}_{R \wedge A} M \simeq {{\,\mathrm{Cell}\,}}_{R/J} M \simeq \Gamma _J M$ by Lemmas 2.2, 2.1 and 2.4. Under slightly different hypotheses we can close the cycle and obtain this conclusion directly.

Lemma 2.5

Let $I = (p^{a_0}, \dots , v_n^{a_n})$ and $J = (x_1, \dots , x_d)$. If

• (1) each $x_i$ acts nilpotently on each $\pi _* F(S/I, R)$, and

• (2) $v_s^{a_s}$ acts nilpotently on $\pi _* F(S/(p^{a_0}, \dots , v_{s-1}^{a_{s-1}}), R/J)$ for each $0 \le s \le n$,

then $C^f_n M \simeq \Gamma _J M$ as R-modules.

Proof

Item (1) ensures that $F(S/I, \Gamma _J R) \simeq \Gamma _J F(S/I, R)$ is equivalent to F(S/I, R) for each I in the cofinal system, and passage to homotopy colimits implies that

$$\begin{aligned} C^f_n \Gamma _J R \overset{\simeq }{\longrightarrow }C^f_n R \end{aligned}$$

is an equivalence. Item (2) ensures that $C^f_n R/J \simeq {{\,\mathrm{hocolim}\,}}_I F(S/I, R/J)$ is equivalent to R/J, which implies that

$$\begin{aligned} C^f_n \Gamma _J R \overset{\simeq }{\longrightarrow }\Gamma _J R \end{aligned}$$

is an equivalence, since R/J builds $\Gamma _J R$. Hence $C^f_n R \simeq \Gamma _J R$, and more generally $C^f_n M = C^f_n R \wedge _R M \simeq \Gamma _J R \wedge _R M = \Gamma _J M$. $\square$

2.5 A composite functor spectral sequence

Let $I, J \subset R_*$ be finitely generated ideals in a graded commutative ring, and let $M_*$ be an $R_*$-module. If $I = (x)$ we write $\Gamma _x M_* = H^0_I(M_*)$ and $M_*/x^\infty = H^1_I(M_*)$ for the kernel and the cokernel of the localisation homomorphism $\gamma$ below.

$$\begin{aligned} 0 \rightarrow \Gamma _x M_* \longrightarrow M_* \overset{\gamma }{\longrightarrow }M_*[1/x] \longrightarrow M_*/x^\infty \rightarrow 0. \end{aligned}$$

More generally, let $\Gamma _I M_* = H^0_I(M_*)$ denote the I-power torsion submodule of $M_*$. The identity $\Gamma _I(\Gamma _J M_*) = \Gamma _{I+J} M_*$ leads to a composite functor spectral sequence

$$\begin{aligned} E_2^{i,j} = H^i_I(H^j_J(M_*)) \Longrightarrow _i H^{i+j}_{I+J}(M_*). \end{aligned}$$

This is a case of the double complex spectral sequence of [17,  §xXV.6]. When $I = (x)$ and $J = (y)$, it arises by horizontally filtering the condensation of the central commutative square below, leading to an $E_1$-term given by the inner modules in the upper and lower rows, and an $E_2$-term given by the modules at the four corners.

For bidegree reasons, the spectral sequence collapses at this stage, so that

$$\begin{aligned} E_2^{i,j} = E_\infty ^{i,j} = {\left\{ \begin{array}{ll} \Gamma _x(\Gamma _y M_*) &{}\quad \text {for } (i,j) = (0,0), \\ (\Gamma _y M_*)/x^\infty &{}\quad \text {for } (i,j) = (1,0), \\ \Gamma _x (M_*/y^\infty ) &{}\quad \text {for } (i,j) = (0,1), \\ (M_*/y^\infty )/x^\infty &{}\quad \text {for } (i,j) = (1,1). \end{array}\right. } \end{aligned}$$

It follows that we have identities

$$\begin{aligned} \Gamma _x(\Gamma _y M_*)&= \Gamma _{(x,y)} M_* = H^0_{(x,y)}(M_*) \\ (M_*/y^\infty )/x^\infty&= M_*/(x^\infty , y^\infty ) = H^2_{(x,y)}(M_*), \end{aligned}$$

and a natural short exact sequence

$$\begin{aligned} 0 \rightarrow (\Gamma _y M_*)/x^\infty \longrightarrow H^1_{(x,y)}(M_*) \longrightarrow \Gamma _x(M_*/y^\infty ) \rightarrow 0. \end{aligned}$$

If $\Gamma _y M_* \subset \Gamma _x M_*$, so that the y-power torsion is entirely x-power torsion, then $(\Gamma _y M_*)/x^\infty = 0$ and $H^1_{(x,y)}(M_*) \cong \Gamma _x(M_*/y^\infty )$.

3 Dualities

3.1 Artinian and Noetherian $S_p$-modules

Let $S_p$ denote the p-completed sphere spectrum. The category of $S_p$-modules contains a subcategory of p-power torsion modules satisfying $\Gamma _p M \simeq M$, and a subcategory of p-complete modules for which $M \simeq M^\wedge _p$. The (covariant) functors $\Gamma _p$ and $(-)^\wedge _p$ give mutually inverse equivalences between these full subcategories, cf. [34,  Thm. 3.3.5].

We say that a p-power torsion module M is Artinian if each homotopy group $\pi _t(M)$ is an Artinian ${\mathbb {Z}}_p$-module, i.e., a finite direct sum of modules of the form ${\mathbb {Q}}_p/{\mathbb {Z}}_p$ or ${\mathbb {Z}}/p^a$ for $a\ge 1$. Dually, we say that a p-complete module M is Noetherian if each homotopy group $\pi _t(M)$ is a Noetherian ${\mathbb {Z}}_p$-module, i.e., a finite direct sum of modules of the form ${\mathbb {Z}}_p$ or ${\mathbb {Z}}/p^a$ for $a\ge 1$. The latter are the same as the finitely generated ${\mathbb {Z}}_p$-modules. The simultaneously Artinian and Noetherian $S_p$-modules M are those for which each $\pi _t(M)$ is finite.

3.2 Brown–Comenetz duality

We recall two related dualities. First, working in $S_p$-modules, the Brown–Comenetz duality spectrum I represents the cohomology theory

$$\begin{aligned} I^t(M) = {{\,\mathrm{Hom}\,}}_{{\mathbb {Z}}_p}(\pi _t(M), {\mathbb {Q}}_p/{\mathbb {Z}}_p), \end{aligned}$$

cf. [12]. This makes sense because ${\mathbb {Q}}_p/{\mathbb {Z}}_p$ is an injective ${\mathbb {Z}}_p$-module. Letting $IM = F_{S_p}(M, I)$ we obtain a contravariant endofunctor I of $S_p$-modules, with $\pi _{-t}(IM) = I^t(M)$. It maps p-power torsion modules to p-complete modules, and restricts to a functor from Artinian $S_p$-modules to Noetherian $S_p$-modules, since

$$\begin{aligned} {{\,\mathrm{Hom}\,}}_{{\mathbb {Z}}_p}({\mathbb {Q}}_p/{\mathbb {Z}}_p, {\mathbb {Q}}_p/{\mathbb {Z}}_p) \cong {\mathbb {Z}}_p \quad \text {and}\quad {{\,\mathrm{Hom}\,}}_{{\mathbb {Z}}_p}({\mathbb {Z}}/p^a, {\mathbb {Q}}_p/{\mathbb {Z}}_p) \cong {\mathbb {Z}}/p^a. \end{aligned}$$

In general, it does not map p-complete modules to p-power torsion modules, but it does restrict to a functor from Noetherian $S_p$-modules to Artinian $S_p$-modules. Moreover, the natural map

$$\begin{aligned} \rho : M \longrightarrow I(IM) \end{aligned}$$

is an equivalence for M that are Artinian or Noetherian. Hence the two restrictions of I are mutually inverse contravariant equivalences between Artinian $S_p$-modules and Noetherian $S_p$-modules. If the $S_p$-module action on M extends to a (left or right) R-module structure, then IM is naturally a (right or left) R-module.

3.3 Anderson duality

Second, the Eilenberg–MacLane spectrum $I_{{\mathbb {Q}}_p} = H{\mathbb {Q}}_p$ represents the ordinary rational cohomology theory

$$\begin{aligned} I_{{\mathbb {Q}}_p}^t(M) = {{\,\mathrm{Hom}\,}}_{{\mathbb {Z}}_p}(\pi _t(M), {\mathbb {Q}}_p) \end{aligned}$$

in $S_p$-modules. The canonical surjection ${\mathbb {Q}}_p \rightarrow {\mathbb {Q}}_p/{\mathbb {Z}}_p$ induces a map of cohomology theories, and a map $I_{{\mathbb {Q}}_p} \rightarrow I$ of representing spectra, whose homotopy fibre defines the Anderson duality spectrum $I_{{\mathbb {Z}}_p}$, cf. [2] and [36]. Letting $I_{{\mathbb {Z}}_p}M = F_{S_p}(M, I_{{\mathbb {Z}}_p})$ and $I_{{\mathbb {Q}}_p}M = F_{S_p}(M, I_{{\mathbb {Q}}_p})$, we obtain a natural homotopy fibre sequence

$$\begin{aligned} \Sigma ^{-1} IM \longrightarrow I_{{\mathbb {Z}}_p}M \longrightarrow I_{{\mathbb {Q}}_p}M \longrightarrow IM, \end{aligned}$$

which lifts the (short) injective resolution $0 \rightarrow {\mathbb {Z}}_p \rightarrow {\mathbb {Q}}_p \rightarrow {\mathbb {Q}}_p/{\mathbb {Z}}_p \rightarrow 0$. The associated long exact sequence in homotopy splits into short exact sequences

$$\begin{aligned} 0 \rightarrow {{\,\mathrm{Ext}\,}}_{{\mathbb {Z}}_p}(\pi _{t-1}(M), {\mathbb {Z}}_p) \longrightarrow \pi _{-t}(I_{{\mathbb {Z}}_p}M) \longrightarrow {{\,\mathrm{Hom}\,}}_{{\mathbb {Z}}_p}(\pi _t(M), {\mathbb {Z}}_p) \rightarrow 0. \end{aligned}$$

(3.1)

If the $S_p$-module action on M extends to a (left or right) R-module structure, then $I_{{\mathbb {Z}}_p} M$ is naturally a (right or left) R-module, and the short exact sequence above is one of $\pi _*(R)$-modules.

The contravariant endofunctor $I_{{\mathbb {Z}}_p}$ on $S_p$-modules is equivalent to $\Sigma ^{-1} I$ on the subcategory of p-power torsion modules, since $I_{{\mathbb {Q}}_p}$ is trivial on these objects. More relevant to us is the fact that it maps Noetherian $S_p$-modules to Noetherian $S_p$-modules, since

$$\begin{aligned}&{{\,\mathrm{Ext}\,}}_{{\mathbb {Z}}_p}({\mathbb {Z}}_p, {\mathbb {Z}}_p) = 0&{{\,\mathrm{Hom}\,}}_{{\mathbb {Z}}_p}({\mathbb {Z}}_p, {\mathbb {Z}}_p) \cong {\mathbb {Z}}_p \\&{{\,\mathrm{Ext}\,}}_{{\mathbb {Z}}_p}({\mathbb {Z}}/p^a, {\mathbb {Z}}_p) \cong {\mathbb {Z}}/p^a&{{\,\mathrm{Hom}\,}}_{{\mathbb {Z}}_p}({\mathbb {Z}}/p^a, {\mathbb {Z}}_p) = 0. \end{aligned}$$

Moreover, the natural map

$$\begin{aligned} \rho : M \longrightarrow I_{{\mathbb {Z}}_p}(I_{{\mathbb {Z}}_p} M) \end{aligned}$$

(3.2)

is an equivalence for Noetherian M, cf. [46,  Thm. 2] and [37,  Cor. 2.8]. Hence $I_{{\mathbb {Z}}_p}$ restricts to a contravariant self-equivalence of Noetherian $S_p$-modules, being its own inverse equivalence.

We emphasise that the Brown–Comenetz and Anderson dual spectra, IM and $I_{{\mathbb {Z}}_p}M$, and the algebraic expressions for their homotopy groups, are contravariantly functorial in the $S_p$- or R-module M.

Lemma 3.1

There are natural equivalences

$$\begin{aligned} I(\Gamma _p M) \simeq (IM)^\wedge _p \simeq \Sigma (I_{{\mathbb {Z}}_p}M)^\wedge _p \end{aligned}$$

Proof

For $S_p$-modules M and N we have $\Gamma _p S \wedge _{S_p} M = \Gamma _p M$ and $F_{S_p}(\Gamma _p S, N) \simeq N^\wedge _p$, since $\Gamma _p S \simeq {{\,\mathrm{hocolim}\,}}_\nu S^{-1}/p^\nu$. The first equivalence then follows from the adjunction $F_{S_p}(\Gamma _p S \wedge _{S_p} M, I) \simeq F_{S_p}(\Gamma _p S, F_{S_p}(M, I))$. The second equivalence follows from the homotopy fibre sequence defining the Anderson dual, since $(I_{{\mathbb {Q}}_p} M)^\wedge _p$ is trivial. If the $S_p$-module structure on M extends to an R-module structure, then this is respected by all of these equivalences. $\square$

4 Mahowald–Rezk duality

4.1 Spectra with finitely presented cohomology

Let A denote the mod p Steenrod algebra, where p is a prime. We write $H^*(X)$ for the mod p cohomology of a spectrum X, with its natural left A-module structure. For $n\ge 0$ let A(n) be the finite sub (Hopf) algebra of A that is generated by $Sq^1, Sq^2, \dots , Sq^{2^n}$ for $p=2$, and by $\beta , P^1, \dots , P^{p^{n-1}}$ for p odd. Also let E(n) be the exterior sub (Hopf) algebra of A(n) generated by $Q_0, Q_1, \dots , Q_n$, where $Q_0 = Sq^1$ and $Q_i = [Sq^{2^i}, Q_{i-1}]$ for $i\ge 1$ and $p=2$, and $Q_0 = \beta$ and $Q_{i+1} = [P^{p^i}, Q_i]$ for $i\ge 0$ and p odd.

Let X be a spectrum that is p-complete and bounded below. Following Mahowald and Rezk [38,  §3] we say that X is an fp-spectrum if $H^*(X)$ is finitely presented as an A-module. This is equivalent to asking that $H^*(X) \cong A \otimes _{A(n)} M$ is induced up from a finite A(n)-module M, for some n. We say that a graded abelian group $\pi _*$ is finite if the direct sum $\bigoplus _t \pi _t$ is finite. The class of p-local finite spectra V such that $\pi _*(V \wedge X)$ is finite generates a thick subcategory of the stable homotopy category, and is therefore equal to the class of p-local finite spectra of type $\ge m+1$ for some well-defined integer $m\ge 0$. We then say that X has fp-type equal to m. In each case $n \ge m$, sometimes with strict inequality, cf. [16,  Prop. 3.9].

Theorem 4.1

[38,  Prop. 4.10, Thm. 8.2] Let X be p-complete and bounded below, with $H^*(X) \cong A \otimes _{A(n)} M$ for some finite A(n)-module M. Then $I C^f_n X$ is p-complete and bounded below, with $H^*(I C^f_n X) \cong A \otimes _{A(n)} {{\,\mathrm{Hom}\,}}_{{\mathbb {F}}_p}(M, \Sigma ^{a(n)} {\mathbb {F}}_p)$, where a(n) is the top degree of a nonzero class in A(n).

Recall that we write $ko = ko^\wedge _p$ and $tm\!f= tm\!f^\wedge _p$ for the p-completed connective real K-theory and topological modular forms spectra, respectively, and that these are commutative S-algebras.

Proposition 4.2

[38,  Cor. 9.3] There is an equivalence of ko-modules

$$\begin{aligned} \Sigma ^6 ko \simeq I C^f_1 ko \end{aligned}$$

(at all primes p), and an equivalence of $tm\!f$-modules

$$\begin{aligned} \Sigma ^{23} tm\!f\simeq I C^f_2 tm\!f\end{aligned}$$

(at $p=2$ and at $p=3)$. The underlying $S_p$-modules are Noetherian and bounded below.

Proof

For $X = ko$ completed at $p=2$ we have $H^*(ko) \cong A/\!/A(1) = A \otimes _{A(1)} {\mathbb {F}}_2$ by Stong [45], so

$$\begin{aligned} H^*(IC^f_1 ko) \cong A \otimes _{A(1)} \Sigma ^6 {\mathbb {F}}_2 = \Sigma ^6 A/\!/A(1), \end{aligned}$$

since $a(1) = 6$. Choosing a map $S^6 \rightarrow IC^f_1 ko$ generating the lowest homotopy (and homology) group, and using the natural ko-module structure on the target, we obtain a ko-module map $\phi : \Sigma ^6 ko \rightarrow IC^f_1 ko$. The induced A-module homomorphism $\phi ^* : H^*(IC^f_1 ko) \rightarrow H^*(\Sigma ^6 ko)$ has the form $\Sigma ^6 A/\!/A(1) \rightarrow \Sigma ^6 A/\!/A(1)$, and is an isomorphism in degree 6, hence is an isomorphism in all degrees. It follows that $\phi$ is an equivalence of 2-complete ko-modules.

