On the Steenrod module structure of $\mathbb{R}$-motivic Spanier-Whitehead duals
By Prasit Bhattacharya, Bertrand J. Guillou, and Ang Li
Abstract
The $\mathbb{R}$-motivic cohomology of an $\mathbb{R}$-motivic spectrum is a module over the $\mathbb{R}$-motivic Steenrod algebra $\mathcal{A}^{\mathbb{R}}$. In this paper, we describe how to recover the $\mathbb{R}$-motivic cohomology of the Spanier–Whitehead dual $\mathrm{DX}$ of an $\mathbb{R}$-motivic finite complex $\mathrm{X}$, as an $\mathcal{A}^{\mathbb{R}}$-module, given the $\mathcal{A}^{\mathbb{R}}$-module structure on the cohomology of $\mathrm{X}$. As an application, we show that 16 out of 128 different $\mathcal{A}^{\mathbb{R}}$-module structures on $\mathcal{A}^{\mathbb{R}}(1)\coloneq \langle \mathrm{Sq}^1, \mathrm{Sq}^2 \rangle$ are self-dual.
1.Introduction
Given a finite cell complex $\mathrm{X}$, it is useful to determine its Spanier–Whitehead dual $\mathrm{DX}$, which is the dual of the suspension spectrum $\Sigma ^\infty \mathrm{X}$ in the stable homotopy category of spectra. For instance, the mod 2 cohomology of $\mathrm{DX}$, as a (left) module over the mod 2 Steenrod algebra $\mathcal{A}$, is an input of Adams spectral sequences computing homotopy class of maps out of $\mathrm{X}$. An interesting case is when $\mathrm{X}$ is self-dual, as it leads to additional symmetries Reference MR often useful for computational purposes.
In the classical case, the $\mathcal{A}$-module structure on $\mathrm{H}^*(\mathrm{DX}; \mathbb{F}_2) \cong \mathrm{H}_*(\mathrm{X}, \mathbb{F}_2)$ is determined easily by the standard formula Equation 1.1 which involves the Kronecker pairing and the antiautomorphism $\chi \colon \mathcal{A}\longrightarrow \mathcal{A}$ of the Steenrod algebra. However, one should not expect an $\mathbb{R}$-motivic generalization of the standard formula because, first, the cohomology of the Spanier–Whitehead dual of an $\mathbb{R}$-motivic finite complex $\mathrm{X}$ is not always the linear dual of the cohomology of $\mathrm{X}$, as the coefficient ring $\mathbb{M}_2^\mathbb{R}$ is not a field (see Equation 2.1). Second, the $\mathbb{R}$-motivic Steenrod algebra is not known to support an antiautomorphism (but see Appendix A).
The ring of stable operations $\mathrm{E}^*\mathrm{E}$ of a cohomology theory $\mathrm{E}$ is not typically equipped with an antiautomorphism Reference B, p. 204. The natural conjugation on $\mathrm{E}_*\mathrm{E}$ is not $\mathrm{E}_*$-linear and so does not pass to the dual $\mathrm{E}^*\mathrm{E}$. Even in the presence of an antiautomorphism, the usual formula Equation 1.1 does not yield an $\mathrm{E}^*\mathrm{E}$-action on duals (see Remark A.3).
This article is concerned with the case $\mathrm{E}=\mathbf{H}_\mathbb{R}\mathbb{F}_2$,$\mathbb{R}$-motivic cohomology with coefficients in $\mathbb{F}_2$. We rely on Boardman’s mandala Reference B (see Figure 1.1) to demonstrate a method that computes the action of the $\mathbb{R}$-motivic Steenrod algebra $\mathcal{A}^\mathbb{R}$ on the Spanier–Whitehead duals of those finite $\mathbb{R}$-motivic spectra whose cohomology is free over $\mathbb{M}_2^\mathbb{R}$.
Let us pause to briefly discuss Boardman’s mandala. Given a finite cell complex $\mathrm{X}$ there are eight ways in which its mod $2$ homology and cohomology interact with the Steenrod algebra and its dual. They represent the vertices of the mandala. Boardman identified the relationships between them, which represent the edges. Each edge of the mandala corresponds to a formula. For example, the edge $\mathrm{D}''$ in Figure 1.1 corresponds to the formula (see Reference B, p. 190)
that relates the left $\mathcal{A}$-module structure on the cohomology $\mathrm{H}^*(\mathrm{X})$ with that of the left $\mathcal{A}$-module structure on the homology of $\mathrm{X}$. However, not all edges of the mandala exist for a general cohomology theory $\mathrm{E}$ (Reference B, Section 6).
