1 Introduction↩︎

The aim of this article is to compute the tri-graded cohomology of \(\mathbb{R}\)-motivic \(\mathcal{A}(2)\), meaning \(\text{Ext}_{\mathcal{A}(2)}(\mathbb{M}_2, \mathbb{M}_2)\). Here, \(\mathcal{A}(2)\) is dual¹ to the subalgebra of the \(\mathbb{R}\)-motivic mod \(2\) Steenrod algebra generated by \(\text{Sq}^1\), \(\text{Sq}^2\), and \(\text{Sq}^4\), and \(\mathbb{M}_2 = \mathbb{F}_2[\rho, \tau]\) is the \(\mathbb{R}\)-motivic homology of a point with \(\mathbb{F}_2\)-coefficients.

Recent breakthroughs in stable homotopy theory [1] [2] have been achieved by making computations in motivic homotopy theory. In [2] the authors employ the \(\mathbb{C}\)-motivic modular forms spectrum \(\textit{mmf}^{\,\,\mathbb{C}}\) as constructed in [3] and use its \(\mathbb{C}\)-motivic stable homotopy groups to approximate the \(\mathbb{C}\)-motivic stable homotopy groups of spheres. One way to compute the stable homotopy groups of \(\textit{mmf}^{\,\,\mathbb{C}}\) is by running the \(\mathbb{C}\)-motivic Adams spectral sequence [4]. The input to this spectral sequence is the cohomology of \(\mathbb{C}\)-motivic \(\mathcal{A}(2)\), as computed in [1]. The logical next step that we take here is to mimic these computations over \(\mathbb{R}\). If we were to assume the existence of an \(\mathbb{R}\)-motivic modular forms spectrum \(\textit{mmf}^{\,\,\mathbb{R}}\), then we could follow the very same program and eventually further our knowledge of the classical stable homotopy groups of spheres. We should emphasize that, as of writing this article, an \(\mathbb{R}\)-motivic modular forms spectrum has not yet been constructed. However, we believe that our computation should serve as further motivation to do so.

As we will see, the cohomology of \(\mathbb{R}\)-motivic \(\mathcal{A}(2)\) has an extremely rich and complicated structure. To compare, the cohomology of classical \(\mathcal{A}(2)^\text{cl}\) has a presentation as a bigraded \(\mathbb{F}_2\)-algebra with \(13\) generators, see e.g. [5]. The cohomology of \(\mathbb{C}\)-motivic \(\mathcal{A}(2)^\mathbb{C}\) can be given a presentation with \(16\) generators [1]². In contrast, table 17 shows that the cohomology of \(\mathbb{R}\)-motivic \(\mathcal{A}(2)\) has \(56\) generators.

We should mention previous work on the cohomology of the \(\mathbb{R}\)-motivic Steenrod algebra: In [6] the cohomology of \(\mathcal{A}(1)\) is computed in its entirety. In [7] and [8] the cohomology of \(\mathcal{A}\) is computed in a range of degrees.

1.1 The \(\rho\)-Bockstein spectral sequence↩︎

We will use the \(\rho\)-Bockstein spectral sequence to compute the cohomology of \(\mathbb{R}\)-motivic \(\mathcal{A}(2)\). This type of spectral sequence has been employed many times before [6] [7] [8] to compute the cohomology of \(\mathbb{R}\)-motivic \(\mathcal{A}(1)\) and \(\mathcal{A}\). To provide a short derivation of this spectral sequence, we can filter \(\mathcal{A}(2)\) by powers of the ideal generated by an element called \(\rho\). This extends to a filtration on the cobar complex which computes the cohomology of \(\mathcal{A}(2)\). The spectral sequence associated to this filtration is called the \(\rho\)-Bockstein spectral sequence. Its signature is \[E_1^{s,f,w} = \text{Ext}_{\mathcal{A}(2)^{\mathbb{C}}}^{s,f,w}(\mathbb{M}_2^{\mathbb{C}},\mathbb{M}_2^{\mathbb{C}})[\rho] \implies \text{Ext}_{\mathcal{A}(2)}^{s,f,w}(\mathbb{M}_2,\mathbb{M}_2),\] i.e. the input is the tri-graded cohomology of \(\mathbb{C}\)-motivic \(\mathcal{A}(2)^{\mathbb{C}}\) with a variable \(\rho\) freely adjoined, and the output is the tri-graded cohomology of \(\mathbb{R}\)-motivic \(\mathcal{A}(2)\).

Our task is to find the differentials in this spectral sequence. We will use two methods:

• \(\rho\)-localization: This idea was first introduced in [7] to study the cohomology of \(\mathbb{R}\)-motivic \(\mathcal{A}\). In our case, localizing the Hopf algebroid \(\mathcal{A}(2)\) with respect to \(\rho\) leads to a splitting. This makes it possible to compute the \(\rho\)-localized cohomology of \(\mathcal{A}(2)\) before running the \(\rho\)-Bockstein spectral sequence. It turns out that the \(\rho\)-localized cohomology of \(\mathcal{A}(2)\) is isomorphic to a shifted version of the cohomology of classical \(\mathcal{A}(1)^{\text{cl}}\), together with a freely adjoined generator \(\tau^8\).

• Internal coweight method: This method was already used in [8] for \(\mathbb{R}\)-motivic \(\mathcal{A}\). The idea is to fix a specific linear combination of degrees, the so-called internal coweight. The \(\rho\)-Bockstein differentials as well as \(\rho\)-multiplication preserve the internal coweight by design. Therefore, the \(\rho\)-Bockstein spectral sequence splits into smaller spectral sequences, each indexed by an internal coweight. These smaller spectral sequences can often be solved individually, for example by appealing to the \(\rho\)-localization.

Due to our knowledge of the \(\rho\)-localization of the cohomology of \(\mathcal{A}(2)\) we will find that the \(\rho\)-Bockstein spectral sequence ends with \(E_{11} \cong E_\infty\), or in other words the longest differential is a \(d_{10}\). This is not apparent from the description of the \(\rho\)-localization as given in Corollary 8, but it is apparent once we have computed \(E_{11}\) and compare it with the \(\rho\)-localization.

After having computed \(E_\infty\) we need to consider hidden extensions. The multiplicative structure of the \(E_\infty\)-page only determines the multiplicative structure on the cohomology of \(\mathcal{A}(2)\) up to higher \(\rho\)-filtration. Due to the fact that we have \(56\) indecomposables on \(E_\infty\), finding all hidden extensions is a daunting task. We will restrict ourselves to only a few indecomposables and describe hidden extensions by them. In particular, we will not give an explicit presentation for the cohomology of \(\mathcal{A}(2)\). We do have all of the indecomposables as they coincide with those of \(E_\infty\) for formal reasons, see table 17. However, giving all relations involves solving all hidden extensions. While the author believes it to be possible to resolve most hidden extensions using the methods of section 6, it would take a prohibitive amount of time.

Two interesting hidden extensions are \(\tau^8 \cdot h_1^4 = \rho^4 \tau^4 P\), and \(\tau^8 \cdot \rho^6 g^2 = \rho^{14} \Delta^2\). For one, the elements involved in these relations are integral to the computation of the Adams spectral sequence for \(\textit{tmf}\). The element \(h_1\) classically represents the Hopf fibration \(\eta\). The elements \(P\), \(g\), and \(\Delta^2\) govern the structure of the Adams spectral sequence for \(\textit{tmf}\), see [9]³. Secondly, these examples show that the existence of the elements \(\tau\) and \(\rho\) in \(\mathbb{R}\)-motivic homotopy theory leads to relations in the cohomology of \(\mathcal{A}(2)\) which have no classical, or even \(\mathbb{C}\)-motivic, analog. The first hidden extension \(\tau^8 \cdot h_1^4 = \rho^4 \tau^4 P\) will prove extremely useful when determining other hidden extensions. It implies that any element which has a non-trivial \(\rho^4 \tau^4 P\)-multiple on \(E_\infty\) also needs to support four \(h_1\)-multiplications, and each one of these \(h_1\)-multiples has to support a \(\tau^8\)-multiplication. Oftentimes, the targets of these multiplications are determined for degree reasons. If they turn out to be in higher \(\rho\)-filtration, then the multiplication in question is in fact a hidden extension. This happens quite frequently.

1.2 An Adams spectral sequence?↩︎

In the last section, we make the assumption that an \(\mathbb{R}\)-motivic modular forms spectrum exists, or more precisely that there exists an object in the \(\mathbb{R}\)-motivic homotopy category such that its mod \(2\) cohomology is \(\mathcal{A}//\mathcal{A}(2)\). By a standard change of rings argument the \(E_2\)-page of the Adams spectral sequence associated to this spectrum is the cohomology of \(\mathcal{A}(2)\). We compute the \(d_2\)-differentials on all indecomposables in this hypothetical Adams spectral sequence. The main method is a comparison to the Adams \(d_2\)-differentials for the \(\mathbb{C}\)-motivic modular forms spectrum. In principle, once we have the \(d_2\)-differentials on the indecomposables, all other \(d_2\)-differentials follow from the Leibniz rule. But since the Leibniz rule involves multiplication, computing the \(d_2\)-differentials of all elements of the Adams \(E_2\)-page means solving all hidden extensions. Therefore, we opt not to push the computation of the Adams spectral sequence further.