For $X = \ell = BP\langle 1\rangle$ completed at any prime p we have $H^*(\ell ) \cong A/\!/E(1)$, essentially by [42], so $H^*(IC^f_1 \ell ) \cong \Sigma ^{2p} A/\!/E(1)$ and $IC^f_1 \ell \simeq \Sigma ^{2p} \ell$. For p odd this implies the claim for $ko \simeq \bigvee _{i=0}^{(p-3)/2} \Sigma ^{4i} \ell$.

For $X = tm\!f$ completed at $p=2$ we have $H^*(tm\!f) \cong A/\!/A(2) = A \otimes _{A(2)} {\mathbb {F}}_2$ by Hopkins–Mahowald [32,  Thm. 4.5], cf. Mathew [40,  Thm. 1.1]. Hence

$$\begin{aligned} H^*(IC^f_2 tm\!f) \cong A \otimes _{A(2)} \Sigma ^{23} {\mathbb {F}}_2 = \Sigma ^{23} A/\!/A(2), \end{aligned}$$

since $a(2) = 23$. Choosing a map $S^{23} \rightarrow IC^f_2 tm\!f$ generating the lowest homotopy group, and using the natural $tm\!f$-module structure on the target, we obtain a $tm\!f$-module map $\phi : \Sigma ^{23} tm\!f\rightarrow IC^f_2 tm\!f$. The induced A-module homomorphism $\phi ^* : H^*(IC^f_2 tm\!f) \rightarrow H^*(\Sigma ^{23} tm\!f)$ has the form $\Sigma ^{23} A/\!/A(2) \rightarrow \Sigma ^{23} A/\!/A(2)$, and is an isomorphism in degree 23, hence is an isomorphism in all degrees. It follows that $\phi$ is an equivalence of 2-complete $tm\!f$-modules.

For $X = tm\!f$ completed at $p=3$ we have $H^*(tm\!f) \cong A \otimes _{A(2)} M$ for a finite A(2)-module M with ${{\,\mathrm{Hom}\,}}_{{\mathbb {F}}_3}(M, {\mathbb {F}}_3) \cong \Sigma ^{-64} M$, by Proposition 4.3 below. It follows that

$$\begin{aligned} H^*(I C^f_2 tm\!f) \cong A \otimes _{A(2)} \Sigma ^{23} M \cong \Sigma ^{23} H^*(tm\!f) \end{aligned}$$

as A-modules, since $a(2) = 87$ and $87-64=23$. Choosing a map $S^{23} \rightarrow IC^f_2 tm\!f$ generating the lowest homotopy group, and using the natural $tm\!f$-module structure on the target, we obtain a $tm\!f$-module map $\phi : \Sigma ^{23} tm\!f\rightarrow IC^f_2 tm\!f$. The induced A-module homomorphism $\phi ^* : H^*(IC^f_2 tm\!f) \rightarrow H^*(\Sigma ^{23} tm\!f)$ has the form $\Sigma ^{23} A \otimes _{A(2)} M \rightarrow \Sigma ^{23} A \otimes _{A(2)} M$, and is an isomorphism in degree 23. It follows from the relation $P^3 g_0 = P^1 g_8$ that $\phi ^*$ is also an isomorphism in degree $23+8=31$, hence in all degrees, and that $\phi$ is an equivalence of 3-complete $tm\!f$-modules. $\square$

Proposition 4.3

At $p=3$ there is an isomorphism $H^*(tm\!f) \cong A \otimes _{A(2)} M$ of A-modules, where

$$\begin{aligned} M = \frac{A(2)/\!/E(2)\{g_0, g_8\}}{(P^1 g_0, P^3 g_0 = P^1 g_8)} \end{aligned}$$

is a finite A(2)-module of dimension 18 satisfying ${{\,\mathrm{Hom}\,}}_{{\mathbb {F}}_3}(M, {\mathbb {F}}_3) \cong \Sigma ^{-64} M$.

Proof

Let all spectra be implicitly completed at $p=3$. Let $\Psi = S \cup _\nu e^4 \cup _\nu e^8$, where $\nu = \alpha _1$ is detected by $P^1$. According to [40,  Thm. 4.16], there is an equivalence $tm\!f\wedge \Psi \simeq tm\!f_0(2)$ of $tm\!f$-module spectra, where $\pi _*(tm\!f_0(2)) = {\mathbb {Z}}_p[a_2, a_4]$ with $|a_2| = 4$ and $|a_4| = 8$. We take as known that $H^*(tm\!f_0(2)) \cong A/\!/E(2)\{g_0, g_8\}$, where $|g_0| = 0$ and $|g_8| = 8$. The homotopy cofibre sequences

$$\begin{aligned} S&\longrightarrow \Psi \longrightarrow \Sigma ^4 C\nu \\ \Sigma ^4 C\nu&\longrightarrow \Sigma ^4 \Psi \longrightarrow S^{12} \end{aligned}$$

induce homotopy cofibre sequences

$$\begin{aligned} tm\!f&\longrightarrow tm\!f_0(2) \longrightarrow \Sigma ^4 tm\!f\wedge C\nu \\ \Sigma ^4 tm\!f\wedge C\nu&\longrightarrow \Sigma ^4 tm\!f_0(2) \longrightarrow \Sigma ^{12} tm\!f\end{aligned}$$

of $tm\!f$-modules. Passing to cohomology we get two short exact sequences, which we splice together to an exact complex

$$\begin{aligned} 0 \rightarrow \Sigma ^{12} H^*(tm\!f) \rightarrow \Sigma ^4 A/\!/E(2)\{g_0, g_8\} \overset{\partial }{\longrightarrow }A/\!/E(2)\{g_0, g_8\} \rightarrow H^*(tm\!f) \rightarrow 0 \end{aligned}$$

of A-modules. Here $\partial (\Sigma ^4 g_0) \in {\mathbb {F}}_3\{P^1 g_0\}$ and $\partial (\Sigma ^4 g_8) \in {\mathbb {F}}_3\{P^3 g_0, P^1 g_8\}$. Hence this complex is induced up from an exact complex

$$\begin{aligned} 0 \rightarrow \Sigma ^{12} M \rightarrow \Sigma ^4 A(2)/\!/E(2)\{g_0, g_8\} \overset{\partial }{\longrightarrow }A(2)/\!/E(2)\{g_0, g_8\} \rightarrow M \rightarrow 0 \end{aligned}$$

of A(2)-modules, where the rank of $\partial$ is twice the dimension of M. The dimension of $A(2)/\!/E(2)$ is 27, so the dimension of M is 18. By exactness, $\partial (\Sigma ^4 g_0)$ and $\partial (\Sigma ^4 g_8)$ are nonzero. From [18,  Cor. 6.7] it follows that we can choose the signs of the generators so that $\partial (\Sigma ^4 g_0) = P^1 g_0$ and $\partial (\Sigma ^4 g_8) = P^3 g_0 - P^1 g_8$. This gives the stated presentation of M.

Applying $D = {{\,\mathrm{Hom}\,}}_{{\mathbb {F}}_3}(-, {\mathbb {F}}_3)$ we find that $D(A(2)/\!/E(2)) \cong \Sigma ^{-64} A(2)/\!/E(2)$, and can calculate that $D\partial$ has the same form as $\partial$, so that the dual of the exact A(2)-module complex above presents $D(\Sigma ^{12} M) = \Sigma ^{-12} DM$ as $\Sigma ^{-76} M$, which implies that M is concentrated in degrees $0 \le * \le 64$, and is self-dual. $\square$

4.2 Local cohomology theorems by Mahowald–Rezk duality

Notation 4.4

The graded ring structure of $\pi _*(ko)$ is well known [11]. We use the notation

$$\begin{aligned} \pi _*(ko) = {\mathbb {Z}}_p[\eta , A, B]/(2 \eta , \eta ^3, \eta A, A^2 = 4 B) \end{aligned}$$

where $|\eta | = 1$, $|A| = 4$ and $|B| = 8$, cf. [15,  Ex. 2.30]. If p is odd this simplifies to $\pi _*(ko) = {\mathbb {Z}}_p[A]$. We call B the Bott element.

Notation 4.5

The graded ring structure of $\pi _*(tm\!f)$ is also known [3, 32, 20,  Ch. 13], [15], apart from a couple of finer points.

For $p=2$, using the naming scheme of [15,  Ch. 9], the graded commutative ${\mathbb {Z}}_2$-algebra $\pi _*(tm\!f)$ is generated by forty classes $x_k$, where

$$\begin{aligned} x \in \{\eta , \nu , \epsilon , \kappa , {\bar{\kappa }}, B, C, D, M\} \end{aligned}$$

and $0 \le k \le 7$. The indices k that occur are shown in Table 1. We abbreviate $x_0$ to x, and note that $|x_k| = |x| + 24 k$ is positive in each case. We call $B \in \pi _8(tm\!f)$ and $M \in \pi _{192}(tm\!f)$ the Bott element and the Mahowald element, respectively. See Fig. 4 for the mod 2 Adams $E_\infty$-term for $tm\!f$ in the range $0 \le t-s \le 192$.

In [20,  Ch. 13] these homotopy elements are instead labelled by their detecting classes in the elliptic (or Adams–Novikov) spectral sequence $E_\infty$-term. The ‘$E_\infty$’ column lists these classes, which, however, only characterise the $x_k$ modulo higher filtrations. For example, the cosets detected by $\nu \Delta ^k$ for k even and $2 \nu \Delta ^k$ for $k \in \{1,5\}$ have indeterminacy ${\mathbb {Z}}/2$ generated by an $\eta$- or $\eta _1$-multiple, the cosets detected by $c_4 \Delta ^k$ have indeterminacy ${\mathbb {Z}}/2\{\epsilon _k\}$ for $k \in \{0,1,4,5\}$, the coset detected by $2 c_6 \Delta ^2$ has indeterminacy ${\mathbb {Z}}/4\{{\bar{\kappa }}^3\}$, and the coset detected by $2 c_6 \Delta ^6$ has indeterminacy ${\mathbb {Z}}/2\{\eta \nu _6 \epsilon \}$. The edge homomorphism $\pi _*(tm\!f) \rightarrow {\mathbb {Z}}[c_4, c_6, \Delta ]/(c_4^3-c_6^2=1728\Delta )$ sends multiples of $c_4$, $c_6$ and $\Delta$ to the corresponding integral modular forms. The notation $q = \epsilon \Delta$ is also in common usage.

For $p=3$, using the naming scheme of [15,  Ch. 13], the graded commutative ${\mathbb {Z}}_3$-algebra $\pi _*(tm\!f)$ is generated by twelve classes $x_k$, where

$$\begin{aligned} x \in \{\nu , \beta , B, C, D, H\} \end{aligned}$$

and $0 \le k \le 2$. The values of k that occur are shown in Table 2. We again abbreviate $x_0$ to x, and note that $|x_k| = |x| + 24 k$ is positive in each case. We call $B \in \pi _8(tm\!f)$ and $H \in \pi _{72}(tm\!f)$ the Bott element and the Hopkins–Miller element, respectively. In [20,  Ch. 13] these homotopy elements are labelled by their detecting classes in the elliptic spectral sequence, as listed in the ‘$E_\infty$’ column. See Fig. 12 for the mod 3 ($tm\!f$-module) Adams $E_\infty$-term for $tm\!f$ in the range $0 \le t-s \le 72$.

To avoid repetitive case distinctions we will sometimes write ${\mathbb {Z}}_p[B, M]$, (p, B, M) or (B, M), both for $p=2$ and for $p=3$, in spite of the fact that the notations just introduced for $p=3$ would be ${\mathbb {Z}}_3[B, H]$, (3, B, H) or (B, H). In effect, the element ‘M’ should be read as ‘H’ for $p=3$.

Table 1 Algebra generators $x_k$ for $\pi _*(tm\!f)$ at $p=2$

Table 2 Algebra generators $x_k$ for $\pi _*(tm\!f)$ at $p=3$

Definition 4.6

For any p-complete connective S-algebra R with $\pi _0(R) = {\mathbb {Z}}_p$ let

$$\begin{aligned} {\mathfrak {n}}_0 = \ker (\pi _*(R) \rightarrow {\mathbb {Z}}_p) \end{aligned}$$

denote the ideal in $\pi _*(R)$ given by the classes in positive degrees, and let

$$\begin{aligned} {\mathfrak {n}}_p = \ker (\pi _*(R) \rightarrow {\mathbb {F}}_p) \end{aligned}$$

denote the maximal ideal generated by ${\mathfrak {n}}_0$ and p.

We shall review the precise structure of $\pi _*(tm\!f)$ as a ${\mathbb {Z}}_p[B, M]$-module in Sect. 8, but for now we will only need the following, more qualitative, consequences of that structure. Their analogues for ko are straightforward.

Lemma 4.7

The following hold for $p=2$ and for $p=3$.

• (1) The graded group $\pi _*(tm\!f)$ is finitely generated as a ${\mathbb {Z}}_p[B, M]$-module.

• (2) The radical of $(B, M) \subset \pi _*(tm\!f)$ is ${\mathfrak {n}}_0$, and the radical of (p, B, M) is ${\mathfrak {n}}_p$.

• (3) The graded group $\pi _*(tm\!f/(p, B, M))$ is finite.

• (4) Each $\pi _t(\Gamma _{(B, M)} tm\!f)$ is a finitely generated ${\mathbb {Z}}_p$-module.

Proof

We use the notation and results of Sects. 8.1 and 8.3, so the careful reader will have to refer forward to those results now, or come back to this proof later. In particular, $N = tm\!f/M$, with $N_* \cong \pi _*(N)$, by Theorems 8.2 and 8.14.

• (1) Since $\pi _*(tm\!f) \cong N_* \otimes {\mathbb {Z}}[M]$, it suffices to check that $N_*$ is finitely generated as a ${\mathbb {Z}}_p[B]$-module, which is clear from the explicit expressions given in Theorems 8.4 and 8.15.

• (2) Because $\pi _*(tm\!f)/(B, M)$ is finitely generated over ${\mathbb {Z}}_p$, each positive degree class is nilpotent, and therefore each class in ${\mathfrak {n}}_0$ lies in the radical of (B, M). The other conclusions follow.

• (3) Since $\Gamma _B N_*$ is finite, each ko[k] is B-torsion free, and each ko[k]/B is a finitely generated ${\mathbb {Z}}_p$-module, it follows that $\pi _*(N/B)$ is a finitely generated ${\mathbb {Z}}_p$-module. Hence $\pi _*(N/(p, B)) \cong \pi _*(tm\!f/(p, B, M))$ is finite.

• (4) Likewise, since $\Gamma _B N_*$ is finite and each $ko[k]/B^\infty$ is bounded above and finitely generated over ${\mathbb {Z}}_p$ in each degree, it follows that $\pi _*(\Gamma _B N)$ is bounded above and finitely generated over ${\mathbb {Z}}_p$ in each degree. Hence $\pi _*(\Gamma _{(B, M)} tm\!f) \cong \pi _*(\Gamma _B N) \otimes {\mathbb {Z}}[M]/M^\infty$ is also bounded above and finitely generated over ${\mathbb {Z}}_p$ in each degree.

$\square$

Theorem 4.8

There are equivalences of ko-modules

$$\begin{aligned} \Gamma _{(p,B)} ko = \Gamma _{{\mathfrak {n}}_p} ko \simeq C^f_1 ko \simeq \Sigma ^{-6} I(ko) \end{aligned}$$

(at all primes p), and equivalences of $tm\!f$-modules

$$\begin{aligned} \Gamma _{(p,B,M)} tm\!f= \Gamma _{{\mathfrak {n}}_p} tm\!f\simeq C^f_2 tm\!f\simeq \Sigma ^{-23} I(tm\!f) \end{aligned}$$

(at $p=2$ and at $p=3)$. The underlying $S_p$-modules are Artinian and bounded above.