1.3 Charts↩︎

There is a sizable amount of charts attached to this article. They depict the \(\rho\)-Bockstein spectral sequence, as well as the \(\rho\)-localization and certain quotients of the cohomology of \(\mathcal{A}(2)\). Due to their size and number, these charts are not contained in this manuscript but instead can be found at [10]. Nevertheless, they are integral to our computation. More details on charts are in section 4.

1.4 Outline↩︎

We begin in section 2 by setting up the relevant notation and defining the algebraic objects of interest.

In section 3 we set up the \(\rho\)-Bockstein spectral sequence and prove some structural theorems. We also compute the \(\rho\)-localization of the cohomology of \(\mathcal{A}(2)\).

In section 4 we provide a guide on how to read the charts attached to this manuscript. This section serves as a reference and should only be consulted once a chart becomes relevant.

Section 5 contains a computation of all differentials in the \(\rho\)-Bockstein spectral sequence.

In section 6 we determine some hidden extensions on the \(\rho\)-Bockstein \(E_\infty\)-page. We are mainly concerned with hidden extensions by \(\tau^8\), \(h_0\), \(h_1\), and \(h_2\).

Section 7 contains a computation of a hypothetical Adams spectral sequence for an \(\mathbb{R}\)-motivic modular forms spectrum. We determine the Adams \(d_2\)-differentials on all indecomposables.

1.5 Future directions↩︎

There are several interesting open problems arising from the work carried out in this manuscript.

1.6 Acknowledgements↩︎

The author would like to thank Dan Isaksen for sharing this problem with him and for many helpful conversations and comments. The author would also like to thank Joey Beauvais-Feisthauer for creating the software (most notably the SeqSee package [18]) used for making the charts accompanying this manuscript.

2 Notation↩︎

Let \(\mathbb{M}_2\) denote the \(\mathbb{R}\)-motivic homology of a point with \(\mathbb{F}_2\) coefficients. Recall from [19] that \(\mathbb{M}_2 \cong \mathbb{F}_2[\rho, \tau]\) where the degree of \(\rho\) is \((-1, -1)\) and the degree of \(\tau\) is \((0, -1)\).

Let \(\mathcal{A}\) denote the \(\mathbb{R}\)-motivic dual Steenrod algebra. Recall from [20] that \(\mathcal{A}\) is described as an algebra by \[\mathcal{A} \cong \frac{\mathbb{M}_2[\tau_0, \tau_1, \dotsc, \xi_0, \xi_1, \dotsc]}{(\xi_0 = 1, \tau_k^2 = \tau \xi_{k+1} + \rho \tau_{k+1} + \rho \tau_0 \xi_{k+1})},\] where the degree of \(\tau_k\) is \((2^{k+1}-1, 2^k-1)\) and the degree of \(\xi_k\) is \((2^{k+1}-2, 2^k-1)\). Furthermore, \(\mathcal{A}\) has the structure of a Hopf algebroid via \[\eta_L(\tau) = \tau,~\,\eta_R(\tau) = \tau + \rho \tau_0,~\,\eta_L(\rho) = \eta_R(\rho) = \rho,\] \[\Delta(\tau_k) = \tau_k \otimes 1 + \sum \xi_{k-i}^{2^i} \otimes \tau_{i},\] \[\Delta(\xi_k) = \sum \xi_{k-i}^{2^i} \otimes \xi_{i}.\] Note that \(\Delta(\tau) = \tau \otimes 1\) and \(\Delta(\rho) = \rho \otimes 1\) are forced by the axioms of a Hopf algebroid.

Then \(\mathcal{A}(2)\) is defined to be the Hopf algebroid obtained by quotienting out the ideal \((\tau_3, \tau_4, \dotsc, \xi_1^4, \xi_2^2, \xi_3, \dotsc)\). This quotient is dual to the subalgebra of the \(\mathbb{R}\)-motivic Steenrod algebra generated by \(\text{Sq}^1\), \(\text{Sq}^2,\) and \(\text{Sq}^4\). As an algebra \(\mathcal{A}(2)\) has the structure \[\mathcal{A}(2) \cong \frac{\mathbb{M}_2[\tau_0, \tau_1, \tau_2, \xi_0, \xi_1, \xi_2]}{(\xi_0 = 1, \xi_1^4 = 0, \xi_2^2 = 0, \tau_k^2 = \tau \xi_{k+1} + \rho \tau_{k+1} + \rho \tau_0 \xi_{k+1})},\] and it is again a Hopf algebroid via the same formulas for the units and coproduct as for \(\mathcal{A}\). In the relations, treat \(\tau_3\) and \(\xi_3\) as zero.

We write a superscript \(\mathbb{C}\) for the same objects in the \(\mathbb{C}\)-motivic world, e.g. \(\mathbb{M}_2^\mathbb{C}\) is the \(\mathbb{C}\)-motivic homology of a point. Then all of the objects \(\mathbb{M}_2^\mathbb{C}\), \(\mathcal{A}^\mathbb{C}\), and \(\mathcal{A}(2)^\mathbb{C}\) can be obtained from the above formulas by setting \(\rho\) equal to \(0\).

We write a superscript \(\text{cl}\) for the same classical objects, e.g. \(\mathcal{A}^\text{cl}\) is the classical dual Steenrod algebra. The classical objects can be obtained from the above by setting \(\rho\) equal to \(0\) and \(\tau\) equal to \(1\).

3 The \(\rho\)-Bockstein spectral sequence↩︎

We recall some basics about the construction of the \(\rho\)-Bockstein spectral sequence. For more details see [6] or [7]. Take the cobar complex that computes the cohomology of \(\mathbb{R}\)-motivic \(\mathcal{A}(2)\). Filter it by powers of the ideal generated by \(\rho\). This gives rise to a spectral sequence. Because of \(\mathcal{A}(2)^{\mathbb{C}} \cong \mathcal{A}(2)/\rho\) and \(\mathbb{M}_2^{\mathbb{C}} \cong \mathbb{M}_2/\rho\) and the fact that \(\rho\)-multiplication is injective on \(\mathcal{A}(2)\) and \(\mathbb{M}_2\), the \(E_1\)-page of this spectral sequence can be identified with \(\text{Ext}_{\mathcal{A}(2)^{\mathbb{C}}}(\mathbb{M}_2^{\mathbb{C}},\mathbb{M}_2^{\mathbb{C}})[\rho]\). So the signature of the spectral sequence is \[E_1^{s,f,w} = \text{Ext}_{\mathcal{A}(2)^{\mathbb{C}}}^{s,f,w}(\mathbb{M}_2^{\mathbb{C}},\mathbb{M}_2^{\mathbb{C}})[\rho] \implies \text{Ext}_{\mathcal{A}(2)}^{s,f,w}(\mathbb{M}_2,\mathbb{M}_2).\]

Note that we do not explicitly record the grading arising from the \(\rho\)-filtration as another superscript. As a consequence, all \(d_r\)-differentials have degree \((-1, 1, 0)\). Note also that by definition of the \(\rho\)-Bockstein spectral sequence the \(d_r\)-differentials increase the \(\rho\)-filtration by \(r\). So for example if \(x \in \text{Ext}_{\mathcal{A}(2)^{\mathbb{C}}}^{s,f,w}(\mathbb{M}_2^{\mathbb{C}},\mathbb{M}_2^{\mathbb{C}})\) then \(d_r(x) = \rho^r y\) for some \(y \in \text{Ext}_{\mathcal{A}(2)^{\mathbb{C}}}^{s+r-1,f+1,w+r}(\mathbb{M}_2^{\mathbb{C}},\mathbb{M}_2^{\mathbb{C}})\). This \(y\) may of course very well be zero. Additionally each \(d_r\) satisfies a Leibniz rule with respect to the multiplication on the \(E_r\)-page. If \(a\) and \(b\) are elements on the \(E_r\)-page then \[d_r(a \cdot b) = a \cdot d_r(b) + d_r(a) \cdot b.\] Note that we are working modulo the prime \(2\) so we do not have to worry about signs.

We list some very important structural Propositions about this spectral sequence.

The second Proposition is a slight variant of [7].

From this point onward we will freely make use of the computation [1] of the cohomology of \(\mathcal{A}(2)^{\mathbb{C}}\), meaning \(\text{Ext}_{\mathcal{A}(2)^{\mathbb{C}}}(\mathbb{M}_2^{\mathbb{C}}, \mathbb{M}_2^{\mathbb{C}})\). In the time that has passed since that computation was carried out, there have been slight changes in notation. We will use the names of the generators as presented in [4]. The only changes are that \(\alpha\) became \(a\) and \(\nu\) became \(n\). We reproduce a table of these generators below in Table 1. The first column provides the name of the generator, the second column describes its stem \(s\), Adams filtration \(f\), and weight \(w\).