Proof

For any $S_p$-module M the natural homomorphism

$$\begin{aligned} \rho : \pi _t(M) \longrightarrow {{\,\mathrm{Hom}\,}}_{{\mathbb {Z}}_p}({{\,\mathrm{Hom}\,}}_{{\mathbb {Z}}_p}(\pi _t(M), {\mathbb {Q}}_p/{\mathbb {Z}}_p), {\mathbb {Q}}_p/{\mathbb {Z}}_p) \end{aligned}$$

is injective. Hence, if $\pi _{-t}(IM) = {{\,\mathrm{Hom}\,}}_{{\mathbb {Z}}_p}(\pi _t(M), {\mathbb {Q}}_p/{\mathbb {Z}}_p)$ is a Noetherian (= finitely generated) ${\mathbb {Z}}_p$-module, then $\rho$ exhibits $\pi _t(M)$ as a submodule of an Artinian ${\mathbb {Z}}_p$-module, which must itself be Artinian. This applies with $M = C^f_1 ko$, since we know from Proposition 4.2 that $IM \simeq \Sigma ^6 ko$, and ko is a Noetherian $S_p$-module. It also applies with $M = C^f_2 tm\!f$, since $IM \simeq \Sigma ^{23} tm\!f$ and $tm\!f$ is a Noetherian $S_p$-module by Lemma 4.7(1). Hence $\rho : M \rightarrow I(IM)$ is in fact an equivalence in these cases, so that

$$\begin{aligned} C^f_1 ko&\simeq I(I C^f_1 ko) \simeq I(\Sigma ^6 ko) \simeq \Sigma ^{-6} I(ko) \\ C^f_2 tm\!f&\simeq I(I C^f_2 tm\!f) \simeq I(\Sigma ^{23} tm\!f) \simeq \Sigma ^{-23} I(tm\!f). \end{aligned}$$

We use Lemma 2.5 with $n=1$, $R = M = ko$ and $J = (p, B)$ to see that

$$\begin{aligned} C^f_1 ko \simeq \Gamma _{(p, B)} ko. \end{aligned}$$

(4.1)

The finite spectra S/I with $I = (p^{a_0}, v_1^{a_1})$, and their Spanier–Whitehead duals, have type 2, so $\pi _* F(S/I, ko)$ is finite, since ko has fp-type 1. Hence both p and B act nilpotently on this graded ${\mathbb {Z}}_p$-module. This confirms the first condition. For the second condition, note that $R/J = ko/(p, B)$ and $F(S/p^{a_0}, R/J)$ have finite graded homotopy groups, hence $p^{a_0}$ acts nilpotently on the former and $v_1^{a_1}$ acts nilpotently on the latter. The radical in $\pi _*(ko)$ of $J = (p, B)$ equals $\sqrt{J} = {\mathfrak {n}}_p$, and as reviewed in Sect. 2.4 this implies the equivalence

$$\begin{aligned} \Gamma _{(p, B)} ko \simeq \Gamma _{{\mathfrak {n}}_p} ko. \end{aligned}$$

Similarly, we use Lemma 2.5 with $n=2$, $R = M = tm\!f$ and $J = (p, B, M)$ to see that

$$\begin{aligned} C^f_2 tm\!f\simeq \Gamma _{(p, B, M)} tm\!f. \end{aligned}$$

(4.2)

The finite spectra S/I with $I = (p^{a_0}, v_1^{a_1}, v_2^{a_2})$, and their Spanier–Whitehead duals, have type 3, so $\pi _* F(S/I, tm\!f)$ is finite, since $tm\!f$ has fp-type 2. Hence p, B and M act nilpotently on this graded ${\mathbb {Z}}_p$-module. Next, note that $R/J = tm\!f/(p, B, M)$, $F(S/p^{a_0}, R/J)$ and $F(S/(p^{a_0}, v_1^{a_1}), R/J)$ have finite homotopy, by Lemma 4.7(3), so that $p^{a_0}$ acts nilpotently on the first, $v_1^{a_1}$ acts nilpotently on the second, and $v_2^{a_2}$ acts nilpotently on the third. The radical in $\pi _*(tm\!f)$ of $J = (p, B, M)$ equals $\sqrt{J} = {\mathfrak {n}}_p$, by Lemma 4.7(2), which implies the equivalence

$$\begin{aligned} \Gamma _{(p, B, M)} tm\!f\simeq \Gamma _{{\mathfrak {n}}_p} tm\!f. \end{aligned}$$

$\square$

Remark 4.9

As an alternative to Lemma 2.5, we could use Lemmas 2.2 and 2.4 to establish (4.1). For $p=2$ the Adams complex $A = S/(2, v_1^4)$ from [1,  §12] has type 2 and satisfies $ko \wedge S/(2, v_1^4) \simeq ko/(2, B)$. Hence $C^f_1 ko \simeq {{\,\mathrm{Cell}\,}}_{ko \wedge A} ko \simeq {{\,\mathrm{Cell}\,}}_{ko/(2, B)} ko \simeq \Gamma _{(2, B)} ko$. For p odd the complex $S/(p, v_1)$ has type 2 and $ko \wedge S/(p, v_1) \simeq ko/(p, A^m)$ with $m = (p-1)/2$. Hence $C^f_1 ko \simeq \Gamma _{(p, A^m)} ko \simeq \Gamma _{(p, B)} ko$, since $(p, A^m)$ and (p, B) have the same radical.

Likewise, we could use the existence [5] of a 2-local finite complex $A = S/(2, v_1^4, v_2^{32})$ of type 3 satisfying $tm\!f\wedge A \simeq tm\!f/(2, B, M)$ to deduce that $C^f_2 tm\!f\simeq {{\,\mathrm{Cell}\,}}_{tm\!f\wedge A} \simeq {{\,\mathrm{Cell}\,}}_{tm\!f/(2, B, M)} tm\!f\simeq \Gamma _{(2, B, M)} tm\!f$. Similar remarks apply at $p=3$, using the type 3 complex $A = S/(3, v_1, v_2^9)$ constructed in [7]. In these cases the argument for (4.2) using Lemma 2.5 is dramatically simpler, as is to be expected, since asking for $R \wedge A$ and R/J to be equivalent is a much more restrictive condition than asking that they build one another.

Theorem 4.10

There are equivalences of ko-modules

$$\begin{aligned} \Gamma _B ko = \Gamma _{{\mathfrak {n}}_0} ko \simeq \Sigma ^{-5} I_{{\mathbb {Z}}_p}(ko) \end{aligned}$$

(at all primes p), and equivalences of $tm\!f$-modules

$$\begin{aligned} \Gamma _{(B,M)} tm\!f= \Gamma _{{\mathfrak {n}}_0} tm\!f\simeq \Sigma ^{-22} I_{{\mathbb {Z}}_p}(tm\!f) \end{aligned}$$

(at $p=2$ and at $p=3)$. The underlying $S_p$-modules are Noetherian and bounded above.

Proof

Proposition 4.2, equivalence (4.1), and Lemma 3.1 applied to the ko-module $M = \Gamma _B ko$, give equivalences

$$\begin{aligned} \Sigma ^6 ko \simeq IC^f_1 ko \simeq I(\Gamma _{(p,B)} ko) \simeq \Sigma (I_{{\mathbb {Z}}_p} \Gamma _B ko)^\wedge _p. \end{aligned}$$

Here $\Gamma _B ko$ is a Noetherian $S_p$-module, which implies that $I_{{\mathbb {Z}}_p} \Gamma _B ko$ is also Noetherian and (in particular) p-complete. Hence $\Sigma ^5 ko \simeq I_{{\mathbb {Z}}_p} \Gamma _B ko$, which implies that

$$\begin{aligned} \Gamma _B ko \simeq I_{{\mathbb {Z}}_p}(I_{{\mathbb {Z}}_p} \Gamma _B ko) \simeq I_{{\mathbb {Z}}_p}(\Sigma ^5 ko) \simeq \Sigma ^{-5} I_{{\mathbb {Z}}_p}(ko). \end{aligned}$$

Furthermore, $\Gamma _B ko = \Gamma _{{\mathfrak {n}}_0} ko$, since the radical of (B) in $\pi _*(ko)$ equals ${\mathfrak {n}}_0$.

Similarly, Proposition 4.2, equivalence (4.2), and Lemma 3.1 applied to the $tm\!f$-module $M = \Gamma _{(B, M)} tm\!f$, give equivalences

$$\begin{aligned} \Sigma ^{23} tm\!f\simeq IC^f_2 tm\!f\simeq I(\Gamma _{(p,B,M)} tm\!f) \simeq \Sigma (I_{{\mathbb {Z}}_p} \Gamma _{(B, M)} tm\!f)^\wedge _p. \end{aligned}$$

Part (4) of Lemma 4.7 asserts that $\Gamma _{(B, M)} tm\!f$ is a Noetherian $S_p$-module, which implies that $I_{{\mathbb {Z}}_p} \Gamma _{(B, M)} tm\!f$ is also Noetherian and p-complete. Hence $\Sigma ^{22} tm\!f\simeq I_{{\mathbb {Z}}_p} \Gamma _{(B, M)} tm\!f$, which implies that

$$\begin{aligned} \Gamma _{(B, M)} tm\!f\simeq I_{{\mathbb {Z}}_p}(I_{{\mathbb {Z}}_p} \Gamma _{(B, M)} tm\!f) \simeq I_{{\mathbb {Z}}_p}(\Sigma ^{22} tm\!f) \simeq \Sigma ^{-22} I_{{\mathbb {Z}}_p}(tm\!f). \end{aligned}$$

Finally, $\Gamma _{(B, M)} tm\!f= \Gamma _{{\mathfrak {n}}_0} tm\!f$, since the radical of (B, M) in $\pi _*(tm\!f)$ equals ${\mathfrak {n}}_0$ by Lemma 4.7(2). $\square$

5 Gorenstein duality

5.1 Gorenstein maps of S-algebras

The original version [21,  Def. 8.1] of the following definition was slightly more restrictive, but by [21,  Prop. 8.4] there is no difference when k is proxy-small as an R-module.

Definition 5.1

Let $R \rightarrow k$ be a map of S-algebras. We say that $R \rightarrow k$ is Gorenstein of shift a if there is an equivalence of left k-modules

$$\begin{aligned} F_R(k, R) \simeq \Sigma ^a k. \end{aligned}$$

Our next aim is to prove Proposition 5.3. We write ${\mathbb {F}}_p$ and ${\mathbb {Z}}_p$ for the mod p and p-adic integral Eilenberg–MacLane spectra, respectively, with their unique (commutative) $S_p$-algebra structures.

Suppose that R is an $S_p$-algebra and that $k = {\mathbb {F}}_p$. There are then equivalences

$$\begin{aligned} k \simeq Ik = F_{S_p}(k, I) \cong F_R(k, F_{S_p}(R, I)) = F_R(k, IR) \end{aligned}$$

of left k-modules, where Ik and IR are the Brown–Comenetz duals of k and R, and where k acts from the right on the domains of the two mapping spectra. Hence, if $R \rightarrow k$ is Gorenstein of shift a, then there is a k-module equivalence

$$\begin{aligned} F_R(k, R) \simeq \Sigma ^a k \simeq F_R(k, \Sigma ^a IR). \end{aligned}$$

Recall the notation ${\mathcal {E}}= F_R(k, k)$ from Sect. 2.3. Restriction along $R \rightarrow k$ defines an S-algebra map $k^{op} \rightarrow {\mathcal {E}}$, and the left ${\mathcal {E}}$-action on k induces right ${\mathcal {E}}$-actions on $F_R(k, R)$ and $F_R(k, \Sigma ^a IR)$. If R is connective with $\pi _0(R) = {\mathbb {Z}}_p$, then ${\mathcal {E}}$ is coconnective with $\pi _0({\mathcal {E}}) \cong k^{op}$ a field. According to [21,  Prop. 3.9] the k-module equivalence above then extends to an ${\mathcal {E}}$-module equivalence, so that

$$\begin{aligned} F_R(k, R) \wedge _{{\mathcal {E}}} k \simeq F_R(k, \Sigma ^a IR) \wedge _{{\mathcal {E}}} k. \end{aligned}$$

Moreover, if k is proxy-small, so that k-cellularisation is effectively constructible by [21,  Thm. 4.10], then we can rewrite this as an equivalence

$$\begin{aligned} {{\,\mathrm{Cell}\,}}_k R \simeq {{\,\mathrm{Cell}\,}}_k(\Sigma ^a IR). \end{aligned}$$

Finally, if $\pi _*(IR)$ is p-power torsion, then k builds IR as the homotopy colimit of a refined Whitehead tower in R-modules, so that ${{\,\mathrm{Cell}\,}}_k(\Sigma ^a IR) \simeq \Sigma ^a IR$. Hence these hypotheses ensure that $R \rightarrow k$ has Gorenstein duality of shift a, in the following sense.

Definition 5.2

A map $R \rightarrow k = {\mathbb {F}}_p$ of $S_p$-algebras has Gorenstein duality of shift a if there is an equivalence

$$\begin{aligned} {{\,\mathrm{Cell}\,}}_k R \simeq \Sigma ^a IR. \end{aligned}$$

Similarly, a map $R \rightarrow k = {\mathbb {Z}}_p$ of $S_p$-algebras has Gorenstein duality of shift a if there is an equivalence

$$\begin{aligned} {{\,\mathrm{Cell}\,}}_k R \simeq \Sigma ^a I_{{\mathbb {Z}}_p} R. \end{aligned}$$

Proposition 5.3

Let $k = {\mathbb {F}}_p$ or ${\mathbb {Z}}_p$, let $R \rightarrow k$ be a map of connective $S_p$-algebras with $\pi _0(R) = {\mathbb {Z}}_p$, and let ${\mathcal {E}}= F_R(k, k)$. Suppose that

• (1) $R \rightarrow k$ is Gorenstein of shift a, and

• (2) k is proxy-small as an R-module.

For $k = {\mathbb {F}}_p$ also assume that

• (3) ${{\,\mathrm{Hom}\,}}_{{\mathbb {Z}}_p}(\pi _*(R), {\mathbb {Q}}_p/{\mathbb {Z}}_p)$ is p-power torsion.

Then $R \rightarrow k$ has Gorenstein duality of shift a.

Proof

The case $k = {\mathbb {F}}_p$ was discussed above. When $k = {\mathbb {Z}}_p$, we have an equivalence

$$\begin{aligned} k \simeq F_{S_p}(k, I_{{\mathbb {Z}}_p}) \cong F_R(k, I_{{\mathbb {Z}}_p} R) \end{aligned}$$

of k-modules, where $I_{{\mathbb {Z}}_p} R = F_{S_p}(R, I_{{\mathbb {Z}}_p})$ is the Anderson dual of R. Hence, if $R \rightarrow k$ is Gorenstein of shift a then there is a k-module equivalence

$$\begin{aligned} F_R(k, R) \simeq \Sigma ^a k \simeq F_R(k, \Sigma ^a I_{{\mathbb {Z}}_p} R). \end{aligned}$$

Since R is connective with $\pi _0(R) = {\mathbb {Z}}_p$, it follows that ${\mathcal {E}}$ is coconnective with $\pi _0({\mathcal {E}}) \cong k^{op}$ and $\pi _{-1}({\mathcal {E}}) = 0$. We prove in Proposition 5.4 below that this implies that the k-module equivalence above extends to an ${\mathcal {E}}$-module equivalence, so that

$$\begin{aligned} F_R(k, R) \wedge _{{\mathcal {E}}} k \simeq F_R(k, \Sigma ^a I_{{\mathbb {Z}}_p} R) \wedge _{{\mathcal {E}}} k. \end{aligned}$$

If k is proxy-small, then we can rewrite this as

$$\begin{aligned} {{\,\mathrm{Cell}\,}}_k R \simeq {{\,\mathrm{Cell}\,}}_k(\Sigma ^a I_{{\mathbb {Z}}_p} R). \end{aligned}$$

Since $\pi _*(I_{{\mathbb {Z}}_p} R)$ is a bounded above graded ${\mathbb {Z}}_p$-module, it follows that $I_{{\mathbb {Z}}_p} R$ is built from k, so that ${{\,\mathrm{Cell}\,}}_k R \simeq \Sigma ^a I_{{\mathbb {Z}}_p} R$. $\square$

5.2 Uniqueness of ${\mathcal {E}}$-module structures

We continue to write ${\mathcal {E}}= F_R(k,k)$ for the R-module endomorphism ring spectrum of k.

Proposition 5.4

Let $k = {\mathbb {F}}_p$ or ${\mathbb {Z}}_p$, and let $k^{op} \rightarrow {\mathcal {E}}$ be a map of coconnective S-algebras inducing an isomorphism on $\pi _0$. For $k = {\mathbb {Z}}_p$ also assume that $\pi _{-1}({\mathcal {E}}) = 0$. Then any two right ${\mathcal {E}}$-module structures on k are equivalent.

Proof

When $k = {\mathbb {F}}_p$, this is a special case of [21,  Prop. 3.9]. When $k = {\mathbb {Z}}_p$, we refine the proof of that proposition. Let $k_1$ and $k_2$ be right ${\mathcal {E}}$-modules whose restricted k-module structures are given by the usual left action on k. Let $M = {\mathcal {E}}$ as an ${\mathcal {E}}$-module, with the usual right action, and choose ${\mathcal {E}}$-module maps $f_1 : M \rightarrow k_1$ and $f_2 : M \rightarrow k_2$ inducing isomorphisms on $\pi _0$. We shall extend M to a cellular ${\mathcal {E}}$-module N such that $f_1$ extends to an ${\mathcal {E}}$-module equivalence $g_1 : N \rightarrow k_1$, and such that $f_2$ extends to an ${\mathcal {E}}$-module map $g_2 : N \rightarrow k_2$. It will then follow that $g_2$ is also an equivalence, and $k_1$ and $k_2$ are equivalent as ${\mathcal {E}}$-modules.