The relations that these generators satisfy are spelled out in [1]. Since the \(E_1\)-page of the \(\rho\)-Bockstein spectral sequence is isomorphic to \(\text{Ext}_{\mathcal{A}(2)^{\mathbb{C}}}(\mathbb{M}_2^{\mathbb{C}},\mathbb{M}_2^{\mathbb{C}})[\rho]\), the generators and relations referred to above also describe that \(E_1\)-page.

Next, we provide a presentation and a table of generators for the \(\rho\)-localized cohomology of \(\mathcal{A}(2)\). The generators \(h_1, h_2, u,\) and \(g\) listed below are the degree-shifted versions (according to Proposition 6) of the classical generators of the cohomology of \(\mathcal{A}(1)^{\text{cl}}\). Similarly, the relations are given by taking the relations from \(\mathcal{A}(1)^{\text{cl}}\) and substituting \(h_1, h_2, u,\) and \(g\) for the usual variables. A chart of this algebra is available at [10] in the folder Cohomology of A(2) rho-localized. For a guide on how to read it see section 4 and specifically section 4.2.

Just as in [7], we can find the \(\rho\)-Bockstein differentials on the powers of \(\tau\) by making cobar complex computations.

4 Charts↩︎

For a computation of this size, charts are indispensable. Due to the large number of charts needed, they will not be included in this manuscript. Instead, the charts can be found at [10].

This section is meant to serve as a reference. We chose to put it here to notify the reader that from now on charts will become relevant. In practice, this section can be skipped until the charts in question are needed.

4.1 Generalities↩︎

Let us first collect some general properties that hold true for all charts:

• The horizontal axis describes the stem \(s\),

• the vertical axis describes the Adams filtration \(f\).

Consequently, we have to plot \(\tau\)-multiples implicitly since \(\tau\) has both stem and Adams filtration equal to \(0\). Most often, we do the following:

• Colors indicate behavior with respect to \(\tau\)-multiplication. The exact meaning of each color depends on the individual chart and is discussed in the sections below.

Another important element is \(\rho\). By Proposition 5 we can work exclusively with \(\rho\)-torsion free classes until we reach \(E_{\infty}\). Therefore, we do the following:

• We try to avoid plotting \(\rho\)-multiples whenever possible.

As a consequence, the only charts where \(\rho\)-multiples are plotted explicitly are the \(\rho\)-power-torsion charts and the charts for the \(\rho\)-free quotient of the cohomology of \(\mathcal{A}(2)\).

For each chart there are two versions available: A pdf version and an html version. The html version always provides more information. Hovering over a dot in an html chart shows its name, degrees and other data. In particular, we can quickly find the weight of an element in the chart, assuming we know its stem and Adams filtration. The drawback of the html charts is that the name of each element is not directly printed onto the chart, so it might be harder to find a desired element at first glance. The pdf version has the name of each element⁴ printed on the chart. That can make it easier to find a desired element, as long as there aren’t too many elements in the same stem and Adams filtration. In summary, the pdf charts are simpler to use when unfamiliar with the names of elements and their degrees, but the html charts are a more efficient tool because the weights are integral to our computation. Ultimately, which version to use depends on the specific chart and personal preference.

We will now give detailed instructions on how to read each set of charts.

4.2 Cohomology of \(\mathcal{A}(2)\) \(\rho\)-localized↩︎

This chart shows the \(\rho\)-localization of the cohomology of \(\mathbb{R}\)-motivic \(\mathcal{A}(2)\) as computed in Corollary 8. It contains the following information:

• A dot represents the module \(\mathbb{F}_2[\rho^{\pm 1}, \tau^8]\),

• lines of slope \(1\) represent \(h_1\)-multiplication, lines of slope \(1/3\) represent \(h_2\)-multiplication,

• an arrow of slope \(1\) indicates that a given dot supports infinitely many \(h_1\)-multiplications.

4.3 \(\rho\)-Bockstein \(E_r\)-pages \(\rho\)-free quotient↩︎

By Proposition 5 there are only \(\rho\)-Bockstein differentials between \(\rho\)-torsion free classes. As such, we can work in the \(\rho\)-free quotient \(E_r/(\text{\rho-power-torsion})\). Since every element in this quotient is \(\rho\)-torsion free, we leave out all \(\rho\)-multiples in the charts.

Generally, the following will hold true for all pages:

• Dots represent a quotient of \(\mathbb{F}_2[\rho, \tau^n]\), where the power \(n\) depends on the page,

• the color of a dot determines the \(\tau^n\)-torsion of that quotient,

• vertical lines represent \(h_0\)-multiplication, lines of slope \(1\) represent \(h_1\)-multiplication, lines of slope \(1/3\) represent \(h_2\)-multiplication,

• the color of a line simply corresponds to the color of its target dot, except when the line is colored magenta⁵, which indicates that the given multiplication hits the \(\tau^n\)-multiple of the target dot,

• two lines originating from the same dot implies that the \(h_i\)-multiplication in question hits the sum of those two targets,

• arrows indicate that a given dot supports infinitely many multiplications of the type corresponding to the slope of the arrow, e.g. a vertical arrow indicates infinitely many \(h_0\)-multiples, an arrow of slope \(1\) indicates infinitely many \(h_1\)-multiples,

• the color of an arrow indicates the \(\tau^n\)-torsion of the implied multiples.

4.3.1 \(E_1\)↩︎

Recall that by construction the \(\rho\)-Bockstein \(E_1\)-page is \(\text{Ext}_{\mathcal{A}(2)^{\mathbb{C}}}(\mathbb{M}_2^{\mathbb{C}},\mathbb{M}_2^{\mathbb{C}})[\rho]\), which is entirely \(\rho\)-torsion free. Hence the \(\rho\)-free quotient of \(E_1\) is just \(E_1\) itself. Since we avoid plotting \(\rho\)-multiples, our chart will look like \(\text{Ext}_{\mathcal{A}(2)^{\mathbb{C}}}(\mathbb{M}_2^{\mathbb{C}},\mathbb{M}_2^{\mathbb{C}})\). Charts of this already exist in the literature, e.g. [4]. Our chart will be modeled exactly after this cited one⁶. We provide a table on how to interpret the colors in our chart.

4.3.2 \(E_2\)↩︎

By Lemma 12 there is no element named \(\tau\) on \(E_2\) because \(\tau\) supports a \(d_1\). However, we do still have \(\tau^2\). So all dots will now represent \(\mathbb{F}_2[\rho, \tau^2]\)-modules. Sometimes, this leads to more dots at some coordinate in the chart than there were before. That is because not every \(\tau\)-multiple on \(E_1\) has to support a differential. For example at the coordinates \((1, 1)\) there are two dots, namely \(h_1\) and \(\tau h_1\). The first one represents the "even exponent \(\tau\)-multiples" of \(h_1\) and the second one the "odd exponent \(\tau\)-multiples" of \(h_1\). Of course one should be careful when saying this as there is no \(\tau\)-multiplication on \(E_2\), only \(\tau^2\)-multiplication.

We will use the same colors as for \(E_1\), but replace \(\tau\) by \(\tau^2\) everywhere. For example, on \(E_1\) red was \(\tau\)-torsion, so on \(E_2\) red now stands for \(\tau^2\)-torsion. Also, a magenta line on \(E_2\) now indicates that the corresponding multiplication hits the \(\tau^2\)-multiple of the target dot.

4.3.3 \(E_3\) and \(E_4\)↩︎

The smallest power of \(\tau\) that exists on these pages is \(\tau^4\). We adjust the meaning of the colors just as we did when changing from \(E_1\) to \(E_2\). A magenta line now indicates that the corresponding multiplication hits the \(\tau^4\)-multiple of the target dot.

4.3.4 \(E_r\) for \(r \geq 5\)↩︎

The smallest power of \(\tau\) that exists on these pages is \(\tau^8\).

4.4 \(\rho\)-Bockstein internal coweight charts↩︎

The internal coweight is defined in Definition 17. We provide charts for fixed values of the internal coweight on various \(E_r\)-page \(\rho\)-free quotients. These charts cover all values relevant to the computations of \(\rho\)-Bockstein differentials as carried out in section 5. It should be noted that these charts can be re-used on higher pages. For example on \(E_3\) we will refer to internal coweight \(7\), so then we can re-use the \(E_2\)-chart for internal coweight \(7\).

The charts contain the following information:

• A dot represents the module \(\mathbb{F}_2[\rho]\),

• the color of a dot follows the coloring scheme of the respective \(E_r\)-page the chart is based on,⁷

• an arrow pointing to the left indicates that the given dot is not involved in any differentials, i.e. it represents a \(\rho\)-torsion free class on \(E_\infty\),

• a cyan line⁸ connecting two dots indicates a \(\rho\)-Bockstein differential. The length of the line (or equivalently the relative position of the two dots it connects) implies which page the differential occurs on. More precisely, a line of slope \(1/(r-1)\) represents a \(d_r\)-differential from the dot in lower Adams filtration to the \(\rho^r\)-multiple of the dot in higher Adams filtration.

Some charts show all differentials, e.g. figure 2, and some charts do not show differentials, e.g. figure 1. We never assume knowledge of a completed chart as in figure 2. Instead, if an internal coweight chart is needed in section 5 we will explicitly determine the differentials for the relevant elements on that chart.

4.5 Cohomology of \(\mathcal{A}(2)\) \(\rho\)-free quotient↩︎

We provide two charts which combine into the \(\rho\)-free quotient of the cohomology of \(\mathbb{R}\)-motivic \(\mathcal{A}(2)\), i.e. \(\text{Ext}_{\mathcal{A}(2)}(\mathbb{M}_2,\mathbb{M}_2)/(\text{\rho-power-torsion})\). The first chart shows elements of coweight \(s - w\) congruent to \(0\), \(1\), and \(2\) mod \(8\). The second chart shows elements of coweight congruent to \(4\) mod \(8\). In particular, there are no \(\rho\)-torsion free elements in other coweights⁹. The main reason we use coweights here is because \(h_1\)- and \(\rho\)-multiplication respect coweights. We consider them modulo \(8\) so that \(\tau^8\)-multiples are on the same chart. The first chart combines the three coweights \(0\), \(1\), and \(2\) so that all \(h_2\)-multiples are on the same chart.

The charts contain the following information:

• A dot represents the module \(\mathbb{F}_2[\tau^8]\),

• lines of slope \(1\) represent \(h_1\)-multiplication, lines of slope \(1/3\) represent \(h_2\)-multiplication, horizontal lines represent \(\rho\)-multiplication,

• horizontal arrows indicate that a given dot supports infinitely many \(\rho\)-multiplications,

• dotted lines indicate that a multiplication comes from a hidden extension,

• magenta lines indicate that a multiplication hits the \(\tau^8\)-multiple of the target dot. All magenta lines are consequences of hidden \(\tau^8\)-extensions.

There are multiplications that are not hidden on the generator of a dot itself, but on the \(\tau^8\)-multiple of that generator. We have chosen not to introduce a separate symbol for this phenomenon, and simply draw a solid line. These types of hidden multiplications are always consequences of other hidden extensions. For example, the \(h_1\)-multiplication from \(h_1^3\) to \(h_1^4\) is not hidden. However, the \(h_1\)-multiplication on \(\tau^8 \cdot h_1^3\) is hidden with value \(\rho^4 \cdot \tau^4 P\). It is a consequence of the horizontal magenta line connecting the dot \(\rho^3 \cdot \tau^4 P\) to the dot \(h_1^4\), indicating a hidden \(\tau^8\)-extension from \(h_1^4\) to \(\rho^4 \cdot \tau^4 P\).

4.6 \(\rho\)-Bockstein \(E_\infty\)-page \(\rho\)-power-torsion↩︎

There are eight charts combining into the \(\rho\)-power-torsion subalgebra of the \(E_\infty\)-page of the \(\rho\)-Bockstein spectral sequence. The charts are divided into coweights \(s - w\) mod \(8\). The main reason we use coweights here is because \(h_1\)- and \(\rho\)-multiplication respect coweights. We consider them modulo \(8\) so that \(\tau^8\)-multiples are on the same chart. These charts do not show any hidden extensions, although some are discussed in section 6.4.

The charts contain the following information:

• A dot represents a quotient of \(\mathbb{F}_2[\tau^8]\),

• the color of a dot determines the \(\tau^8\)-torsion of that quotient, see table 7,

• vertical lines represent \(h_0\)-multiplication, lines of slope \(1\) represent \(h_1\)-multiplication, lines of slope \(1/3\) represent \(h_2\)-multiplication, horizontal lines represent \(\rho\)-multiplication,

• the color of a line simply corresponds to the color of its target dot,

• arrows indicate that a given dot supports infinitely many multiplications of the type corresponding to the slope of the arrow,

• the color of an arrow indicates the \(\tau^8\)-torsion of the implied multiples.

5 \(\rho\)-Bockstein differentials↩︎

In this section, we compute all differentials in the \(\rho\)-Bockstein spectral sequence converging to the cohomology of \(\mathbb{R}\)-motivic \(\mathcal{A}(2)\). There are charts provided at [10] which show the \(\rho\)-free quotient \(E_r/(\text{\rho-power-torsion})\) of each \(E_r\)-page of the \(\rho\)-Bockstein spectral sequence. For a guide on how to read these charts see section 4, and specifically section 4.3.

For the reader’s convenience and for future reference, we begin with a table collecting all differentials in the \(\rho\)-Bockstein spectral sequence on indecomposable elements. From left to right, the columns of table 8 describe

1. the name of the indecomposable supporting a differential,

2. the degree of that indecomposable,

3. the name of the target of the differential,

4. the index of the page where the differential occurs,

5. where the proof of that differential can be found.

All elements listed are indecomposables on the respective \(E_r\)-pages where they support a \(d_r\)-differential. Unlisted indecomposables do not support a differential. In particular, by using the Leibniz rule all other differentials in the spectral sequence can be deduced from the differentials in this table. The rest of this section will then be dedicated to proving the existence of these differentials.

By definition of the \(\rho\)-Bockstein spectral sequence, the differentials \(d_r\) increase the \(\rho\)-filtration by \(r\). Because of Proposition 5 all indecomposables in the \(\rho\)-Bockstein spectral sequence have \(\rho\)-filtration zero. As a consequence, the values of their \(d_r\)-differentials will always contain \(\rho^r\). This is also apparent when comparing the third and fourth column of table 8. Based on this observation, we will make the following notational convention.

5.1 Chart guide↩︎

The relevant charts from [10] for this section can be found in the folders Rho-Bockstein E_r-pages rho-free quotient and Rho-Bockstein internal coweight charts.

The charts in the first-mentioned folder depict the \(\rho\)-free quotient of the \(\rho\)-Bockstein spectral sequence \(E_r\)-page. They aid the reader throughout the different sections by providing an overview of the elements still under consideration for differentials.

The charts in the second folder will be crucial to many arguments in this section. How to use these charts in conjunction with the text is spelled out in detail in section 5.3 for the case of internal coweight \(5\). The internal coweight is defined in Definition 17. Whenever a proof in this section mentions an internal coweight, we encourage the reader to open an associated internal coweight chart. Since each internal coweight chart is sparse, we suggest using the pdf version. To give an example of where to use a chart, in Lemma 21 we want to look at internal coweight \(6\). Since Lemma 21 is located in the \(d_2\)-differentials section, we can use the chart titled Rho-Bockstein E_2 rho-free quotient s+f-w=6. This chart will aid the reader in following the proof of Lemma 21. Note that sometimes we will re-use internal coweight charts from lower pages. For example in Lemma 42 we consider internal coweight \(14\), and we can use the chart Rho-Bockstein E_4 rho-free quotient s+f-w=14 for that purpose. Since Lemma 42 is in the \(d_5\)-differentials section, we can ignore elements in the given chart that are involved in \(d_4\)-differentials. For every internal coweight mentioned in the following sections there is an associated chart.

5.2 \(d_1\)-differentials↩︎

We continue by giving proofs of the differentials one page at a time. We will freely make use of [1], which provides generators and relations for \(\text{Ext}_{\mathcal{A}(2)^{\mathbb{C}}}(\mathbb{M}_2^{\mathbb{C}},\mathbb{M}_2^{\mathbb{C}})\), i.e. for the cohomology of \(\mathbb{C}\)-motivic \(\mathcal{A}(2)^{\mathbb{C}}\). We also recalled these generators earlier in table 1.

We start with table 9 collecting all indecomposables on the \(E_1\)-page, together with the values of their \(d_1\)-differentials. The \(d_1\)-differentials on all other elements are then consequences of the Leibniz rule. An empty entry in the \(d_1\)-column of table 9 implies that \(d_1\) applied to that indecomposable is zero. We leave out the indecomposable \(\rho\) as we know it is a permanent cycle since it is a cycle in the cobar complex. Similarly, following Notation 14 we will also leave out the factor \(\rho\) everywhere. As an example, consider the element \(\tau\) in table 9. The \(d_1\)-column entry is \(h_0\), and that tells us \(d_1(\tau) = \rho h_0\).

5.3 The internal coweight method↩︎

Before moving on to the next page, we will discuss another extremely potent method for deducing differentials in the \(\rho\)-Bockstein spectral sequence. This method is essentially the same as [8], we’re just giving it a name and make separate charts for it. Before going into details, we will collect all the different steps and ideas involved in this method:

1. By Proposition 5 it suffices to consider the \(\rho\)-free quotient \(E_r/(\text{\rho-power-torsion})\) of each \(E_r\)-page when computing \(\rho\)-Bockstein differentials.

2. The hyperplanes given by fixing a constant value for \(s + f - w\) on each \(E_r\)-page form additive sub-spectral sequences. This is because the degree of \(d_r\) is \((-1, 1, 0)\), so it preserves \(s + f - w\).