Note that $\pi _{-1}(M) = 0$. As a first approximation to N we build a cellular ${\mathcal {E}}$-module $M'$ by attaching ${\mathcal {E}}$-cells of dimension $\le 0$ to M, so that $\pi _{-1}(M') = 0$ and $\pi _t(M) \rightarrow \pi _t(M')$ is trivial for each $t\le -2$. More precisely, for each $t\le -2$ choose an ${\mathcal {E}}$-module map

$$\begin{aligned} \bigvee _\alpha \Sigma ^t {\mathcal {E}}\overset{\phi _t}{\longrightarrow }\bigvee _\beta \Sigma ^t {\mathcal {E}}\end{aligned}$$

such that

$$\begin{aligned} 0 \rightarrow \bigoplus _\alpha {\mathbb {Z}}_p \overset{\pi _t(\phi _t)}{\longrightarrow }\bigoplus _\beta {\mathbb {Z}}_p \longrightarrow \pi _t(M) \rightarrow 0 \end{aligned}$$

is a free ${\mathbb {Z}}_p$-resolution of $\pi _t(M)$. There is then a map $C\phi _t \rightarrow M$ from the homotopy cofibre of $\phi _t$, inducing an isomorphism on $\pi _t$. The composite $C\phi _t \rightarrow M \rightarrow k_1$ is null-homotopic, since $C\phi _t$ has cells in dimensions t and $t+1 \le -1$ only, while $k_1$ is connective. Let $M'$ be the mapping cone of the sum over t of the maps $C\phi _t \rightarrow M$, and let $f'_1 : M' \rightarrow k_1$ extend $f_1$. Then $M'$ has the stated properties.

Iterating the process infinitely often, and letting N be the (homotopy) colimit of the sequence $M \subset M' \subset \dots$, we calculate that $\pi _t(N) = 0$ for $t \ne 0$, while $g_1 : N \rightarrow k_1$ is a $\pi _0$-isomorphism, and therefore an equivalence.

We obtained N from M by attaching cells of dimensions $\le 0$, so the obstructions to extending $f_2 : M \rightarrow k_2$ lie in the negative homotopy groups of $k_2$, which are trivial. Hence an extension $g_2 : N \rightarrow k_2$ exists. It must be a $\pi _0$-isomorphism, since $f_1$, $f_2$ and $g_1$ have this property, and is therefore an equivalence, as claimed. $\square$

Remark 5.5

The hypothesis on $\pi _{-1}({\mathcal {E}})$ can in general not be omitted; see [21,  Rem. 3.11].

5.3 Gorenstein descent

Suppose that we are given maps $R \rightarrow T \rightarrow k$ of S-algebras, and that T is somehow easier to work with than R. A descent theorem for a property P gives hypotheses under which P for $T \rightarrow k$ implies P for $R \rightarrow k$. We first apply this idea in the case of the Gorenstein property.

Lemma 5.6

Let $T \rightarrow k$ be a map of S-algebras, and suppose that the homomorphism of coefficient rings $\pi _*(T) \rightarrow \pi _*(k)$ is (algebraically) Gorenstein in the sense that ${{\,\mathrm{Ext}\,}}_{\pi _*(T)}^{*,*}(\pi _*(k), \pi _*(T))$ is a free $\pi _*(k)$-module of rank 1, on a generator in bidegree (s, t). Then $T \rightarrow k$ is Gorenstein of shift $a = t-s$.

Grothendieck’s definition given in [29,  p. 63] is more restrictive, but this condition suffices for our purposes.

Proof

The conditionally convergent ${{\,\mathrm{Ext}\,}}$ spectral sequence

$$\begin{aligned} E_2^{s,t} = {{\,\mathrm{Ext}\,}}_{\pi _*(T)}^{s,t}(\pi _*(k), \pi _*(T)) \Longrightarrow _s \pi _{t-s} F_T(k, T) \end{aligned}$$

of [22,  Thm. IV.4.1] is a $\pi _*(k)$-module spectral sequence that collapses at the $E_2$-term, hence is strongly convergent. It follows that $\pi _* F_T(k, T) \cong \Sigma ^a \pi _*(k)$ as $\pi _*(k)$-modules, so that $F_T(k, T) \simeq \Sigma ^a k$ as k-modules. $\square$

Example 5.7

${\mathbb {Z}}_p \rightarrow {\mathbb {F}}_p$ is Gorenstein of algebraic shift $(s,t) = (1,0)$ and of topological shift $a = -1$.

Lemma 5.8

Let $R \rightarrow T \rightarrow k$ be maps of S-algebras, and suppose that $R \rightarrow T$ is Gorenstein of shift b. Then $T \rightarrow k$ is Gorenstein of shift a if and only if $R \rightarrow k$ is Gorenstein of shift $a+b$.

Proof

By hypothesis, $F_R(T, R) \simeq \Sigma ^b T$ as left T-modules. It follows that

$$\begin{aligned} F_R(k, R) \cong F_T(k, F_R(T, R)) \simeq F_T(k, \Sigma ^b T) \cong \Sigma ^b F_T(k, T) \end{aligned}$$

as left k-modules. Hence $F_T(k, T) \simeq \Sigma ^a k$ if and only if $F_R(k, R) \simeq \Sigma ^{a+b} k$. $\square$

Example 5.9

If $\pi _*(T) \cong {\mathbb {Z}}_p[y_1, \dots , y_d]$ is polynomial on finitely many generators, and $k = {\mathbb {Z}}_p$, then the ring homomorphism $\pi _*(T) \rightarrow {\mathbb {Z}}_p$ is Gorenstein of shift $(s, t) = (d, - \sum _{i=1}^d |y_i|)$. Hence the S-algebra map $T \rightarrow {\mathbb {Z}}_p$ is Gorenstein of shift

$$\begin{aligned} a = - d - \sum _{i=1}^d |y_i| = - \sum _{i=1}^d (|y_i|+1). \end{aligned}$$

Moreover, $\pi _*(T) \rightarrow {\mathbb {F}}_p$ is Gorenstein of shift $(d + 1, - \sum _{i=1}^d |y_i|)$ and $T \rightarrow {\mathbb {F}}_p$ is Gorenstein of shift $- d - 1 - \sum _{i=1}^d |y_i|$.

Proposition 5.10

The S-algebra maps

$$\begin{aligned} ko \longrightarrow ku \longrightarrow {\mathbb {Z}}_p \longrightarrow {\mathbb {F}}_p \end{aligned}$$

are Gorenstein of shift $-2$, $-3$ and $-1$, respectively. Hence $ko \rightarrow {\mathbb {Z}}_p$ is Gorenstein of shift $-5$ and $ko \rightarrow {\mathbb {F}}_p$ is Gorenstein of shift $-6$.

At $p=2$ the S-algebra maps

$$\begin{aligned} tm\!f\longrightarrow tm\!f_1(3) \longrightarrow {\mathbb {Z}}_2 \longrightarrow {\mathbb {F}}_2 \end{aligned}$$

are Gorenstein of shift $-12$, $-10$ and $-1$, respectively. At $p=3$ the S-algebra maps

$$\begin{aligned} tm\!f\longrightarrow tm\!f_0(2) \longrightarrow {\mathbb {Z}}_3 \longrightarrow {\mathbb {F}}_3 \end{aligned}$$

are Gorenstein of shift $-8$, $-14$ and $-1$, respectively. At $p\ge 5$ the S-algebra maps

$$\begin{aligned} tm\!f\longrightarrow {\mathbb {Z}}_p \longrightarrow {\mathbb {F}}_p \end{aligned}$$

are Gorenstein of shift $-22$ and $-1$, respectively. Hence $tm\!f\rightarrow {\mathbb {Z}}_p$ is Gorenstein of shift $-22$, and $tm\!f\rightarrow {\mathbb {F}}_p$ is Gorenstein of shift $-23$, uniformly at all primes.

Proof

The homotopy rings

$$\begin{aligned} \begin{aligned} \pi _*(ku)&\cong {\mathbb {Z}}_p[u] \\ \pi _*(tm\!f_1(3))&\cong {\mathbb {Z}}_2[a_1, a_3] \\ \pi _*(tm\!f_0(2))&\cong {\mathbb {Z}}_3[a_2, a_4] \\ \pi _*(tm\!f)&\cong {\mathbb {Z}}_p[c_4, c_6], \end{aligned} \end{aligned}$$

(5.1)

where $p\ge 5$ in the last case, are all polynomial, with $|u| = 2$, $|a_i| = 2i$ and $|c_i| = 2i$. See [11, 39,  Prop. 3.2], [4,  §1.3], [19,  Prop. 6.1] or [15,  §9.3, Thm. 13.4]. This accounts for the Gorenstein shifts by $-3$, $-10$, $-14$ and $-22$, as in Example 5.9.

The shifts by $-1$ are covered by Example 5.7.

By Wood’s theorem [14,  Lem. 4.1.2], and its parallels [40,  Thm. 4.12, Thm. 4.15] for topological modular forms, there are equivalences

$$\begin{aligned} \begin{aligned} ko \wedge C\eta&\simeq ku \\ tm\!f\wedge \Phi&\simeq tm\!f_1(3) \\ tm\!f\wedge \Psi&\simeq tm\!f_0(2) \end{aligned} \end{aligned}$$

(5.2)

of ko- or $tm\!f$-modules, according to the case. Here $C\eta = S \cup _\eta e^2$ is a 2-cell, 2-dimensional Spanier–Whitehead self-dual spectrum, $\Phi$ is an 8-cell, 12-dimensional Spanier–Whitehead self-dual 2-local spectrum [15,  Lem. 1.42] with mod 2 cohomology $H^*(\Phi ) \cong A(2)/\!/E(2) \cong \Phi A(1)$ realising the double of A(1), and $\Psi = S \cup _\nu e^4 \cup _\nu e^8$ is a 3-cell 8-dimensional Spanier–Whitehead self-dual 3-local spectrum [15,  Def. 13.3] with mod 3 cohomology $H^*(\Psi ) \cong P(0) = \langle P^1\rangle$. The duality equivalences $D(C\eta ) \simeq \Sigma ^{-2} C\eta$, $D\Phi \simeq \Sigma ^{-12} \Phi$ and $D\Psi \simeq \Sigma ^{-8} \Psi$ account for the Gorenstein shifts by $-2$, $-12$ and $-8$, respectively. For example, in the case of $tm\!f$ at $p=2$ we have equivalences

$$\begin{aligned} F_{tm\!f}(tm\!f_1(3), tm\!f)&\simeq F_{tm\!f}(tm\!f\wedge \Phi , tm\!f) \simeq F(\Phi , tm\!f) \\&\simeq tm\!f\wedge D\Phi \simeq tm\!f\wedge \Sigma ^{-12} \Phi \simeq \Sigma ^{-12} tm\!f_1(3) \end{aligned}$$

of $tm\!f$-modules. $\square$

5.4 Small descent

We can use descent to verify that k is a (proxy-)small R-module in the cases relevant for Sects. 7 and 8.

Lemma 5.11

Let $R \rightarrow T \rightarrow k$ be maps of S-algebras, such that T is small as an R-module and k is small as a T-module. Then k is small as an R-module.

Proof

Since T finitely builds k as a T-module, this remains true as R-modules. Hence R finitely builds k. $\square$

Lemma 5.12

Let T be a commutative S-algebra with $\pi _*(T) \cong {\mathbb {Z}}_p[y_1, \dots , y_d]$. Then ${\mathbb {Z}}_p$ and ${\mathbb {F}}_p$ are small as T-modules.

Proof

$T/y_1 \wedge _T \dots \wedge _T T/y_d \simeq {\mathbb {Z}}_p$ and ${\mathbb {Z}}_p/p \simeq {\mathbb {F}}_p$ are finitely built from T. $\square$

Remark 5.13

More generally, if $\pi _*(k)$ is a perfect $\pi _*(T)$-module, meaning that it admits a finite length resolution by finitely generated projective $\pi _*(T)$-modules, then k is small as a T-module.

Corollary 5.14

${\mathbb {Z}}_p$ and ${\mathbb {F}}_p$ are small as ko-modules and as $tm\!f$-modules, at all primes p.

Proof

Lemma 5.12 applies to the commutative $S_p$-algebras ko for $p\ge 3$, ku for all p, $tm\!f_1(3)$ for $p=2$, $tm\!f_0(2)$ for $p=3$, and $tm\!f$ for $p\ge 5$. Lemma 5.11 then covers the cases of ko at $p=2$, and $tm\!f$ at $p \in \{2,3\}$, in view of (5.2). $\square$

5.5 Descent of algebraic cellularisation

Definition 5.15

Let R be a commutative S-algebra and k an R-module. We say that R has algebraic k-cellularisation by J if $J = (x_1, \dots , x_d) \subset \pi _*(R)$ is a finitely generated ideal with

$$\begin{aligned} {{\,\mathrm{Cell}\,}}_k M \simeq \Gamma _J M \end{aligned}$$

for all R-modules M.

This condition only depends on the radical $\sqrt{J}$ of J, and by Lemmas 2.1 and 2.4 it is equivalent to asking that the R-modules k and R/J mutually build one another.

Lemma 5.16

Let T be a commutative S-algebra with $\pi _*(T) \cong {\mathbb {Z}}_p[y_1, \dots , y_d]$. Then T has algebraic ${\mathbb {Z}}_p$-cellularisation by $(y_1, \dots , y_d)$, and algebraic ${\mathbb {F}}_p$-cellularisation by $(p, y_1, \dots , y_d)$.

Proof

Letting $J' = (y_1, \dots , y_d)$ or $J' = (p, y_1, \dots , y_d)$ we have $T/J' \simeq k = {\mathbb {Z}}_p$ or $T/J' \simeq k = {\mathbb {F}}_p$, according to the case. Hence ${{\,\mathrm{Cell}\,}}_k M \simeq {{\,\mathrm{Cell}\,}}_{T/J'} M \simeq \Gamma _{J'} M$. $\square$

Lemma 5.17

Let $\phi : R \rightarrow T$ be a map of commutative S-algebras, where R is connective with $\pi _0(R) = {\mathbb {Z}}_p$. Let $J = (x_1, \dots , x_d) \subset \pi _*(R)$ be such that $\pi _*(R/J)$ is bounded above, and suppose that T has algebraic ${\mathbb {Z}}_p$-cellularisation by

$$\begin{aligned} J' = (\phi (x_1), \dots , \phi (x_d)) \subset \pi _*(T). \end{aligned}$$

Then R has algebraic ${\mathbb {Z}}_p$-cellularisation by J.

Similarly, if $\pi _*(R/J)$ is p-power torsion and bounded above, and T has algebraic ${\mathbb {F}}_p$-cellularisation by $J'$, then R has algebraic ${\mathbb {F}}_p$-cellularisation by J.

Proof

In the case $k = {\mathbb {Z}}_p$, the R-module ${\mathbb {Z}}_p$ builds R/J since $\pi _*(R/J)$ is bounded above. Conversely, R builds T so R/J builds $T \wedge _R R/J = T/J'$. By hypothesis, $T/J'$ builds ${\mathbb {Z}}_p$ in T-modules, hence also in R-modules. Thus R/J builds ${\mathbb {Z}}_p$ in R-modules.

Similarly, for $k = {\mathbb {F}}_p$ the R-module ${\mathbb {F}}_p$ builds R/J since $\pi _*(R/J)$ is p-power torsion and bounded above. Conversely, R/J builds $T/J'$ as before. By hypothesis, $T/J'$ builds ${\mathbb {F}}_p$ in T-modules, hence also in R-modules. Thus R/J builds ${\mathbb {F}}_p$ in R-modules. $\square$

Recall from Notation 4.5 that $B \in \pi _8(tm\!f)$ (together with $B + \epsilon$) is detected by the modular form $c_4$, while we write M for $M \in \pi _{192}(tm\!f)$ detected by $\Delta ^8$ when $p=2$, and for $H \in \pi _{72}(tm\!f)$ detected by $\Delta ^3$ when $p=3$. For uniformity of notation, let us also write M for the class in $\pi _{24}(tm\!f)$ detected by $\Delta$ when $p\ge 5$.

Proposition 5.18

The commutative $S_p$-algebra ko has algebraic ${\mathbb {Z}}_p$-cellularisation by (B), and algebraic ${\mathbb {F}}_p$-cellularisation by (p, B), for all primes p.

$$\begin{aligned} {{\,\mathrm{Cell}\,}}_{{\mathbb {Z}}_p} ko&\simeq \Gamma _B ko \\ {{\,\mathrm{Cell}\,}}_{{\mathbb {F}}_p} ko&\simeq \Gamma _{(p, B)} ko \end{aligned}$$

Likewise, $tm\!f$ has algebraic ${\mathbb {Z}}_p$-cellularisation by (B, M), and algebraic ${\mathbb {F}}_p$-cellularisation by (p, B, M), for all primes p.

$$\begin{aligned} {{\,\mathrm{Cell}\,}}_{{\mathbb {Z}}_p} tm\!f&\simeq \Gamma _{(B, M)} tm\!f\\ {{\,\mathrm{Cell}\,}}_{{\mathbb {F}}_p} tm\!f&\simeq \Gamma _{(p, B, M)} tm\!f\end{aligned}$$

Proof

For ko, we apply Lemma 5.17 to the complexification map $\phi : ko \rightarrow ku$ with $J = (B)$, where $\phi (B) = u^4$. Then $\pi _*(ko/B) \cong {\mathbb {Z}}_p\{1, \eta , \eta ^2, A\}/(2 \eta , 2 \eta ^2)$ is finitely generated over ${\mathbb {Z}}_p$. Moreover, $J' = (u^4)$ has radical $(u) \subset \pi _*(ku)$. According to Lemma 5.16, ku has algebraic ${\mathbb {Z}}_p$-cellularisation by (u), hence it also has algebraic ${\mathbb {Z}}_p$-cellularisation by $J'$.

Similarly, with $J = (p, B)$ we see that $\pi _*(ko/(p, B))$ is finite and $J' = (p, u^4)$ has radical $(p, u) \subset \pi _*(ku)$, so ku has algebraic ${\mathbb {F}}_p$-cellularisation by (p, u) and by $J'$.

For $tm\!f$ at $p=2$ we apply Lemma 5.17 to the map $\phi : tm\!f\rightarrow tm\!f_1(3)$ with $J = (B, M)$, cf. (5.1), where

$$\begin{aligned} \phi (B)&= c_4&\qquad c_4&= a_1 (a_1^3 - 24 a_3) \\ \phi (M)&= \Delta ^8&\qquad \Delta&= a_3^3 (a_1^3 - 27 a_3), \end{aligned}$$

according to the formulas for $\Gamma _1(3)$-modular forms. See [15,  §9.3] and the more detailed references therein. It is clear from Theorem 8.4 that $\pi _*(tm\!f/(B, M)) \cong \pi _*(N/B)$ is finitely generated over ${\mathbb {Z}}_2$. Moreover, $J' = (c_4, \Delta ^8)$ has radical $(a_1, a_3) \subset \pi _*(tm\!f_1(3))$, so $tm\!f_1(3)$ has algebraic ${\mathbb {Z}}_2$-cellularisation by $(a_1, a_3)$ according to Lemma 5.16, hence also by $J'$.

Similarly, with $J = (2, B, M)$ we see that $\pi _*(tm\!f/(2, B, M)) \cong \pi _*(N/(2, B))$ is finite and $J' = (2, c_4, \Delta ^8)$ has radical $(2, a_1, a_3)$, so $tm\!f$ has algebraic ${\mathbb {F}}_2$-cellularisation by $(2, a_1, a_3)$ and by $J'$.

For $tm\!f$ at $p=3$ we apply Lemma 5.17 to the map $\phi : tm\!f\rightarrow tm\!f_0(2)$ with $J = (B, H)$, cf. (5.1), where

$$\begin{aligned} \phi (B)&= c_4&\qquad c_4&= 16 (a_2^2 - 3 a_4) \\ \phi (H)&= \Delta ^3&\qquad \Delta&= 16 a_4^2 (a_2^2 - 4 a_4) \end{aligned}$$

according to the formulas for $\Gamma _0(2)$-modular forms. See [15,  §13.1] and the more detailed references therein. It is clear from Theorem 8.15 that $\pi _*(tm\!f/(B, H)) \cong \pi _*(N/B)$ is finitely generated over ${\mathbb {Z}}_3$. Moreover, $J' = (c_4, \Delta ^3)$ has radical $(a_2, a_4) \subset \pi _*(tm\!f_0(2))$, so $tm\!f_0(2)$ has algebraic ${\mathbb {Z}}_3$-cellularisation by $(a_2, a_4)$ according to Lemma 5.16, hence also by $J'$.

Similarly, with $J = (3, B, H)$ we see that $\pi _*(tm\!f/(3, B, H)) \cong \pi _*(N/(3, B))$ is finite and $J' = (3, c_4, \Delta ^3)$ has radical $(3, a_2, a_4)$, so $tm\!f$ has algebraic ${\mathbb {F}}_3$-cellularisation by $(3, a_2, a_4)$ and by $J'$.

For $tm\!f$ at $p\ge 5$, the ideal $J' = (B, M) = (c_4, \Delta )$, with $\Delta = (c_4^3 - c_6^2)/1728$, has radical $(c_4, c_6)$. Hence $tm\!f$ has algebraic ${\mathbb {Z}}_p$-cellularisation by $(c_4, c_6)$ and by $J'$.

Similarly, the ideal $J' = (p, B, M) = (p, c_4, \Delta )$ has radical $(p, c_4, c_6)$, so $tm\!f$ has algebraic ${\mathbb {F}}_p$-cellularisation by $(p, c_4, c_6)$ and by $J'$. $\square$

5.6 Local cohomology theorems by Gorenstein duality

Theorem 5.19

There are equivalences of ko-modules

$$\begin{aligned} \Gamma _B ko = \Gamma _{{\mathfrak {n}}_0} ko \simeq \Sigma ^{-5} I_{{\mathbb {Z}}_p}(ko) \end{aligned}$$

and equivalences of $tm\!f$-modules

$$\begin{aligned} \Gamma _{(B,M)} tm\!f= \Gamma _{{\mathfrak {n}}_0} tm\!f\simeq \Sigma ^{-22} I_{{\mathbb {Z}}_p}(tm\!f) \end{aligned}$$

at all primes p.

Proof

We apply Proposition 5.3 to $R \rightarrow k$ with $R = ko$ or $R = tm\!f$ and $k = {\mathbb {Z}}_p$. Then $R \rightarrow k$ is Gorenstein of shift $a = -5$ or $a = -22$ by Proposition 5.10, and k is small, hence proxy-small, as an R-module by Corollary 5.14. Hence ${{\,\mathrm{Cell}\,}}_k R \simeq \Sigma ^a I_{{\mathbb {Z}}_p} R$ in each case. Moreover, ${{\,\mathrm{Cell}\,}}_k R \simeq \Gamma _J R$ for $J = (B) \subset \pi _*(ko)$ or $J = (B, M) \subset \pi _*(tm\!f)$, by Proposition 5.18. Finally, $\Gamma _J R \simeq \Gamma _{{\mathfrak {n}}_0} R$ since J has radical ${\mathfrak {n}}_0$ in each case, cf. Lemma 4.7(2). $\square$

Theorem 5.20

There are equivalences of ko-modules

$$\begin{aligned} \Gamma _{(p,B)} ko = \Gamma _{{\mathfrak {n}}_p} ko \simeq \Sigma ^{-6} I(ko) \end{aligned}$$

and equivalences of $tm\!f$-modules

$$\begin{aligned} \Gamma _{(p,B,M)} tm\!f= \Gamma _{{\mathfrak {n}}_p} tm\!f\simeq \Sigma ^{-23} I(tm\!f) \end{aligned}$$

at all primes p.

Proof

We apply Proposition 5.3 to $R \rightarrow k$ with $R = ko$ or $R = tm\!f$ and $k = {\mathbb {F}}_p$. Then $R \rightarrow k$ is Gorenstein of shift $a = -6$ or $a = -23$ by Proposition 5.10, and k is small, hence proxy-small, as an R-module by Corollary 5.14. Furthermore, $\pi _t(R)$ is a finitely generated ${\mathbb {Z}}_p$-module for each t, as is clear from Theorems 8.2 and 8.4 below, so ${{\,\mathrm{Hom}\,}}_{{\mathbb {Z}}_p}(\pi _*(R), {\mathbb {Q}}_p/{\mathbb {Z}}_p)$ is p-power torsion in each degree. Hence ${{\,\mathrm{Cell}\,}}_k R \simeq \Sigma ^a I R$ in each case. Moreover, ${{\,\mathrm{Cell}\,}}_k R \simeq \Gamma _J R$ for $J = (p, B) \subset \pi _*(ko)$ or $J = (p, B, M) \subset \pi _*(tm\!f)$, by Proposition 5.18. Finally, $\Gamma _J R \simeq \Gamma _{{\mathfrak {n}}_p} R$ since J has radical ${\mathfrak {n}}_p$ in each case, cf. Lemma 4.7(2). $\square$

5.7 ko- and tmf-module Steenrod algebras

For completeness, we record the structure of $\pi _*({\mathcal {E}})$ in our main cases of interest, where ${\mathcal {E}}= F_R(k, k)$, $R = ko$ or $tm\!f$, and $k = {\mathbb {F}}_p$.

Proposition 5.21

[15, 20, 30] For $p=2$ there are algebra isomorphisms

$$\begin{aligned}&\pi _* F_{ko}({\mathbb {F}}_2, {\mathbb {F}}_2) \cong A(1) \\&\pi _* F_{tm\!f}({\mathbb {F}}_2, {\mathbb {F}}_2) \cong A(2). \end{aligned}$$

For $p=3$ there is a square-zero quadratic extension

$$\begin{aligned} \pi _* F_{tm\!f}({\mathbb {F}}_3, {\mathbb {F}}_3) = A_{tm\!f} \longrightarrow A(1), \end{aligned}$$

where $A_{tm\!f}$ is generated by classes $\beta$ and $P^1$ in cohomological degrees 1 and 4, subject to $\beta ^2 = 0$, $\beta (P^1)^2 \beta = (\beta P^1)^2 + (P^1 \beta )^2$ and $(P^1)^3 = 0$. In each case, classes in homotopical degree $-m$ correspond to classes in cohomological degree m.

Proof

Restriction along $S \rightarrow tm\!f$ induces an S-algebra map

$$\begin{aligned} {\mathcal {E}}= F_{tm\!f}({\mathbb {F}}_2, {\mathbb {F}}_2) \longrightarrow F_S({\mathbb {F}}_2, {\mathbb {F}}_2) \end{aligned}$$

and an algebra homomorphism $\pi _*({\mathcal {E}}) \rightarrow A$ to the mod 2 Steenrod algebra. Base change along $S \rightarrow {\mathbb {F}}_2$ lets us rewrite the S-algebra map as

$$\begin{aligned} {\mathcal {E}}\cong F_{{\mathbb {F}}_2 \wedge tm\!f}({\mathbb {F}}_2 \wedge {\mathbb {F}}_2, {\mathbb {F}}_2) \longrightarrow F_{{\mathbb {F}}_2}({\mathbb {F}}_2 \wedge {\mathbb {F}}_2, {\mathbb {F}}_2). \end{aligned}$$

Since the dual Steenrod algebra $A_* = \pi _*({\mathbb {F}}_2 \wedge {\mathbb {F}}_2)$ is free as an $H_*(tm\!f) = \pi _*({\mathbb {F}}_2 \wedge tm\!f)$-module, the ${{\,\mathrm{Ext}\,}}$ spectral sequences for these two function spectra collapse, and let us rewrite the algebra homomorphism as the monomorphism

$$\begin{aligned} \pi _*({\mathcal {E}}) \cong {{\,\mathrm{Hom}\,}}_{H_*(tm\!f)}(A_*, {\mathbb {F}}_2) \longrightarrow {{\,\mathrm{Hom}\,}}_{{\mathbb {F}}_2}(A_*, {\mathbb {F}}_2) \cong A. \end{aligned}$$

By duality, this identifies $\pi _*({\mathcal {E}})$ with the $H^*(tm\!f) = A/\!/A(2)$-comodule primitives in A, which is precisely the subalgebra A(2).

The proof for ko is the same, replacing A(2) with A(1).

The result for $p=3$ is due to Henriques and Hill [30,  Thm. 2.2], [20,  §13.3], except for the comment that the extension is square-zero, which appears in [15,  §13.1]. $\square$

6 Thera duality

A third line of proof is discussed in [15,  §10.3, §10.4, §13.5], yielding the following theorems.

Theorem 6.1

[15,  Thm. 10.6, Prop. 10.12] There are equivalences of 2-complete $tm\!f$-modules

$$\begin{aligned} \Sigma ^{23} tm\!f&\simeq I(\Gamma _{(2, B, M)} tm\!f) \\ \Sigma ^{22} tm\!f&\simeq I_{{\mathbb {Z}}_2}(\Gamma _{(B, M)} tm\!f). \end{aligned}$$

Theorem 6.2

[15,  Thm. 13.20, Prop. 13.21] There are equivalences of 3-complete $tm\!f$-modules

$$\begin{aligned} \Sigma ^{23} tm\!f&\simeq I(\Gamma _{(3, B, H)} tm\!f) \\ \Sigma ^{22} tm\!f&\simeq I_{{\mathbb {Z}}_3}(\Gamma _{(B, H)} tm\!f). \end{aligned}$$

This approach combines descent with a strengthening of the Cohen–Macaulay property, equivalent to the Gorenstein property. Recall (5.1). One first observes that

$$\begin{aligned} \Sigma ^{11} tm\!f_1(3) \simeq I(\Gamma _{(2, a_1, a_3)} tm\!f_1(3)), \end{aligned}$$

because the local cohomology of $\pi _*(tm\!f_1(3)) = {\mathbb {Z}}_2[a_1, a_3]$ at the maximal ideal ${\mathfrak {n}}_2 = (2, a_1, a_3)$ is concentrated in a single cohomological degree, and, moreover, its ${\mathbb {Z}}_2$-module Pontryagin dual is a free $\pi _*(tm\!f_1(3))$-module on one generator. The conclusion for $tm\!f$ follows by faithful descent along $tm\!f\rightarrow tm\!f_1(3) \simeq tm\!f\wedge \Phi$, since $\Phi$ is Spanier–Whitehead self-dual.

7 Topological K-theory

As a warm-up to Sect. 8, we spell out the structure of the local cohomology spectral sequences

$$\begin{aligned} E_2^{s,t} = H^s_{(B)}(\pi _*(ko))_t \Longrightarrow _s \pi _{t-s}(\Gamma _{B} ko) \cong \pi _{t-s}(\Sigma ^{-5} I_{{\mathbb {Z}}_2}(ko)) \end{aligned}$$

and

$$\begin{aligned} E_2^{s,t} = H^s_{(2, B)}(\pi _*(ko))_t \Longrightarrow _s \pi _{t-s}(\Gamma _{(2, B)} ko) \cong \pi _{t-s}(\Sigma ^{-6} I(ko)). \end{aligned}$$

Multiplication by B acts injectively on the depth 1 graded commutative ring

$$\begin{aligned} \pi _*(ko) = {\mathbb {Z}}_2[\eta , A, B]/(2 \eta , \eta ^3, \eta A, A^2 - 4 B) \end{aligned}$$

and we let $N_*$ denote a basic block for this action.

Definition 7.1

In this section only, let $N_* \subset \pi _*(ko)$ be the ${\mathbb {Z}}_2$-submodule of classes in degrees $0 \le * < 8$, and let $N = ko/B$.

Lemma 7.2

The composite $N_* \otimes {\mathbb {Z}}[B] \rightarrow \pi _*(ko) \otimes {\mathbb {Z}}[B] \overset{\cdot }{\rightarrow }\pi _*(ko)$ is an isomorphism. As a ${\mathbb {Z}}_2$-module, $N_* = {\mathbb {Z}}_2\{1, \eta , \eta ^2, A\}/(2 \eta , 2 \eta ^2)$ is a split extension by the 2-torsion submodule $\Gamma _2 N_* = {\mathbb {Z}}/2\{\eta , \eta ^2\}$ of the 2-torsion free quotient $N_* / \Gamma _2 N_* = {\mathbb {Z}}_2\{1, A\}$.

Lemma 7.3

$H^0_{(B)}(\pi _*(ko)) = 0$ and $H^1_{(B)}(\pi _*(ko)) \cong N_* \otimes {\mathbb {Z}}[B]/B^\infty$.