3. We already have complete knowledge of the amount of \(\mathbb{F}_2[\rho]\)-modules on the \(\rho\)-free quotient of \(E_\infty\). This follows from Corollary 8 and the fact that the kernel of \(\rho\)-localization is precisely the \(\rho\)-power-torsion elements. So the \(\rho\)-free quotient injects into the \(\rho\)-localization and in particular the \(\mathbb{F}_2[\rho]\)-modules on the \(\rho\)-free quotient of \(E_\infty\) correspond to the \(\mathbb{F}_2[\rho^{\pm 1}]\)-modules in the \(\rho\)-localization.

We will make some additional remarks: For the first point, it is worth noting that the \(\rho\)-free quotient of the \(E_r\)-page is obtained from the \(E_1\)-page by removing elements that either supported or were the target of a differential. For sources of differentials that is just the usual way a spectral sequence works, for targets of differentials that is because they are now \(\rho\)-power-torsion. For the second point, we choose \(s + f - w\) specifically so that \(\rho\)-multiplication also preserves this linear combination. This makes it easier to create charts, as we can have a dot represent an entire \(\mathbb{F}_2[\rho]\)-module. Note that multiplication by any other indecomposable does not preserve this linear combination. For the third point, we even know the \(f\)-values of the \(\mathbb{F}_2[\rho]\)-modules on the \(\rho\)-free quotient because \(\rho\)-multiplication preserves \(f\).

As we will have to refer to the linear combination \(s + f - w\) a lot, we give it a name.

To explain the name, the quantity \(s - w\) is typically referred to as coweight, and the quantity \(s + f\) describes the internal degree if one were to compute \(\text{Ext}\) via a projective resolution.

Calculating the internal coweight of each element in table 9 shows that every indecomposable of \(E_1\) (and hence of each \(E_r\)-page) has a non-negative internal coweight. More precisely, only \(\rho\) has internal coweight \(0\) and all other indecomposables have a strictly positive internal coweight.

We will illustrate the internal coweight method with an example in internal coweight \(5\).

In figure 1 the horizontal axis represents the stem \(s\) and the vertical axis represents the Adams filtration \(f\). Each dot represents a free \(\mathbb{F}_2[\rho]\)-module with internal coweight \(5\). The gray dots correspond to \(\tau^2\)-free classes whereas the red dot corresponds to a \(\tau^2\)-torsion class, but that is not relevant for the following arguments. By Corollary 8 we have a presentation of the \(\rho\)-localized abutment of the spectral sequence. It is easy to check that the only element there with internal coweight \(5\) is \(h_1^5\) (and its \(\rho\)-multiples). We can also find a dot with that label in the provided figure. We see that \(h_1^5\) cannot be involved in any differentials, as incoming differentials would have to come from filtration \(4\) and outgoing differentials would end up in filtration \(6\) and both of those are empty. That means \(h_1^5\) survives the \(\rho\)-Bockstein spectral sequence and thus represents a \(\rho\)-torsion free class at \(E_{\infty}\), namely the one that is called \(h_1^5\) in Corollary 8. Next consider the other four gray dots. If one of them is not involved in any differentials in our spectral sequence, then it will represent a \(\rho\)-torsion free class on \(E_\infty\). That \(\rho\)-torsion free class would map to a non-zero element in the \(\rho\)-localized cohomology of \(\mathcal{A}(2)\). But as we have seen above, there is only one element with that property in internal coweight \(5\), and that element is \(h_1^5\). So all four gray dots must be involved in some differentials. There is now exactly two things that can happen to each dot: Either it supports a differential so that it disappears before \(E_{\infty}\), or it gets hit by a differential and becomes \(\rho\)-power-torsion, hence zero in the \(\rho\)-localization. It is a quick and pleasant exercise to convince oneself that there is only one possible set of differentials. The solution is described in the next paragraph. The main point is that all \(d_r\)-differentials go one to the left and one up in the chart, but they also have to hit a \(\rho^r\)-multiple and \(\rho\)-multiplication moves one to the left.

For the readers’ convenience we include another chart of internal coweight \(5\) in figure 2.

This chart shows each class in the \(\rho\)-free quotient of the \(E_2\)-page in internal coweight \(5\), and describes their behavior in the \(\rho\)-Bockstein spectral sequence. The arrow on \(h_1^5\) means that it survives the \(\rho\)-Bockstein spectral sequence, it is not hit by a differential and it does not support a differential. A line connecting two dots implies that there is a differential from the class that is lower on the \(y\)-axis to the other one. The length of the line determines the index of the differential and hence which power of \(\rho\) is hit, but we will continue leaving out said power of \(\rho\) from the notation, cf. Notation 14. For example the line from \(\tau^3 h_1^2\) to \(\tau^2 h_1^3\) implies \(d_2(\tau^3 h_1^2) = \tau^2 h_1^3\), where we have left out the factor \(\rho^2\) on the right-hand side. To see that this differential has to occur, recall that \(\tau^2 h_1^3\) has to be involved in some differential as there cannot be any \(\rho\)-torsion free classes on \(E_\infty\) in internal coweight \(5\) and Adams filtration \(3\) by Corollary 8. Certainly \(\tau^2 h_1^3\) cannot support a differential as the target would have to be in Adams filtration \(4\). So some \(\rho\)-multiple of \(\tau^2 h_1^3\) has to get hit by a differential. The source of that differential must be in Adams filtration \(2\). A priori there are two elements there, \(\tau^3 h_1^2\) and \(\tau h_2^2\). But note that any \(d_r\) on \(\tau h_2^2\) would land at the coordinates \((5, 3)\) in this chart. Since the \(\rho\)-multiples of \(\tau^2 h_1^3\) live to its own left in the chart, we see that \(\tau h_2^2\) cannot hit them with a differential. Hence \(\tau^3 h_1^2\) must support a differential hitting a \(\rho\)-multiple of \(\tau^2 h_1^3\). Since \(d_r(\tau^3 h_1^2)\) is at the coordinates \((1, 3)\) and \(\tau^2 h_1^3\) is at the coordinates \((3, 3)\) and \(3 - 1 = 2\), we must have \(d_2(\tau^3 h_1^2) = \tau^2 h_1^3\). Now the other differential \(d_6(\tau^4 h_1) = \tau h_2^2\) is forced.

In a completed internal coweight chart as in figure 2, every dot falls into exactly one of the two following categories:

• There is a horizontal arrow pointing to its left, indicating that the class corresponding to this dot is not involved in any differentials in the \(\rho\)-Bockstein spectral sequence. Hence it is \(\rho\)-torsion free on \(E_\infty\). The arrow points to where its \(\rho\)-multiples are.

• There is a line of height \(1\) and positive slope connecting the given dot with exactly one other dot. This indicates that there is a differential from the class that is lower on the vertical axis to the other one. The index of the differential is implicitly given by the length of the line (or equivalently by the general position of the two dots connected by the line). More precisely, a line of slope \(1/(r-1)\) from \(x\) to \(y\) implies \(d_r(x) = \rho^r y\). Note that we only use the internal coweight method starting on \(E_2\) and higher pages, so there will never be vertical lines which would indicate \(d_1\)’s.

There are charts for all relevant internal coweights at [10]. For a guide on how to read them see section 4, and specifically section 4.4.

By playing the same game as we just did for other values of the internal coweight, we could already compute a lot of \(d_r\)-differentials for \(r > 2\) on \(E_2\). We will not do this. Instead, we will keep moving from page to page and only apply the internal coweight method when it is needed to prove existence or non-existence of some differential.

5.4 \(d_2\)-differentials↩︎

From this point onward we continue with describing the differentials page by page. Remember that we listed all non-zero differentials on indecomposables in table 8. Our goal is to prove that this list is both complete and correct.

Moving forward, we will not give tables of all indecomposables on each page as we did for \(E_1\). Instead, inside of each Proposition we will only list those indecomposables that could possibly support a differential for degree reasons. Note that all other differentials on each page can then be deduced via the Leibniz rule. We argue separately for each indecomposable whether or not the given differential occurs.

The tables inside of each Proposition have columns describing

1. the name of the indecomposable possibly supporting a differential,

2. the degree of said indecomposable,

3. the target of the differential (leaving out the factors of \(\rho\) as in table 9),

4. whether or not the given differential occurs,

5. where the proof can be found.

It will be convenient now and at later stages to refer to the \(\rho\)-Bockstein differentials hitting \(c\) and \(a\), so we state them here. As it becomes relevant now, we remind the reader of Notation 15. We also refer the reader back to the chart guide in section 5.1.

5.5 \(d_3\)-differentials↩︎

The next few Lemmas build up to the proof of \(d_3(\tau^3 n^2) = \tau^2 a g + \tau \Delta c\). We should note that in the following Lemma 25 we are making a one-time exception to Convention 18. As of now, \(\tau^2 a g\) could be hit by a \(d_3\) which would make the claimed \(d_6\) trivial. Note that this \(d_3\) is the only shorter differential that \(\tau^2 a g\) could be hit by. However, in the proof of Lemma 30 we will be able to lead the assumption \(d_3(\tau^3 n^2) = \tau^2 a g\) to a contradiction by using Lemma 25.

5.6 \(d_4\)-differentials↩︎

It will be convenient to refer to the \(d_6\)-differential hitting \(e\) many times, so we state it here first.

The next Lemma will show that \(\tau^4 P\) is a permanent cycle. Along the way we will analyze internal coweight \(12\) and prove the differential on \(\tau^2 n\). For future reference, we include another finding in the statement of the Lemma as well.

To determine the differential on \(\tau^4 \Delta h_1 h_1\) and for many other future differentials it will be convenient to refer to the differential that \(n\) is involved in, so we will state it here.

Along the way of the proof of the next differential \(d_4(\tau^4 \Delta c h_1) = a^3\), we will analyze internal coweight \(16\). For future reference, we include some of the findings in the statement of the Lemma.

5.7 \(d_5\)-differentials↩︎

For the differential on \(\tau^4 c\) we want to refer to a \(d_7\) hitting \(d\), so we state it first.

5.8 \(d_6\)-differentials↩︎

5.9 \(d_7\)-differentials↩︎

5.10 \(d_8\)-differentials↩︎

5.11 \(d_9\)-differentials↩︎

5.12 \(d_{10}\)-differentials↩︎

5.13 \(d_r\)-differentials for \(r \geq 11\)↩︎

5.14 Indecomposables on \(E_\infty\)↩︎

By inspection, there are \(56\) indecomposables on the \(\rho\)-Bockstein \(E_\infty\)-page. We list them here in table 17. In some cases, a choice has to be made regarding the name of the indecomposable. For example in degree \((32, 7, 12)\) we have an \(\mathbb{F}_2\)-basis consisting of \(\tau^5 \Delta c\) and \(\tau^6 a g\). The element \(\tau^6 a g\) is decomposable, but the other two elements \(\tau^5 \Delta c\) and \(\tau^5 \Delta c+ \tau^6 a g\) are indecomposable. We choose to include \(\tau^5 \Delta c+ \tau^6 a g\) in our table. The reason for this is that we believe that this element better represents the structure of the \(E_\infty\)-page because \(\tau^5 \Delta c+ \tau^6 a g\) is the unique \(\rho\)-power-torsion element in degree \((32, 7, 12)\).

Based on the following remark, table 17 also describes the indecomposables in the cohomology of \(\mathcal{A}(2)\).

6 Hidden extensions on the \(\rho\)-Bockstein \(E_\infty\)-page↩︎

For any multiplicative spectral sequence, the multiplicative structure of the abutment of the spectral sequence may differ from that of \(E_\infty\). On the \(E_\infty\)-page, the filtration degree of a product is the sum of the filtration degrees of the factors. In the abutment, the filtration degree of a product can be larger than the sum of the filtration degrees of the factors. Roughly speaking, if this happens one typically says that there is a hidden extension. For a more precise definition of the term hidden extension see [22]. We will use the cited definition here.

We will consider some but not all hidden extensions. The reason being that \(E_\infty\) has too many indecomposables. For now, we will only consider hidden extensions by \(\tau^8\), \(h_0\), \(h_1\), and \(h_2\). There is a slight amount of care needed when talking about \(h_0\) because the \(E_\infty\) class of the same name represents two elements in \(\text{Ext}\) due to the presence of \(\rho h_1\) in the same degree. On \(E_\infty\), \(\rho h_1\) is \(\rho\)-torsion free and \(h_0\) is \(\rho\)-torsion, so one of the elements represented by \(h_0\) is \(\rho\)-torsion free and the other one is \(\rho\)-torsion. When we talk about hidden \(h_0\)-extensions we specifically mean hidden extensions by the \(\rho\)-torsion element represented by \(h_0\) on \(E_\infty\). The other three \(E_\infty\) classes \(\tau^8\), \(h_1\), and \(h_2\) each only represent one element of \(\text{Ext}\) which we denote by the same symbol.

Hidden extensions can be separated into three categories:

1. The source of the hidden extension is \(\rho\)-torsion free and the target is \(\rho\)-torsion free,

2. the source is \(\rho\)-torsion free and the target is \(\rho\)-power-torsion,

3. both source and target are \(\rho\)-power-torsion.

The fourth case where the source is \(\rho\)-power-torsion and the target is \(\rho\)-torsion free cannot occur for multiplicative reasons. We study each of these three types of hidden extensions separately, one subsection at a time.

6.1 Chart guide↩︎

The relevant charts from [10] vary for each subsection. For section 6.2 the reader should first open the chart Rho-Bockstein E_11 rho-free quotient from the folder Rho-Bockstein E_r-pages rho-free quotient. By Proposition 71 this chart depicts the \(\rho\)-free quotient of the \(E_\infty\)-page of the \(\rho\)-Bockstein spectral sequence. So the hidden extensions considered in section 6.2 will be between elements on this chart. Once we are done with determining hidden extensions, the two charts in the folder Cohomology of A(2) rho-free quotient show all of these hidden extensions. They were separated into two charts for visual clarity.

In section 6.3 we should again consider the chart Rho-Bockstein E_11 rho-free quotient for the sources of the hidden extensions. However, since the targets of the hidden extensions are \(\rho\)-power-torsion they will be located on charts in the folder Rho Bockstein E_infinity rho-torsion.

For section 6.4 we only consider \(\rho\)-power-torsion elements of coweights congruent to \(1\) mod \(8\), and hidden extensions by elements preserving the coweight. So the relevant chart is Rho-Bockstein E_infinity rho-torsion part with s-w == 1 mod 8 from the folder Rho Bockstein E_infinity rho-torsion. Whenever we invoke the map \(p\) from Lemma 77 in section 6.4 it is helpful to consider the chart Rho-Bockstein E_1 rho-free quotient from the folder Rho-Bockstein E_r-pages rho-free quotient. That is because it coincides as a chart with the cohomology of \(\mathbb{C}\)-motivic \(\mathcal{A}(2)\) since \(E_1\) is the same as the cohomology of \(\mathbb{C}\)-motivic \(\mathcal{A}(2)\) adjoined \(\rho\), but we don’t plot \(\rho\)-multiples in the given chart.

6.2 Hidden extensions on the \(\rho\)-free quotient↩︎

By Corollary 8 we know the multiplicative structure of the cohomology of \(\mathcal{A}(2)\) after \(\rho\)-localization. This completely determines the multiplication on the \(\rho\)-free quotient.

In order to make use of Lemma 73 we need to understand the localization map. To do so, we need to determine some hidden extensions on the \(\rho\)-free quotient of the \(\rho\)-Bockstein \(E_\infty\)-page. Using our knowledge of the \(\rho\)-localization of the cohomology of \(\mathcal{A}(2)\), we can find all hidden extensions by \(\tau^8\), \(h_1\), and \(h_2\) on the \(\rho\)-free quotient. There are two charts of the \(\rho\)-free quotient \(\text{\rm Ext}_{\mathcal{A}(2)}(\mathbb{M}_2,\mathbb{M}_2)/(\text{\rho-power-torsion})\) with the aforementioned hidden extensions at [10]. For a guide on how to read them see section 4, and specifically section 4.5. Our task now is to prove the existence of these hidden extensions.

The following table 18 describes all hidden extensions by \(\tau^8\), \(h_1\), and \(h_2\) on the \(\rho\)-free quotient of the \(\rho\)-Bockstein \(E_\infty\)-page. We have inserted dashed horizontal lines to divide the table into different families. Hidden extensions in one family are closely connected, for example by multiplication with \(h_1\) or \(\tau^4 P\). Inside each family, the \(h_1\)-extensions are roughly located on a line of slope \(1/2\), corresponding to the slope of \(\tau^4 P\)-multiplication. The columns of table 18 describe:

1. the name of the element supporting a hidden extension,

2. the degree of the element supporting a hidden extension,

3. which element we are multiplying with,

4. the name of the target of the hidden extension,

5. the periodicity of the hidden extension in terms of \(E_\infty\)-multiplication on the \(\rho\)-free quotient. Meaning if we multiply both the source and target on the \(\rho\)-free quotient of \(E_\infty\) arbitrarily often by the given periodicity operator then we will see the same hidden extension.

We choose \(E_\infty\)-multiplication in the periodicity column to avoid talking about hidden extensions other than \(\tau^8\), \(h_1\), and \(h_2\) at this point. For example, by Corollary 8 every element in the \(\rho\)-free quotient supports infinitely many \(g\)-multiplications. However, some hidden extensions in the given table are not \(g\)-periodic with respect to \(E_\infty\)-multiplication. For example the \(\tau^8\)-extension on \(h_1^4\) is not \(g\)-periodic with respect to \(E_\infty\)-multiplication because the target of the hidden \(\tau^8\)-extension on \(h_1^4 g^2\) is \(\rho^8 \Delta^2 h_1^4\), which is not the \(g^2\)-multiple of \(\rho^4 \tau^4 P\) on \(E_\infty\).

Also, the periodicity column only concerns multiplication on the \(\rho\)-free quotient. We are not making any statement on possible \(\rho\)-power-torsion summands of any of these hidden extensions.