Proof

These are the cohomology groups of the complex

$$\begin{aligned} 0 \rightarrow \pi _*(ko) \overset{\gamma }{\longrightarrow }\pi _*(ko)[1/B] \rightarrow 0, \end{aligned}$$

which we may rewrite as $0 \rightarrow N_* \otimes {\mathbb {Z}}[B] \overset{\gamma }{\longrightarrow }N_* \otimes {\mathbb {Z}}[B^{\pm 1}] \rightarrow 0$. $\square$

Fig. 1

$E_2^{s,t} = H^s_{(B)}(\pi _*(ko))_t \Longrightarrow _s \pi _{t-s}(\Gamma _B ko)$

Proposition 7.4

The local cohomology spectral sequence

$$\begin{aligned} E_2^{s,t} = H^s_{(B)}(\pi _*(ko))_t \Longrightarrow _s \pi _{t-s}(\Gamma _B ko) \cong \pi _{t-s}(\Sigma ^{-5} I_{{\mathbb {Z}}_2} ko) \end{aligned}$$

has $E_2$-term concentrated on the $s=1$ line, with $E_2^{1,*} = N_* \otimes {\mathbb {Z}}[B]/B^\infty$. There is no room for differentials or hidden extensions, so $E_2 = E_\infty$. Hence there are isomorphisms

$$\begin{aligned} \Sigma ^{-1} {\mathbb {Z}}_2\{1, A\} \otimes {\mathbb {Z}}[B]/B^\infty \cong \Sigma ^{-5} {{\,\mathrm{Hom}\,}}_{{\mathbb {Z}}_2}({\mathbb {Z}}_2\{1, A\} \otimes {\mathbb {Z}}[B], {\mathbb {Z}}_2) \end{aligned}$$

and

$$\begin{aligned} \Sigma ^{-1} {\mathbb {Z}}/2\{\eta , \eta ^2\} \otimes {\mathbb {Z}}[B]/B^\infty \cong \Sigma ^{-6} {{\,\mathrm{Ext}\,}}_{{\mathbb {Z}}_2}({\mathbb {Z}}/2\{\eta , \eta ^2\} \otimes {\mathbb {Z}}[B], {\mathbb {Z}}_2). \end{aligned}$$

Proof

Recall the short exact sequence (3.1) and see Fig. 1, where the vertical and slope 1 solid lines show multiplications by 2 and $\eta$, respectively, while the roughly horizontal dotted paths show multiplications by B. $\square$

Lemma 7.5

$H^s_{(2, B)}(\pi _*(ko)) \cong H^{s-1}_{(2)}(N_*) \otimes {\mathbb {Z}}[B]/B^\infty$ where

$$\begin{aligned} H^0_{(2)}(N_*)&= {\mathbb {Z}}/2\{\eta , \eta ^2\} \\ H^1_{(2)}(N_*)&= {\mathbb {Q}}_2/{\mathbb {Z}}_2\{1, A\}. \end{aligned}$$

Proof

See Lemma 8.6 for the proof of the first isomorphism. The $H^*_{(2)}(N_*)$ are the cohomology groups of the complex

$$\begin{aligned} 0 \rightarrow N_* \overset{\gamma }{\longrightarrow }N_*[1/2] \rightarrow 0. \end{aligned}$$

$\square$

Fig. 2

$E_2^{s,t} = H^s_{(2, B)}(\pi _*(ko))_t \Longrightarrow _s \pi _{t-s}(\Gamma _{(2, B)} ko)$

Proposition 7.6

The local cohomology spectral sequence

$$\begin{aligned} E_2^{s,t} = H^s_{(2, B)}(\pi _*(ko))_t \Longrightarrow _s \pi _{t-s}(\Gamma _{(2, B)} ko) \cong \pi _{t-s}(\Sigma ^{-6} I(ko)) \end{aligned}$$

has $E_2$-term

$$\begin{aligned} E_2^{s,t} \cong H^{s-1}_{(2)}(N_*) \otimes {\mathbb {Z}}[B]/B^\infty . \end{aligned}$$

There is no room for differentials, so $E_2 = E_\infty$. Hence there are isomorphisms

$$\begin{aligned} \Sigma ^{-2} {\mathbb {Q}}_2/{\mathbb {Z}}_2\{1, A\} \otimes {\mathbb {Z}}[B]/B^\infty \cong \Sigma ^{-6} {{\,\mathrm{Hom}\,}}_{{\mathbb {Z}}_2}({\mathbb {Z}}_2\{1, A\} \otimes {\mathbb {Z}}[B], {\mathbb {Q}}_2/{\mathbb {Z}}_2) \end{aligned}$$

and

$$\begin{aligned} \Sigma ^{-1} {\mathbb {Z}}/2\{\eta , \eta ^2\} \otimes {\mathbb {Z}}[B]/B^\infty \cong \Sigma ^{-6} {{\,\mathrm{Hom}\,}}_{{\mathbb {Z}}_2}({\mathbb {Z}}/2\{\eta , \eta ^2\} \otimes {\mathbb {Z}}[B], {\mathbb {Q}}_2/{\mathbb {Z}}_2). \end{aligned}$$

Moreover, there are hidden $\eta$-extensions as shown by sloping dashed red lines.

Proof

See Fig. 2. The hidden $\eta$-extensions are deduced from the known $\pi _*(ko)$-module structure of the abutment. $\square$

8 Topological modular forms

We can now work out the structure of the local cohomology spectral sequences

$$\begin{aligned} E_2^{s,t}&= H^s_{(B, M)}(\pi _*(tm\!f))_t \\&\Longrightarrow _s \pi _{t-s}(\Gamma _{(B, M)} tm\!f) \cong \pi _{t-s}(\Sigma ^{-22} I_{{\mathbb {Z}}_p}(tm\!f)) \end{aligned}$$

and

$$\begin{aligned} E_2^{s,t}&= H^s_{(p, B, M)}(\pi _*(tm\!f))_t \\&\Longrightarrow _s \pi _{t-s}(\Gamma _{(p, B, M)} tm\!f) \cong \pi _{t-s}(\Sigma ^{-23} I(tm\!f)) \end{aligned}$$

for $p=2$ and for $p=3$. Recall the algebra generators for $\pi _*(tm\!f)$ listed in Table 1 for $p=2$ and in Table 2 for $p=3$. In each case multiplication by M acts injectively on the depth 1 graded commutative ring $\pi _*(tm\!f)$, and we let $N_*$ denote a basic block for this action. (The notation BB is used for a similar object in [27].) To begin, we review the ${\mathbb {Z}}_p[B, M]$-module structure on $\pi _*(tm\!f)$ and the ${\mathbb {Z}}_p[B]$-module structure on $N_*$, in the notation from [15,  Ch. 9].

8.1 (B, M)-local cohomology of tmf

Let $p=2$ in this subsection and the next. See Fig. 4 for the mod 2 Adams $E_\infty$-term for $tm\!f$ in the range $0 \le t-s \le 192$, with all hidden 2-, $\eta$- and $\nu$-extensions shown. There are no hidden B- or M-extensions in this spectral sequence.

Definition 8.1

Let $N_* \subset \pi _*(tm\!f)$ be the ${\mathbb {Z}}_2[B]$-submodule generated by all classes in degrees $0 \le * < 192$, and let $N = tm\!f/M$.

Theorem 8.2

[15,  Thm. 9.27] The composite homomorphisms

$$\begin{aligned}&N_* \otimes {\mathbb {Z}}[M] \longrightarrow \pi _*(tm\!f) \otimes \pi _*(tm\!f) \overset{\cdot }{\longrightarrow }\pi _*(tm\!f) \\&N_* \subset \pi _*(tm\!f) \longrightarrow \pi _*(N) \end{aligned}$$

are isomorphisms. Hence $\pi _*(tm\!f)$ is a (split) extension of ${\mathbb {Z}}_2[B, M]$-modules

$$\begin{aligned} 0 \rightarrow \Gamma _B N_* \otimes {\mathbb {Z}}[M] \longrightarrow \pi _*(tm\!f) \longrightarrow \frac{N_*}{\Gamma _B N_*} \otimes {\mathbb {Z}}[M] \rightarrow 0. \end{aligned}$$

Definition 8.3

Let $\nu _3 = \eta _1^3$ and $\nu _7 = 0$, and set $d_k = 8/\gcd (k,8)$, so that $d_0 = 1$, $d_4 = 2$, $d_2 = d_6 = 4$, $d_1 = d_3 = d_5 = d_7 = 8$ and $d_{7-k} \nu _k = 0$ for $0 \le k \le 7$.

Theorem 8.4

[15,  Thm. 9.26] As a ${\mathbb {Z}}_2[B]$-module, $N_*$ is a split extension

$$\begin{aligned} 0 \rightarrow \Gamma _B N_* \longrightarrow N_* \longrightarrow \frac{N_*}{\Gamma _B N_*} \rightarrow 0. \end{aligned}$$

The B-power torsion submodule $\Gamma _B N_*$ is given in Table 3. It is concentrated in degrees $3 \le * \le 164$, and is finite in each degree. The action of B is as indicated in the table, together with $B \cdot \epsilon _1 = 2 {\bar{\kappa }}^2$, $B \cdot \eta \nu _2 = 2 {\bar{\kappa }}^3$ and $B \cdot \epsilon _5 \kappa = 4 \nu \nu _6$.

Table 3 B-power torsion in $N_*$ at $p=2$

The B-torsion free quotient of $N_*$ is the direct sum

$$\begin{aligned} \frac{N_*}{\Gamma _B N_*} = \bigoplus _{k=0}^7 ko[k] \end{aligned}$$

of the following eight ${\mathbb {Z}}_2[B]$-modules, with ko[k] concentrated in degrees $* \ge 24k$:

$$\begin{aligned} ko[0]&= {\mathbb {Z}}_2[B]\{1, C\} \oplus {\mathbb {Z}}/2[B]\{\eta , \eta ^2\} \\ ko[1]&= {\mathbb {Z}}_2\{D_1\} \oplus {\mathbb {Z}}_2[B]\{B_1, C_1\} \oplus {\mathbb {Z}}/2[B]\{\eta _1, \eta \eta _1\} \\ ko[2]&= {\mathbb {Z}}_2\{D_2\} \oplus {\mathbb {Z}}_2[B]\{B_2, C_2\} \oplus {\mathbb {Z}}/2[B]\{\eta B_2, \eta _1^2\} \\ ko[3]&= {\mathbb {Z}}_2\{D_3\} \oplus {\mathbb {Z}}_2[B]\{B_3, C_3\} \oplus {\mathbb {Z}}/2[B]\{\eta B_3, \eta ^2 B_3\} \\ ko[4]&= {\mathbb {Z}}_2\{D_4\} \oplus {\mathbb {Z}}_2[B]\{B_4, C_4\} \oplus {\mathbb {Z}}/2[B]\{\eta _4, \eta \eta _4\} \\ ko[5]&= {\mathbb {Z}}_2\{D_5\} \oplus {\mathbb {Z}}_2[B]\{B_5, C_5\} \oplus {\mathbb {Z}}/2[B]\{\eta B_5, \eta _1 \eta _4\} \\ ko[6]&= {\mathbb {Z}}_2\{D_6\} \oplus {\mathbb {Z}}_2[B]\{B_6, C_6\} \oplus {\mathbb {Z}}/2[B]\{\eta B_6, \eta ^2 B_6\} \\ ko[7]&= {\mathbb {Z}}_2\{D_7\} \oplus {\mathbb {Z}}_2[B]\{B_7, C_7\} \oplus {\mathbb {Z}}/2[B]\{\eta B_7, \eta ^2 B_7\}. \end{aligned}$$

The ${\mathbb {Z}}_2[B]$-module structures are specified by $B \cdot D_k = d_k B_k$ for each $1 \le k \le 7$.

Remark 8.5

The submodule $N_* \subset \pi _*(tm\!f)$ is preserved by the action of $\eta$, $\nu$, $\epsilon$, $\kappa$ and ${\bar{\kappa }}$. To check this, note that the $B^2$-torsion classes $\kappa C_7$, ${\bar{\kappa }} B_7$ and ${\bar{\kappa }} C_7$ are zero. It follows that the isomorphism $N_* \otimes {\mathbb {Z}}[M] \cong \pi _*(tm\!f)$ also respects the action by these elements.

Lemma 8.6

$$\begin{aligned} H^s_{(B,M)}(\pi _*(tm\!f)) \cong H^{s-1}_{(B)}(N_*) \otimes {\mathbb {Z}}[M]/M^\infty . \end{aligned}$$

Proof

The spectral sequence

$$\begin{aligned} E_2^{i,j} = H^i_{(B)}(H^j_{(M)}(N_* \otimes {\mathbb {Z}}[M])) \Longrightarrow _i H^{i+j}_{(B, M)}(N_* \otimes {\mathbb {Z}}[M]) \end{aligned}$$

collapses at the $j=1$ line, where $H^1_{(M)}({\mathbb {Z}}[M]) = {\mathbb {Z}}[M]/M^\infty = {\mathbb {Z}}[M^{-1}]\{1/M\}$. $\square$

Lemma 8.7

$$\begin{aligned} H^0_{(B)}(N_*) = \Gamma _B N_* \end{aligned}$$

and

$$\begin{aligned} H^1_{(B)}(N_*) = N_*/B^\infty = \bigoplus _{k=0}^7 ko[k]/B^\infty \end{aligned}$$

is the direct sum of the following eight ${\mathbb {Z}}_2[B]$-modules, with $ko[k]/B^\infty$ concentrated in degrees $* \le 24k+4$:

$$\begin{aligned} ko[0]/B^\infty&= {\mathbb {Z}}_2[B]/B^\infty \{1, C\} \oplus {\mathbb {Z}}/2[B]/B^\infty \{\eta , \eta ^2\} \\ ko[1]/B^\infty&= {\mathbb {Z}}_2[B]/B^\infty \{B_1, C_1\}/(8 B_1/B) \oplus {\mathbb {Z}}/2[B]/B^\infty \{\eta _1, \eta \eta _1\} \\ ko[2]/B^\infty&= {\mathbb {Z}}_2[B]/B^\infty \{B_2, C_2\}/(4 B_2/B) \oplus {\mathbb {Z}}/2[B]/B^\infty \{\eta B_2, \eta _1^2\} \\ ko[3]/B^\infty&= {\mathbb {Z}}_2[B]/B^\infty \{B_3, C_3\}/(8 B_3/B) \oplus {\mathbb {Z}}/2[B]/B^\infty \{\eta B_3, \eta ^2 B_3\} \\ ko[4]/B^\infty&= {\mathbb {Z}}_2[B]/B^\infty \{B_4, C_4\}/(2 B_4/B) \oplus {\mathbb {Z}}/2[B]/B^\infty \{\eta _4, \eta \eta _4\} \\ ko[5]/B^\infty&= {\mathbb {Z}}_2[B]/B^\infty \{B_5, C_5\}/(8 B_5/B) \oplus {\mathbb {Z}}/2[B]/B^\infty \{\eta B_5, \eta _1 \eta _4\} \\ ko[6]/B^\infty&= {\mathbb {Z}}_2[B]/B^\infty \{B_6, C_6\}/(4 B_6/B) \oplus {\mathbb {Z}}/2[B]/B^\infty \{\eta B_6, \eta ^2 B_6\} \\ ko[7]/B^\infty&= {\mathbb {Z}}_2[B]/B^\infty \{B_7, C_7\}/(8 B_7/B) \oplus {\mathbb {Z}}/2[B]/B^\infty \{\eta B_7, \eta ^2 B_7\}. \end{aligned}$$

Here ${\mathbb {Z}}_2[B]/B^\infty = {\mathbb {Z}}_2[B^{-1}]\{1/B\}$ and ${\mathbb {Z}}/2[B]/B^\infty = {\mathbb {Z}}/2[B^{-1}]\{1/B\}$.

Proof

The relations $B \cdot \eta _k = \eta B_k$ from [15,  Def. 7.22(7)] ensure that

$$\begin{aligned} ko[k][1/B] = {\mathbb {Z}}_2[B^{\pm 1}] \{B_k, C_k\} \oplus {\mathbb {Z}}/2[B^{\pm 1}] \{\eta B_k, \eta ^2 B_k\} \end{aligned}$$

for each $0 \le k \le 7$, from which the formulas for $ko[k]/B^\infty$ follow. Note that $B \cdot D_k = d_k B_k$ in ko[k] implies the relation $d_k \cdot B_k/B = 0$ in $ko[k]/B^\infty$. $\square$

Theorem 8.8

At $p=2$, the local cohomology spectral sequence

$$\begin{aligned} E_2^{s,t} = H^s_{(B,M)}(\pi _*(tm\!f))_t \Longrightarrow _s \pi _{t-s}(\Gamma _{(B, M)} tm\!f) \cong \pi _{t-s}(\Sigma ^{-22} I_{{\mathbb {Z}}_2}(tm\!f)) \end{aligned}$$

has $E_2$-term

$$\begin{aligned} H^s_{(B,M)}(\pi _*(tm\!f))_* \cong H^{s-1}_{(B)}(N_*) \otimes {\mathbb {Z}}[M]/M^\infty \end{aligned}$$

where $H^*_{(B)}(N_*)$ is displayed in Figs. 5 and 6. There is no room for differentials, so $E_2 = E_\infty$. There are hidden additive extensions

$$\begin{aligned} d_{7-k} \cdot \nu _k \doteq C_k/B \end{aligned}$$

(multiplied by all negative powers of M) for $0 \le k \le 6$, indicated by vertical dashed red lines in the figures. Moreover, there are hidden $\eta$- and $\nu$-extensions as shown by sloping dashed and dotted red lines in these figures.