6.3 Hidden extensions from \(\rho\)-torsion free elements to \(\rho\)-power-torsion elements↩︎

As discussed in the introduction of this section, there can also be hidden extensions from \(\rho\)-torsion free elements to \(\rho\)-power-torsion elements on the \(\rho\)-Bockstein \(E_\infty\)-page. As an example, we will consider hidden \(h_0\)-extensions. Recall from the introduction that our choice of the \(\text{Ext}\) element \(h_0\) is \(\rho\)-torsion. It follows that all targets of hidden \(h_0\)-extensions will also be \(\rho\)-torsion.

For the first two hidden \(h_0\)-extensions we mention that they are \(P^2\)-periodic but the \(P^2\)-multiples of the sources are \(\rho\)-power-torsion. Similarly, the \(g\)-multiple of \(\tau^4 n^2\) is \(\rho\)-power-torsion. There is also a hidden \(g\)-extension on \(\tau^5 \Delta h_1 h_1\) making the \(h_0\)-extension on \(\tau^5 \Delta h_1\) \(g\)-periodic with respect to multiplication in \(\text{Ext}\).

To deduce these hidden extensions, we will use an analogue of [8] relying on the map \(p\) from the following Lemma. For brevity, we will write \(\text{Ext}_{\mathcal{A}(2)^k}\) for \(\text{Ext}_{\mathcal{A}(2)^k}(\mathbb{M}_2^k, \mathbb{M}_2^k)\), where \(k\) is \(\mathbb{R}\) or \(\mathbb{C}\).

Whenever we invoke Lemma 77, we are using the structure of \(\text{Ext}_{\mathcal{A}(2)^\mathbb{C}}\). It is worthwhile noting that we do have a chart of \(\text{Ext}_{\mathcal{A}(2)^\mathbb{C}}\). It is the chart depicting the \(\rho\)-free quotient of the \(\rho\)-Bockstein \(E_1\)-page. This chart agrees with \(\text{Ext}_{\mathcal{A}(2)^\mathbb{C}}\) because the \(E_1\)-page is just \(\text{Ext}_{\mathcal{A}(2)^\mathbb{C}}[\rho]\), and we do not plot \(\rho\)-multiples in the given chart.

Hidden extensions by \(h_0\) are relatively easy to deduce because of the somewhat tautological observation that a hidden \(h_0\)-extension from \(x\) to \(y\) is the same as a hidden \(x\)-extension from \(h_0\) to \(y\), under some mild hypotheses.

6.4 Hidden extensions on \(\rho\)-power-torsion elements↩︎

The third and last possible type of hidden extension is where both the source and the target are \(\rho\)-power-torsion. We provide some examples of such hidden extensions by \(\tau^8\), \(h_0\), and \(h_1\).

At [10] eight charts are provided which combine into the \(\rho\)-power-torsion subalgebra of the \(\rho\)-Bockstein \(E_\infty\)-page. Each chart consists of elements whose coweight \(s - w\) is congruent to some fixed value modulo \(8\). For a guide on how to read the charts see section 4, and specifically section 4.6. In this section, we will simply pick one coweight (or equivalently one chart) and illustrate how to compute hidden extensions there. For the rest of this section we will only consider elements of coweight \(1\) mod \(8\). Note that all of \(\tau^8\), \(h_0\), and \(h_1\) have coweight \(0\) mod \(8\), meaning that source and target of every hidden extension we consider are in the same coweight, and hence on the same chart.

There are two general methods we will use to deduce hidden extensions: The first method involves using hidden extensions from the \(\rho\)-free quotient together with the multiplication on \(E_{\infty}\). The second method uses the map \(p\) from Lemma 77.

What follows is a table describing all hidden \(\tau^8\)- and \(h_1\)-extensions between \(\rho\)-power-torsion elements in coweight \(1\) mod \(8\). We have inserted dashed horizontal lines to divide the table into different families. Hidden extensions in the same family are closely connected, for example by multiplication with \(h_1\) or \(\tau^8\) or \(\tau^4 P\). Inside each family, the \(h_1\)-extensions are roughly located on a line of slope \(1/2\), corresponding to the slope of \(\tau^4 P\)-multiplication. The columns of table 20 describe:

1. the name of the element supporting a hidden extension,

2. the degree of the element supporting a hidden extension,

3. which element we are multiplying with,

4. the name of the target of the hidden extension,

5. the periodicity of the hidden extension in terms of \(E_\infty\)-multiplication.

We choose \(E_\infty\)-multiplication in the periodicity column to avoid talking about hidden extensions other than \(\tau^8\) and \(h_1\). For example \(\tau^6 P h_1 e\) supports a hidden \(g\)-extension to \(\rho \tau^3 P \Delta h_1 d\), so the hidden \(h_1\)-extension on \(\tau^6 P h_1 e\) is \(g\)-periodic in Ext while not being \(g\)-periodic with respect to \(E_\infty\)-multiplication.

Another interesting observation is the following: In the \(\rho\)-free quotient we saw that there is a hidden \(\tau^8\)-extension from \(g^2\) to \(\rho^8 \Delta^2\). In the \(\rho\)-power-torsion part we do not see any connection between \(g^2\)- and \(\Delta^2\)-multiples. That is because the hidden \(\tau^8\)-extension in \(\text{Ext}\) goes from \(\rho^6 g^2\) to \(\rho^{14} \Delta^2\), but \(\rho^{14}\)-multiplication is trivial in the \(\rho\)-power-torsion part. Therefore, the patterns of hidden extensions for \(g^2\)- and \(\Delta^2\)-multiples of \(\rho\)-power-torsion elements do not have to look the same. For example the pattern of hidden \(h_1\)- and \(\tau^8\)-extensions on \(\tau^2 e g^2\) resembles the pattern on \(\tau^2 e g\), whereas the pattern on \(\tau^2 \Delta^2 e\) resembles the one on \(\tau^2 e\).

The next table collects all hidden \(h_0\)-extensions on \(\rho\)-power-torsion elements in the \(\rho\)-Bockstein spectral sequence in coweight \(1\) mod \(8\). The columns are the same as those in table 20. Again, we have chosen to separate families of hidden extensions by dashed horizontal lines. Hidden extensions inside each family are typically connected via \(\tau^4 P\)- or \(\tau^5 c\)-multiplication. They are roughly located on a line of slope \(1/2\), corresponding to the slope of \(\tau^4 P\)-multiplication and almost the slope of \(\tau^5 c\)-multiplication. We note that \(\tau^5 \Delta h_1\) is also in coweight \(1\) mod \(8\) and supports a hidden \(h_0\)-extension, but it is \(\rho\)-torsion free so it should not be part of this table.

7 An Adams spectral sequence?↩︎

In the classical setting \(\text{Ext}_{\mathcal{A}(2)^{\text{cl}}}(\mathbb{F}_2, \mathbb{F}_2)\) is the \(E_2\)-page of the Adams spectral sequence computing the homotopy groups of the topological modular forms spectrum \(\textit{tmf}\). Similarly, in the \(\mathbb{C}\)-motivic setting \(\text{Ext}_{\mathcal{A}(2)^\mathbb{C}}(\mathbb{M}_2^\mathbb{C}, \mathbb{M}_2^\mathbb{C})\) is the \(E_2\)-page of the \(\mathbb{C}\)-motivic Adams spectral sequence computing the homotopy groups of the \(\mathbb{C}\)-motivic modular forms spectrum \(\textit{mmf}^{\,\,\mathbb{C}}\). Speculatively, one could assume that there is an \(\mathbb{R}\)-motivic Adams spectral sequence with \(E_2\)-page \(\text{Ext}_{\mathcal{A}(2)}(\mathbb{M}_2, \mathbb{M}_2)\), associated to a speculative \(\mathbb{R}\)-motivic modular forms spectrum \(\textit{mmf}^{\,\,\mathbb{R}}\). Under this assumption, we will describe the \(d_2\)-differentials of that Adams spectral sequence on all indecomposables.

We will not attempt to compute the entire Adams spectral sequence. The issue is that we do not have all hidden extensions that are necessary to pass from the \(\rho\)-Bockstein \(E_\infty\)-page to the cohomology of \(\mathcal{A}(2)\), as there are too many of them. While we do have decompositions for all elements of \(\text{Ext}_{\mathcal{A}(2)}(\mathbb{M}_2, \mathbb{M}_2)\), the Leibniz rule for Adams differentials \(d_r(x \cdot y) = x \cdot d_r(y) + d_r(x) \cdot y\) may involve hidden extensions, i.e. both \(x \cdot d_r(y)\) and \(d_r(x) \cdot y\) may be zero on the Bockstein \(E_\infty\)-page but non-zero in . Hence, knowledge of the differentials on indecomposables does not immediately translate to knowledge of all differentials.

A notational issue that we also need to consider is the passage from the \(\rho\)-Bockstein \(E_\infty\)-page to \(\text{Ext}\). If a given class on the \(E_\infty\)-page has other classes in the same tridegree that are of higher \(\rho\)-filtration, then that class represents multiple elements in \(\text{Ext}\). Oftentimes we can distinguish these elements by their multiplicative decompositions. For example, we already saw in Remark 76 that the \(E_\infty\) class \(\tau^8 g^2\) represents two elements in \(\text{Ext}\) which can be distinguished by their decompositions \(\tau^8 \cdot g^2\) and \(\tau^4 g \cdot \tau^4 g\). We will use such multiplicative decompositions whenever there are multiple elements represented by a single class on \(E_\infty\). In particular, we will make the following notational convention in this section.

Using multiplicative decompositions to distinguish between different elements in \(\text{Ext}\) that are represented by the same class on \(E_\infty\) obviously does not work for indecomposables.

Let us first consider \(\rho\)-torsion free indecomposables. Since the map from the \(\rho\)-free quotient to the \(\rho\)-localization is injective, we can define \(\rho\)-free classes (up to \(\rho\)-power-torsion summands) by their images in the \(\rho\)-localization. We will assign names as they appear on our \(\rho\)-Bockstein \(E_\infty\)-page. For example, we let \(\tau^4 P\) be the element whose image in the \(\rho\)-localization is \(\rho^{-4} \cdot \tau^8 \cdot h_1^4\), or we let \(\tau^6 a\) be the element which localizes to \(\rho^{-1} \cdot \tau^8 \cdot u\). This kind of assignment works as long as there are no \(\rho\)-power-torsion elements in higher \(\rho\)-filtration. That almost never occurs for the elements we care about. Exceptions are \(\tau^2 P a\) which also has \(\rho^6 a d\) in higher \(\rho\)-filtration, and \(P a^2\) which has \(\rho^5 \tau h_0^2 e g\). For the first exception the choice of element representing \(\tau^2 P a\) makes no difference. While \(\rho^6 a d = \rho^6 \cdot a \cdot d\) contains \(a\) which supports an Adams \(d_2\), the target is \(\rho^3\)-torsion so \(\rho^6 a d\) is a \(d_2\)-cycle. For \(P a^2\) we have to be more careful a priori. The element \(\rho^5 \tau h_0^2 e g\) has the decomposition \(\rho^5 \cdot c \cdot a \cdot e\), and as we will see \(d_2(e) = h_1^2 d\) supports a non-trivial \(\rho^5\)-multiplication. However, \(h_1^2 d \cdot c\) is \(\rho^3\)-torsion, so that \(\rho^5 \tau h_0^2 e g\) is again a \(d_2\)-cycle. So the existence of \(\rho\)-power-torsion elements in higher \(\rho\)-filtration is also immaterial for \(P a^2\), at least for the purpose of computing Adams \(d_2\)-differentials.

When an indecomposable class on \(E_\infty\) is \(\rho\)-power-torsion we can typically abuse notation and use the same letters to denote the corresponding element of \(\text{Ext}\) whose \(\rho\)-power-torsion exactly matches that of the class on \(E_\infty\). For example, \(\tau h_1\) has the class \(\rho^2 h_2\) in higher \(\rho\)-filtration, but only one of the elements it represents is \(\rho^2\)-torsion because \(h_2\) is \(\rho\)-torsion free. So we let \(\tau h_1\) denote the unique \(\rho^2\)-torsion element in \(\text{Ext}\) represented by the element on \(E_\infty\) with the same name. Again, there are two indecomposables where this approach does not define them uniquely, namely \(\tau P h_1\) with \(\rho^2 P h_2\) in higher \(\rho\)-filtration and \(\tau P \Delta h_1\) with \(\rho \tau^2 \Delta c h_1^2\). Much like for the \(\rho\)-torsion free exceptions, one can argue that the choices of representatives here make no difference.

7.1 Chart guide↩︎

For section 7.2 the relevant charts from [10] are in the two folders Cohomology of A(2) rho-free quotient and Rho Bockstein E_infinity rho-torsion. The first is relevant for differentials whose sources are \(\rho\)-torsion free, the second for differentials where either the target or the source is \(\rho\)-power-torsion. Note that the Adams \(d_2\)-differential has degree \((-1, 2, 0)\), so in particular coweight \(-1\). Since the charts in the aforementioned folders each depict a certain set of coweights, the source and the target of an Adams \(d_2\)-differential are likely to be on separate charts.

7.2 Adams \(d_2\)-differentials↩︎

What follows is a table of Adams \(E_2\)-page indecomposables, their degrees and the values of their \(d_2\)-differentials. If the \(d_2\)-column is empty, that means that \(d_2\) on that indecomposable is zero. As mentioned in Remark 72, this table contains exactly the same indecomposables as table 17.

We proceed by giving proofs of the claimed \(d_2\)-differentials. The following is an analogue of [8].

Our approach to the Adams spectral sequence will be similar to our approach to the \(\rho\)-Bockstein spectral sequence in the following sense: We provide a table of all Adams \(d_2\)-differentials on indecomposables that could exist solely for degree reasons. Then, we consider whether each differential does or does not occur individually. Note that the degree of the Adams \(d_2\)-differential is \((-1, 2, 0)\). In particular its coweight is \(-1\), meaning the source of a differential and its target are likely to be on different coweight charts.

Sometimes we will make reference to hidden extensions on the \(\rho\)-free quotient of the \(\rho\)-Bockstein \(E_\infty\)-page as proven in section 6.2. In every case, these hidden extensions lift uniquely to the non-quotiented \(E_\infty\)-page because there are no \(\rho\)-power-torsion elements in the relevant degrees.

As charts will become more relevant now, we remind the reader of the chart guide given in section 7.1. We also remind the reader of Notation 15. Sometimes we will need to determine a hidden extension in order to use it in the proof of an Adams differential. In that case, we will list both the degree of the source and the degree of the target. See for example Lemma 98.

To compute the differential on \(\tau^2 a^3\), we need to compute the differential on the decomposable element \(\tau^6 d g = \tau^6 d \cdot g\) first. For that differential we need to establish a hidden extension on the \(\rho\)-Bockstein \(E_\infty\)-page first.

We need yet another hidden extension for the Adams \(d_2\) on \(\tau^2 a^3\).

For the Adams \(d_2\) on \(\tau^2 P n^2\) we need another hidden extension.

This concludes the computation of Adams \(d_2\)-differentials on \(\rho\)-torsion free indecomposables. Now we consider sources that are \(\rho\)-power-torsion.

To prove that \(\tau^2 h_2\) is a \(d_2\)-cycle, we would like to make use of the fact that \(\tau \Delta h_1\) is a \(d_2\)-cycle. So we will prove that first.

We will also need a hidden extension for the Adams \(d_2\) on \(\tau^2 h_2\).

The next two differentials are on \(\tau^5 \Delta c+ \tau^6 a g\) and \(\tau \Delta c+ \tau^2 a g\). In both cases, we could deduce their differentials by using \(g\)-multiplication. For the latter element this works fine, for the first there is a non-trivial amount of care needed due to the presence of \(\rho\)-torsion free elements in higher \(\rho\)-filtration. Therefore, we will prove the \(d_2\) on \(\tau \Delta c+ \tau^2 a g\) first, then prove the one on \(\tau^5 \Delta c+ \tau^6 a g\) without having to refer to \(g\)-multiplication and instead referring to the previous \(d_2\).

To keep the proceeding proof shorter, we will first prove that the decomposable element \(\tau^2 a g = \tau^2 e \cdot n\) is an Adams \(d_2\)-cycle. This in turn requires a hidden extension first.

For the Adams \(d_2\) on \(\tau^5 \Delta c+ \tau^6 a g\) we need one more preparatory hidden extension.

References↩︎

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1. We avoid using additional subscripts or superscripts for dual objects. Everything is phrased in terms of the dual Steenrod algebra.↩︎

2. The table in question contains \(15\) elements because the generator \(\tau\) is not listed explicitly.↩︎

3. Our names \(P\), \(g\), and \(\Delta^2\) translate to \(w_1\), \(g\), and \(w_2\) in [9].↩︎

4. Except for \(h_0\)-, \(h_1\)-, and \(h_2\)-multiples, but the fact that they are multiples is typically visualized by a line in the chart so their name can be inferred easily from the dot that they are connected to via that line.↩︎

5. Here is an example of what a magenta line looks like.

   ↩︎

6. The cited chart also includes \(\mathbb{C}\)-motivic Adams differentials. Those will of course not be in our chart.↩︎

7. But it should be noted that a dot does not represent any \(\tau^n\)-multiples here, as \(\tau^n\) does not have internal coweight \(0\).↩︎

8. Here is an example of what a cyan line looks like.

   ↩︎

9. That is a direct consequence of Corollary 8.↩︎

10. At least in the \(\rho\)-free quotient. Technically only \(\rho^6 g^2\) is \(\tau^8\)-torsion, so to get the precise relations in \(\text{Ext}\) one should multiply everything by \(\rho^6\).↩︎

11. Technically, this relation depends on the choice of the \(\text{Ext}\) element \(P a^2\). However, as we discussed in the introduction to this section, the choice of \(P a^2\) does not affect its Adams \(d_2\).↩︎