Proof

See [15,  §9.2] for the $\eta$- and $\nu$-multiplications in $\Gamma _B N_*$ that are not evident from the notation. We note in particular the relation $\nu ^2 \nu _4 = \eta \epsilon _4 + \eta _1 {\bar{\kappa }}^4$ in degree 105. The dotted black lines show B-multiplications. The homotopy cofibre (and fibre) sequences

$$\begin{aligned}&\Sigma ^{192} \Gamma _{(B, M)} tm\!f\overset{M}{\longrightarrow }\Gamma _{(B, M)} tm\!f\longrightarrow \Gamma _B N \\&I_{{\mathbb {Z}}_2} N \longrightarrow I_{{\mathbb {Z}}_2}(tm\!f) \overset{M}{\longrightarrow }I_{{\mathbb {Z}}_2}(\Sigma ^{192} tm\!f) \end{aligned}$$

and the equivalence $\Gamma _{(B, M)} tm\!f\simeq \Sigma ^{-22} I_{{\mathbb {Z}}_2}(tm\!f)$ imply an equivalence

$$\begin{aligned} \Gamma _B N \simeq \Sigma ^{171} I_{{\mathbb {Z}}_2} N \end{aligned}$$

of $tm\!f$-modules. For each $0 \le k \le 6$ the group $\pi _{24k+3}(\Gamma _B N) \cong \pi _{-24(7-k)}(I_{{\mathbb {Z}}_2} N)$ sits in a short exact sequence

$$\begin{aligned}&0 \rightarrow {{\,\mathrm{Ext}\,}}_{{\mathbb {Z}}_2}(\pi _{24(7-k)-1}(N), {\mathbb {Z}}_2) \longrightarrow \pi _{-24(7-k)}(I_{{\mathbb {Z}}_2} N) \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \longrightarrow {{\,\mathrm{Hom}\,}}_{{\mathbb {Z}}_2}(\pi _{24(7-k)}(N), {\mathbb {Z}}_2) \rightarrow 0, \end{aligned}$$

cf. (3.1). Here $\pi _{24(7-k)-1}(N) = 0$ and

$$\begin{aligned} \pi _{24(7-k)}(N) \cong {\mathbb {Z}}_2\{B^{3(7-k)}, \dots , B^3 D_{6-k}, D_{7-k}\} \cong {\mathbb {Z}}_2^{8-k}, \end{aligned}$$

so $\pi _{24k+3}(\Gamma _B N) \cong {\mathbb {Z}}_2^{8-k}$ is 2-torsion free. In each case this implies that $\nu _k$, which generates a cyclic group $\langle \nu _k\rangle$ of order $d_{7-k}$ in $\Gamma _B N_*$, lifts to a class of infinite order in $\pi _{24k+3}(\Gamma _B N)$. Since $\nu _k$ is (B- or) $B^2$-torsion in $\Gamma _B N_*$, its lift must also be (B- or) $B^2$-torsion, and the only possibility is that $d_{7-k}$ times the lift of $\nu _k$ is a 2-adic unit times the image of $C_k/B \in N_*/B^\infty$. Hence there is a hidden 2-extension from $\frac{1}{2} d_{7-k} \nu _k$ in Adams bidegree $(t-s,s) = (24k+3, 0)$ to $C_k/B$ in bidegree $(t-s,s) = (24k+3,1)$, in the local cohomology spectral sequence

$$\begin{aligned} E_2^{s,t} = H^s_{(B)}(N_*)_t \Longrightarrow _s \pi _{t-s}(\Gamma _B N) \cong \pi _{t-s}(\Sigma ^{171} I_{{\mathbb {Z}}_2} N). \end{aligned}$$

This translates to a hidden 2-extension from $\frac{1}{2} d_{7-k} \nu _k/M$ in bidegree $(t-s,s) = (24k-190, 1)$ to $C_k/BM$ in bidegree $(t-s,s) = (24k-190,2)$ in the local cohomology spectral sequence for $\Gamma _{(B, M)} tm\!f$, together with its multiples by all negative powers of M.

There is no room for further hidden 2-extensions, by elementary $\eta$-, $\nu$- and B-linearity considerations. The hidden $\eta$- and $\nu$-extensions are present in $\pi _*(I_{{\mathbb {Z}}_2} N)$, hence also in $\pi _*(\Gamma _B N)$ and in $\pi _*(\Gamma _{(B, M)} tm\!f)$, with the appropriate degree shifts. $\square$

8.2 (2, B, M)-local cohomology of tmf

Lemma 8.9

$$\begin{aligned} H^s_{(2,B,M)}(\pi _*(tm\!f)) \cong H^{s-1}_{(2,B)}(N_*) \otimes {\mathbb {Z}}[M]/M^\infty . \end{aligned}$$

Proof

Replace (B) by (2, B) in the proof of Lemma 8.6. $\square$

Proposition 8.10

All B-power torsion in $N_*$ is 2-power torsion, so

$$\begin{aligned} H^0_{(2, B)}(N_*)&= \Gamma _B N_* \\ H^1_{(2, B)}(N_*)&= \Gamma _2(N_*/B^\infty ) \\ H^2_{(2, B)}(N_*)&= N_*/(2^\infty , B^\infty ) \end{aligned}$$

with a short exact sequence

$$\begin{aligned} 0 \rightarrow (\Gamma _2 N_*)/B^\infty \longrightarrow \Gamma _2(N_*/B^\infty ) \longrightarrow \Gamma _B(N_*/2^\infty ) \rightarrow 0. \end{aligned}$$

Here

$$\begin{aligned} (\Gamma _2 N_*)/B^\infty= & {} {\mathbb {Z}}/2[B]/B^\infty \{\eta , \eta ^2, \eta _1, \eta \eta _1, \eta B_2, \eta _1^2, \eta B_3, \eta ^2 B_3, \\&\qquad \qquad \qquad \quad \,\,\eta _4, \eta \eta _4, \eta B_5, \eta _1 \eta _4, \eta B_6, \eta ^2 B_6, \eta B_7, \eta ^2 B_7\}, \end{aligned}$$

and

$$\begin{aligned} \Gamma _B(N_*/2^\infty ) = \bigoplus _{k=1}^7 {\mathbb {Z}}/d_k \{B_k/B\}, \end{aligned}$$

while

$$\begin{aligned} N_*/(2^\infty , B^\infty ) = \bigoplus _{k=0}^7 {\mathbb {Z}}_2[B]/(2^\infty , B^\infty ) \{B_k/B, C_k\}, \end{aligned}$$

where ${\mathbb {Z}}_2[B]/(2^\infty , B^\infty ) = {\mathbb {Q}}_2/{\mathbb {Z}}_2[B^{-1}]\{1/B\}$.

Proof

This follows from the composite functor spectral sequence of Sect. 2.5 with $R_* = \pi _*(tm\!f)/M$, first applied with $x = B$ and $y = 2$, and thereafter with $x = 2$ and $y = B$. The formulas for $(\Gamma _2 N_*)/B^\infty$ and $N_*/(2^\infty , B^\infty )$ follow from the expressions for $N_*$ and $N_*/B^\infty$ in Theorem 8.4 and Lemma 8.7. Only the summands ${\mathbb {Z}}_2[D_k] \oplus {\mathbb {Z}}_2[B]\{B_k\} \subset ko[k]$ of $N_*$ contribute to $\Gamma _B(N_*/2^\infty )$, where $B \cdot D_k = d_k B_k$. The B-power torsion in $ko[k]/2^\infty$ equals ${\mathbb {Z}}/d_k\{D_k/d_k\} \subset {\mathbb {Q}}_2/{\mathbb {Z}}_2\{D_k\}$, which we can rewrite as ${\mathbb {Z}}/d_k\{B_k/B\}$. $\square$

Theorem 8.11

The local cohomology spectral sequence

$$\begin{aligned} E_2^{s,t} = H^s_{(2, B, M)}(\pi _*(tm\!f))_t \Longrightarrow _s \pi _{t-s}(\Gamma _{(2, B, M)} tm\!f) \cong \pi _{t-s}(\Sigma ^{-23} I(tm\!f)) \end{aligned}$$

has $E_2$-term

$$\begin{aligned} H^s_{(2, B, M)}(\pi _*(tm\!f))_* \cong H^{s-1}_{(2, B)}(N_*) \otimes {\mathbb {Z}}[M]/M^\infty \end{aligned}$$

where $H^*_{(2, B)}(N_*)$ is displayed in Figs. 7, 8, 9, 10. There are $d_2$-differentials

$$\begin{aligned} d_2(\nu _k) \doteq C_k/d_{7-k}B \end{aligned}$$

(multiplied by all negative powers of M) for $0 \le k \le 6$, indicated by the green zigzag arrows increasing the filtration by 2. There are no hidden additive extensions, but several hidden $\eta$- and $\nu$-extensions, as shown by sloping dashed and dotted red lines in these figures.

Proof

The homotopy (co-)fibre sequences

$$\begin{aligned}&\Sigma ^{192} \Gamma _{(2, B, M)} tm\!f\overset{M}{\longrightarrow }\Gamma _{(2, B, M)} tm\!f\longrightarrow \Gamma _{(2, B)} N \\&I N \longrightarrow I(tm\!f) \overset{M}{\longrightarrow }I(\Sigma ^{192} tm\!f) \end{aligned}$$

and the equivalence $\Gamma _{(2, B, M)} tm\!f\simeq \Sigma ^{-23} I(tm\!f)$ imply an equivalence

$$\begin{aligned} \Gamma _{(2, B)} N \simeq \Sigma ^{170} I N \end{aligned}$$

of $tm\!f$-modules. For each $0 \le k \le 6$ the group

$$\begin{aligned} \pi _{24k+3}(\Gamma _{(2,B)} N) \cong \pi _{-24(7-k)+1}(I N) \cong {{\,\mathrm{Hom}\,}}_{{\mathbb {Z}}_2}(\pi _{24(7-k)-1}(N), {\mathbb {Q}}_2/{\mathbb {Z}}_2) \end{aligned}$$

is trivial, since $\pi _{24(7-k)-1}(N) = 0$. Hence the group $\langle \nu _k\rangle = {\mathbb {Z}}/d_{7-k} \{\nu _k\}$ in degree $24k+3$ of $\Gamma _{(2, B)} N_* = \Gamma _B N_*$ cannot survive to $E_\infty$ in the local cohomology spectral sequence

$$\begin{aligned} E_2^{s,t} = H^s_{(2, B)}(N_*)_t \Longrightarrow _s \pi _{t-s}(\Gamma _{(2, B)} N) \cong \pi _{t-s}(\Sigma ^{170} I N). \end{aligned}$$

This means that $d_2$ must act injectively on $\langle \nu _k\rangle$. Since $\nu _k$ is (B- or) $B^2$-torsion, the only possible target in bidegree $(t-s,s) = (24k+2, 2)$ is ${\mathbb {Q}}_2/{\mathbb {Z}}_2 \{C_k/B\}$, and therefore $d_2$ maps $\langle \nu _k\rangle$ isomorphically to the subgroup of this target that is generated by $C_k/d_{7-k} B$.

This translates to a $d_2$-differential in the local cohomology spectral sequence for $\Gamma _{(2, B, M)} tm\!f$ from $\nu _k/M$ in bidegree $(t-s,s) = (24k-190, 1)$ to $C_k/d_{7-k} BM$ in bidegree $(t-s,s) = (24k-191,3)$, together with its multiples by all negative powers of M. The previously known 2-, $\eta$- and $\nu$-extensions in $\pi _*(N)$ and $\pi _*(I N)$ are also present in $\pi _*(\Gamma _{(2, B)} N)$ and in $\pi _*(\Gamma _{(2, B, M)} tm\!f)$, with the appropriate degree shifts, and those that increase the local cohomology filtration degree are displayed with red lines. $\square$

Remark 8.12

Let $\Theta N_* \subset \Gamma _B N_*$ be the part of the B-power torsion in $N_*$ that is not in degrees $* \equiv 3 \mod 24$, omitting the subgroups $\langle \nu _k\rangle$ for $0 \le k \le 6$ from Table 3. This equals the kernel of the $d_2$-differential in the (2, B)-local cohomology spectral sequence, which is also the image of the edge homomorphism $\pi _*(\Gamma _{(2, B)} N) \rightarrow \Gamma _B N_*$. Furthermore, let $\Theta \pi _*(tm\!f)$ be the part of $\Gamma _B \pi _*(tm\!f)$ that is not in degrees $* \equiv 3 \mod 24$, which equals the image of the edge homomorphism $\pi _*(\Gamma _{(2, B)} tm\!f) \rightarrow \Gamma _B \pi _*(tm\!f)$.

The image of the 2-complete $tm\!f$-Hurewicz homomorphism $\pi _*(S) \rightarrow \pi _*(tm\!f)$ is the direct sum of ${\mathbb {Z}}$ in degree 0, the 8-periodic groups ${\mathbb {Z}}/2\{\eta B^k\}$ and ${\mathbb {Z}}/2\{\eta ^2 B^k\}$ for $k\ge 0$, the group ${\mathbb {Z}}/8\{\nu \}$ in degree 3, and the 192-periodic groups $\Theta \pi _*(tm\!f) \cong \Theta N_* \otimes {\mathbb {Z}}[M]$. This was conjectured by Mahowald, was proved for degrees $n \le 101$ and $n=125$ in [15,  Thm. 11.89], and has now been proved in all degrees by Behrens, Mahowald and Quigley [6]. The three first summands of the $tm\!f$-Hurewicz image are also detected by the Adams d- and e-invariants [1]. To see that the fourth summand is contained in the image from $\pi _*(\Gamma _{(2, B)} tm\!f)$, one can use the commutative diagram

and the equivalence $C^f_1 tm\!f\simeq \Gamma _{(2, B)} tm\!f$ from Lemmas 2.2 and 2.4.

8.3 (B, H)-local cohomology of tmf

Let $p=3$ in this subsection and the next. See Fig. 12 for the mod 3 ($tm\!f$-module) Adams $E_\infty$-term for $tm\!f$ in the range $0 \le t-s \le 72$, with all hidden $\nu$-extensions shown. There are no hidden B- or H-extensions in this spectral sequence.

Definition 8.13

Let $N_* \subset \pi _*(tm\!f)$ be the ${\mathbb {Z}}_3[B]$-submodule generated by all classes in degrees $0 \le * < 72$, and let $N = tm\!f/H$.

Theorem 8.14

[15,  Lem. 13.16] The composite homomorphisms

$$\begin{aligned}&N_* \otimes {\mathbb {Z}}[H] \longrightarrow \pi _*(tm\!f) \otimes \pi _*(tm\!f) \overset{\cdot }{\longrightarrow }\pi _*(tm\!f) \\&N_* \subset \pi _*(tm\!f) \longrightarrow \pi _*(N) \end{aligned}$$

are isomorphisms. Hence $\pi _*(tm\!f)$ is a (split) extension of ${\mathbb {Z}}_3[B, H]$-modules

$$\begin{aligned} 0 \rightarrow \Gamma _B N_* \otimes {\mathbb {Z}}[H] \longrightarrow \pi _*(tm\!f) \longrightarrow \frac{N_*}{\Gamma _B N_*} \otimes {\mathbb {Z}}[H] \rightarrow 0. \end{aligned}$$

Theorem 8.15

[15,  Thm. 13.18] As a ${\mathbb {Z}}_3[B]$-module, $N_*$ is a split extension

$$\begin{aligned} 0 \rightarrow \Gamma _B N_* \longrightarrow N_* \longrightarrow \frac{N_*}{\Gamma _B N_*} \rightarrow 0. \end{aligned}$$

The B-power torsion submodule $\Gamma _B N_*$ is given in Table 4. It is concentrated in degrees $3 \le * \le 40$, and is annihilated by (3, B).

Table 4 B-power torsion in $N_*$ at $p=3$

The B-torsion free quotient of $N_*$ is the direct sum

$$\begin{aligned} \frac{N_*}{\Gamma _B N_*} = \bigoplus _{k=0}^2 ko[k] \end{aligned}$$

of the following three ${\mathbb {Z}}_3[B]$-modules, with ko[k] concentrated in degrees $* \ge 24k$:

$$\begin{aligned} ko[0]&= {\mathbb {Z}}_3[B]\{1, C\} \\ ko[1]&= {\mathbb {Z}}_3\{D_1\} \oplus {\mathbb {Z}}_3[B]\{B_1, C_1\} \\ ko[2]&= {\mathbb {Z}}_3\{D_2\} \oplus {\mathbb {Z}}_3[B]\{B_2, C_2\}. \end{aligned}$$

The ${\mathbb {Z}}_3[B]$-module structures are specified by $B \cdot D_1 = 3 B_1$ and $B \cdot D_2 = 3 B_2$.

Lemma 8.16

$$\begin{aligned} H^s_{(B,H)}(\pi _*(tm\!f)) \cong H^{s-1}_{(B)}(N_*) \otimes {\mathbb {Z}}[H]/H^\infty . \end{aligned}$$

Proof

Replace M by H in the proof of Lemma 8.6. $\square$

Lemma 8.17

$$\begin{aligned} H^0_{(B)}(N_*) = \Gamma _B N_* \end{aligned}$$

and

$$\begin{aligned} H^1_{(B)}(N_*) = N_*/B^\infty = \bigoplus _{k=0}^2 ko[k]/B^\infty \end{aligned}$$

is the direct sum of the following three ${\mathbb {Z}}_3[B]$-modules, with $ko[k]/B^\infty$ concentrated in degrees $* \le 24k+4$:

$$\begin{aligned} ko[0]/B^\infty&= {\mathbb {Z}}_3[B]/B^\infty \{1, C\} \\ ko[1]/B^\infty&= {\mathbb {Z}}_3[B]/B^\infty \{B_1, C_1\}/(3 B_1/B) \\ ko[2]/B^\infty&= {\mathbb {Z}}_3[B]/B^\infty \{B_2, C_2\}/(3 B_2/B). \end{aligned}$$

Proof

For $k \in \{1,2\}$, the relation $B \cdot D_k = 3 B_k$ in ko[k] implies the relation $3 \cdot B_k/B = 0$ in $ko[k]/B^\infty$. $\square$

Theorem 8.18

At $p=3$, the local cohomology spectral sequence

$$\begin{aligned} E_2^{s,t} = H^s_{(B,H)}(\pi _*(tm\!f))_t \Longrightarrow _s \pi _{t-s}(\Gamma _{(B, H)} tm\!f) \cong \pi _{t-s}(\Sigma ^{-22} I_{{\mathbb {Z}}_3}(tm\!f)) \end{aligned}$$

has $E_2$-term

$$\begin{aligned} H^s_{(B,H)}(\pi _*(tm\!f))_* \cong H^{s-1}_{(B)}(N_*) \otimes {\mathbb {Z}}[H]/H^\infty \end{aligned}$$

where $H^*_{(B)}(N_*)$ is displayed in Fig. 13. There is no room for differentials, so $E_2 = E_\infty$. There are hidden additive extensions

$$\begin{aligned} 3 \cdot \nu \doteq C/B \quad \text {and}\quad 3 \cdot \nu _1 \doteq C_1/B \end{aligned}$$

(multiplied by all negative powers of H), indicated by vertical dashed red lines in the figure. Moreover, there is a hidden $\nu$-extension from $\beta ^2$ to $B_1/B$, shown by a sloping dotted red line.

Proof

We refer to [15,  Prop. 13.14] for the relation $\nu \nu _1 \doteq \beta ^3$. The equivalence $\Gamma _{(B, H)} tm\!f\simeq \Sigma ^{-22} I_{{\mathbb {Z}}_3}(tm\!f)$ implies an equivalence $\Gamma _B N \simeq \Sigma ^{51} I_{{\mathbb {Z}}_3} N$ of $tm\!f$-modules. For $k \in \{0,1\}$ the group $\pi _{24k+3}(\Gamma _B N) \cong \pi _{-24(2-k)}(I_{{\mathbb {Z}}_3} N)$ sits in an extension

$$\begin{aligned} 0 \rightarrow {{\,\mathrm{Ext}\,}}_{{\mathbb {Z}}_3}(\pi _{24(2-k)-1}(N), {\mathbb {Z}}_3) \rightarrow \pi _{-24(2-k)}(I_{{\mathbb {Z}}_3} N) \rightarrow {{\,\mathrm{Hom}\,}}_{{\mathbb {Z}}_3}(\pi _{24(2-k)}(N), {\mathbb {Z}}_3) \rightarrow 0. \end{aligned}$$

Here $\pi _{24(2-k)-1}(N) = 0$ and

$$\begin{aligned} \pi _{24(2-k)}(N) \cong {\mathbb {Z}}_3\{B^{3(2-k)}, \dots , D_{2-k}\} \cong {\mathbb {Z}}_3^{3-k}, \end{aligned}$$

so $\pi _{24k+3}(\Gamma _B N) \cong {\mathbb {Z}}_3^{3-k}$. Since $\nu _k$ is B-torsion, there must be a 3-extension in $\pi _{24k+3}(\Gamma _B N)$ from $\nu _k$ to a 3-adic unit times $C_k/B$. The $\nu$-extension from $\nu _1$ to $\beta ^3$ in $\pi _*(N)$ appears in dual form in $\pi _*(I_{{\mathbb {Z}}_3} N)$, $\pi _*(\Gamma _B N)$ and $\pi _*(\Gamma _{(B, H)} tm\!f)$, and appears as a hidden $\nu$-extension from $\beta ^2$ to $B_1/B$ in the second of these. $\square$

8.4 (3, B, H)-local cohomology of tmf

Lemma 8.19

$$\begin{aligned} H^s_{(3,B,H)}(\pi _*(tm\!f)) \cong H^{s-1}_{(3,B)}(N_*) \otimes {\mathbb {Z}}[H]/H^\infty . \end{aligned}$$

Proof

Replace (B) by (3, B) in the proof of Lemma 8.16. $\square$

Proposition 8.20

$$\begin{aligned} H^0_{(3, B)}(N_*)&= \Gamma _B N_* = {\mathbb {Z}}/3\{\nu , \beta , \nu \beta , \beta ^2, \nu _1, \beta ^3, \nu _1\beta , \beta ^4\} \\ H^1_{(3, B)}(N_*)&= \Gamma _3(N_*/B^\infty ) = \Gamma _B(N_*/3^\infty ) = {\mathbb {Z}}/3 \{B_1/B, B_2/B\} \\ H^2_{(3, B)}(N_*)&= N_*/(3^\infty , B^\infty ) = \bigoplus _{k=0}^2 {\mathbb {Z}}_3[B]/(3^\infty , B^\infty ) \{B_k/B, C_k\}. \end{aligned}$$

Proof

This follows from the composite functor spectral sequence of Sect. 2.5 with $R_* = \pi _*(tm\!f)/H$, first applied with $x = 3$ and $y = B$, and thereafter with $x = B$ and $y = 3$. The groups $(\Gamma _B N_*)/3^\infty$ and $(\Gamma _3 N_*)/B^\infty$ vanish. The 3-power torsion in $ko[k]/B^\infty$ is trivial for $k=0$, and equals ${\mathbb {Z}}/3\{B_k/B\}$ for $k \in \{1,2\}$. $\square$

Theorem 8.21

The local cohomology spectral sequence

$$\begin{aligned} E_2^{s,t} = H^s_{(3, B, H)}(\pi _*(tm\!f))_t \Longrightarrow _s \pi _{t-s}(\Gamma _{(3, B, H)} tm\!f) \cong \pi _{t-s}(\Sigma ^{-23} I(tm\!f)) \end{aligned}$$

has $E_2$-term

$$\begin{aligned} H^s_{(3, B, H)}(\pi _*(tm\!f))_* \cong H^{s-1}_{(3, B)}(N_*) \otimes {\mathbb {Z}}[H]/H^\infty \end{aligned}$$

where $H^*_{(3, B)}(N_*)$ is displayed in Fig. 14. There are $d_2$-differentials

$$\begin{aligned} d_2(\nu _k) \doteq C_k/3 B \end{aligned}$$

(multiplied by all negative powers of M) for $k \in \{0,1\}$, indicated by the green zigzag arrows increasing the filtration by 2. There are no hidden additive extensions, but hidden $\nu$-extensions from $\beta ^2$ to $B_1/B$ and from $B_2/B$ to $C_2/3B$, as shown by sloping dashed red lines in this figure.

Proof

The equivalence $\Gamma _{(3, B, M)} tm\!f\simeq \Sigma ^{-23} I(tm\!f)$ implies an equivalence

$$\begin{aligned} \Gamma _{(3, B)} N \simeq \Sigma ^{50} I N \end{aligned}$$

of $tm\!f$-modules. For each $k \in \{0,1\}$ the group

$$\begin{aligned} \pi _{24k+3}(\Gamma _{(3,B)} N) \cong \pi _{-24(2-k)+1}(I N) \cong {{\,\mathrm{Hom}\,}}_{{\mathbb {Z}}_3}(\pi _{24(2-k)-1}(N), {\mathbb {Q}}_3/{\mathbb {Z}}_3) \end{aligned}$$

is trivial, since $\pi _{24(2-k)-1}(N) = 0$. Hence the group $\langle \nu _k\rangle \cong {\mathbb {Z}}/3$ in degree $24k+3$ of $\Gamma _{(3, B)} N_* = \Gamma _B N_*$ cannot survive to $E_\infty$ in the local cohomology spectral sequence

$$\begin{aligned} E_2^{s,t} = H^s_{(3, B)}(N_*)_t \Longrightarrow _s \pi _{t-s}(\Gamma _{(3, B)} N) \cong \pi _{t-s}(\Sigma ^{50} I N). \end{aligned}$$

Since $\nu _k$ is B-torsion, it follows that $d_2$ maps $\langle \nu _k\rangle$ isomorphically to the subgroup of ${\mathbb {Q}}_3/{\mathbb {Z}}_3\{C_k/B\}$ that is generated by $C_k/3 B$. This translates to a $d_2$-differential in the local cohomology spectral sequence for $\Gamma _{(3, B, H)} tm\!f$ from $\nu _k/H$ in bidegree $(t-s,s) = (24k-70, 1)$ to $C_k/3 B H$ in bidegree $(t-s,s) = (24k-71,3)$, together with its multiples by all negative powers of H. The previously known $\nu$-extensions in $\pi _*(N)$ and $\pi _*(I N)$ are also present in $\pi _*(\Gamma _{(3, B)} N)$ and in $\pi _*(\Gamma _{(3, B, H)} tm\!f)$, with the appropriate degree shifts, and those that increase the local cohomology filtration degree are displayed in red. $\square$

Remark 8.22

Let $\Theta N_* \subset \Gamma _B N_*$ be the part of the B-power torsion in $N_*$ that is not in degrees $* \equiv 3 \mod 24$, omitting ${\mathbb {Z}}/3\{\nu \}$ and ${\mathbb {Z}}/3\{\nu _1\}$ from Table 4. Likewise, let $\Theta \pi _*(tm\!f)$ be the part of $\Gamma _B \pi _*(tm\!f)$ that is not in degrees $* \equiv 3 \mod 24$, which equals the image of the (3, B)-local cohomology spectral sequence edge homomorphism $\pi _*(\Gamma _{(3, B)} tm\!f) \rightarrow \Gamma _B \pi _*(tm\!f)$.

Mahowald conjectured that the image of the 3-complete $tm\!f$-Hurewicz homomorphism $\pi _*(S) \rightarrow \pi _*(tm\!f)$ is the direct sum of ${\mathbb {Z}}$ in degree 0, the group ${\mathbb {Z}}/3\{\nu \}$ in degree 3, and the 72-periodic groups $\Theta \pi _*(tm\!f) \cong \Theta N_* \otimes {\mathbb {Z}}[H]$. This was proved for degrees $n < 154$ in [15,  Prop. 13.29].

8.5 Charts

Figure 3 shows $N_* \cong \pi _*(N)$, $\pi _*(\Gamma _B N)$ and $\pi _*(\Sigma ^{171} I_{{\mathbb {Z}}_2} N)$ in the range $-9 \le * \le 192$, visible as three horizontal wedges. The vertical direction has no intrinsic meaning. Circled numbers represent finite cyclic groups of that order, squares represent infinite cyclic groups, and each ellipse containing ‘22’ represents a Klein Vierergruppe. Horizontal dashed lines show multiplication by B, which extends indefinitely to the right in the upper wedge, and indefinitely to the left in the middle and lower wedges. Thick vertical lines indicate additive extensions, by which a square and a circle combine to an infinite cyclic group. The passage from the upper to the middle wedge is given by taking the homotopy fibre of the localisation map $\gamma : N \rightarrow N[1/B]$, leaving the B-power torsion (shown in red) in place and replacing copies of ${\mathbb {Z}}[B]$ or ${\mathbb {Z}}/2[B]$ (shown in blue) by desuspended copies of ${\mathbb {Z}}[B]/B^\infty$ or ${\mathbb {Z}}/2[B]/B^\infty$ (shown in green), respectively. The passage from the upper to the lower wedge takes the torsion-free part of $\pi _*(N)$ to its linear dual in degree $171-*$, and takes the torsion in $\pi _*(N)$ to its Pontryagin dual in degree $170-*$. The local cohomology theorem asserts that the middle and lower wedges are isomorphic. Note in particular how this is achieved in degrees $* \equiv -1, 3 \mod 24$.

Figure 4 shows the $E_\infty$-term of the mod 2 Adams spectral sequence

$$\begin{aligned} E_2^{s,t} = {{\,\mathrm{Ext}\,}}_{A(2)}^{s,t}({\mathbb {F}}_2, {\mathbb {F}}_2) \Longrightarrow _s \pi _{t-s}(tm\!f) \end{aligned}$$

for $0 \le t-s \le 192$, together with all hidden 2-, $\eta$- and $\nu$-extensions in this range. There is also a more subtle multiplicative relation in degree 105, cf. the proof of Theorem 8.8. The vertical coordinate gives the Adams filtration s. The B-power torsion classes are shown in red, and selected product factorisations in terms of the algebra indecomposables in $\pi _*(tm\!f)$ are shown. The B-periodic classes are shown in black, and usually only the ${\mathbb {Z}}[B]$-module generators are labelled. The ${\mathbb {Z}}[B]$-submodule generated by the classes in degrees $0 \le * < 192$ defines the basic block $N_*$, which repeats M-periodically, so that $\pi _*(tm\!f) \cong N_* \otimes {\mathbb {Z}}[M]$. Note how the additive structure of $N_*$ also appears in the upper wedge of Fig. 3.

Fig. 3

Homotopy of the basic block N of $tm\!f$ at $p=2$, of its B-local cohomology, and of its shifted Anderson dual

Fig. 4

$\pi _*(tm\!f)$ at $p=2$ for $0 \le * \le 192$

Figures 5 and 6 show the collapsing local cohomology spectral sequence for $\Gamma _B N$, in the range $-20 \le t-s \le 172$, broken into four sections. In each section the lower row shows $H^0_{(B)}(N_*) = \Gamma _B N_*$, while the upper row shows $H^1_{(B)}(N_*) = N_*/B^\infty$ shifted one unit to the left. Multiplication by 2, $\eta$, $\nu$ and B is shown by lines increasing the topological degree by 0, 1, 3 and 8, respectively. The dotted arrows coming from the left indicate classes that are infinitely divisible by B. Multiplications that connect the lower and upper rows increase the local cohomology filtration, hence are hidden, and are shown in red. The additive extensions in degrees $* \equiv 3 \mod 24$ are also carried over to the central wedge of Fig. 3.

It may be easiest to study these charts by starting in high degrees and descending from there. The top terms in $N_*$ that are not B-divisible are ${\mathbb {Z}}_2\{C_7\}$, ${\mathbb {Z}}/2\{\eta ^2 B_7\}$, ${\mathbb {Z}}/2\{\eta B_7\}$ and ${\mathbb {Z}}/8\{B_7\}$ in degrees 180 and 178 to 176, while the topmost B-power torsion in $N_*$ is ${\mathbb {Z}}/2\{\nu \nu _6 \kappa \}$ in degree 164. These contribute ${\mathbb {Z}}_2\{C_7/B\}$, ${\mathbb {Z}}/2\{\eta ^2 B_7/B\}$, ${\mathbb {Z}}/2\{\eta B_7/B\}$ and ${\mathbb {Z}}/8\{B_7/B\}$ to $N_*/B^\infty$ in internal degrees 172 and 170 to 168, shifted to topological degrees 171 and 169 to 167 in $\pi _*(\Gamma _B N)$, together with ${\mathbb {Z}}/2\{\nu \nu _6 \kappa \}$ in degree 164 of $\Gamma _B N_*$ and $\pi _*(\Gamma _B N)$. In the Anderson dual, the bottom term ${\mathbb {Z}}_2\{1\}$ of $\pi _*(N)$ contributes a copy of ${\mathbb {Z}}_2$ in degree 171 of $\pi _*(\Sigma ^{171} I_{{\mathbb {Z}}_2} N)$, while the terms ${\mathbb {Z}}/2\{\eta \}$, ${\mathbb {Z}}/2\{\eta ^2\}$, ${\mathbb {Z}}/8\{\nu \}$ and ${\mathbb {Z}}/2\{\nu ^2\}$ contribute the groups ${\mathbb {Z}}/2$, ${\mathbb {Z}}/2$, ${\mathbb {Z}}/8$ and ${\mathbb {Z}}/2$ in degrees 169 to 167 and 164. The duality theorem matches these groups isomorphically.

Fig. 5

$E_2^{s,t} = H^s_{(B)}(N_*)_t \Longrightarrow _s \pi _{t-s}(\Gamma _B N)$ at $p=2$

Fig. 6

$E_2^{s,t} = H^s_{(B)}(N_*)_t \Longrightarrow _s \pi _{t-s}(\Gamma _B N)$ at $p=2$

Fig. 7

$E_2^{s,t} = H^s_{(2, B)}(N_*)_t \Longrightarrow _s \pi _{t-s}(\Gamma _{(2, B)} N)$

Fig. 8

$E_2^{s,t} = H^s_{(2, B)}(N_*)_t \Longrightarrow _s \pi _{t-s}(\Gamma _{(2, B)} N)$

Fig. 9

$E_2^{s,t} = H^s_{(2, B)}(N_*)_t \Longrightarrow _s \pi _{t-s}(\Gamma _{(2, B)} N)$

Fig. 10

$E_2^{s,t} = H^s_{(2, B)}(N_*)_t \Longrightarrow _s \pi _{t-s}(\Gamma _{(2, B)} N)$

Fig. 11

Homotopy of the basic block N of $tm\!f$ at $p=3$, of its B-local cohomology, and of its shifted Anderson dual

Fig. 12

$\pi _*(tm\!f)$ at $p=3$ for $0 \le * \le 72$

Fig. 13

$E_2^{s,t} = H^s_{(B)}(N_*)_t \Longrightarrow _s \pi _{t-s}(\Gamma _B N)$ at $p=3$

Fig. 14

$E_2^{s,t} = H^s_{(3, B)}(N_*)_t \Longrightarrow _s \pi _{t-s}(\Gamma _{(3, B)} N)$

Figures 7, 8, 9 and 10 show the local cohomology spectral sequence for $\Gamma _{(2, B)} N$, in the range $-20 \le t-s \le 172$. In each figure the lower row shows $H^0_{(2, B)}(N_*) = \Gamma _B N_*$, the middle row shows $H^1_{(2, B)}(N_*) = \Gamma _2(N_*/B^\infty )$ shifted one unit to the left, and the upper row shows $H^2_{(2, B)}(N_*) = N_*/(2^\infty , B^\infty )$ shifted two units to the left. There are nonzero $d_2$-differentials from topological degrees $* \equiv 3 \mod 24$, leaving $E_3 = E_\infty$. Multiplications by 2, $\eta$, $\nu$ and B, infinitely B-divisible towers, and hidden extensions, are shown as in the previous figures. Note how the abutment $\pi _*(\Gamma _{(2, B)} N)$ is Pontryagin dual to $\pi _{170-*}(N)$.

The charts for $p=3$ follow the same conventions as for $p=2$.