When $\mathrm{H}^{\star }(\mathrm{X})\coloneq [\mathrm{X}, \mathbf{H}_\mathbb{R}\mathbb{F}_2]^{\star }$ is free and finitely generated over $\mathbb{M}_2^\mathbb{R}$,$\mathrm{H}_{\star }(\mathrm{X})$ is the $\mathbb{M}_2^\mathbb{R}$-linear dual of $\mathrm{H}^{\star }(\mathrm{X})$, as the relevant universal coefficient spectral sequence collapses. Consequently, the work in Reference B relates the left action of $\mathcal{A}^{\mathbb{R}}$ on $\mathrm{H}^{\star }(\mathrm{X})$ as well as the left action of $\mathcal{A}^\mathbb{R}$ on $\mathrm{H}_\star (\mathrm{X})$, to the $\mathcal{A}^\mathbb{R}_{\star }$-comodule structure on $\mathrm{H}^{\star }(\mathrm{X})$ (see Proposition 3.4, Proposition 3.10, and Proposition 3.13). These relations are the green dashed edges in Figure 1.1. As a result, one deduces the left $\mathcal{A}^\mathbb{R}$-module structure on $\mathrm{H}_{\star }(\mathrm{X})$ from that of $\mathrm{H}^{\star }(\mathrm{X})$ without resorting to an antiautomorphism (unlike Equation 1.1).
Our main application is concerned with identifying the $\mathbb{R}$-motivic spectra in the class $\mathcal{A}^\mathbb{R}_1$ introduced in Reference BGL. Each spectrum in $\mathcal{A}^\mathbb{R}_1$ is a realization of some $\mathcal{A}^\mathbb{R}$-module structure on the subalgebra $\mathcal{A}^\mathbb{R}(1)\coloneq \mathbb{M}_2^\mathbb{R}\langle \mathrm{Sq}^1, \mathrm{Sq}^2 \rangle \subset \mathcal{A}^\mathbb{R}$ (see Figure 4.1). In the classical case, Davis and Mahowald Reference DM showed that the subalgebra $\mathcal{A}(1)$ of the Steenrod algebra admits four different left $\mathcal{A}$-module structures, of which two are self-dual (see also Reference BEM, Remark 1.1). In Reference BGL, it was shown that $\mathcal{A}^\mathbb{R}(1)$ admits 128 different $\mathcal{A}^\mathbb{R}$-module structures. In this paper, we show Theorem 1.2.
Theorem 1.2.
Among the 128 different $\mathcal{A}^\mathbb{R}$-module structures on $\mathcal{A}^\mathbb{R}(1)$, only 16 are self-dual.
Remark 1.3.
In Reference BGL it was shown that every $\mathcal{A}^\mathbb{R}$-module structure on $\mathcal{A}^\mathbb{R}(1)$ can be realized as a finite $\mathbb{R}$-motivic spectrum, but we do not know if the realizations are unique. Hence, the spectra realizing a self-dual $\mathcal{A}^\mathbb{R}$-module structure on $\mathcal{A}^\mathbb{R}(1)$ may not be Spanier–Whitehead self-dual.
Davis and Mahowald also showed Reference DM that each realization of $\mathcal{A}(1)$ is the cofiber of a self-map of the spectrum $\mathcal{Y}\coloneq \mathbb{S}/2 \wedge \mathbb{S}/\eta$, where $\eta$ is the first Hopf element in the stable stems. In the $\mathbb{R}$-motivic stable stems, both $2$ and $\mathsf{h}$ in $\pi _{0,0}(\mathbb{S}_\mathbb{R})$ are lifts of $2 \in \pi _0(\mathbb{S})$ in the classical stable stems, and $\eta _{1,1} \in \pi _{1,1}(\mathbb{S}_\mathbb{R})$ is the unique lift of $\eta$ in bidegree $(1,1)$ (up to a unit). This results in two different $\mathbb{R}$-motivic lifts of $\mathcal{Y}$, namely
It was shown in Reference BGL, Theorem 1.8 that each $\mathcal{A}^\mathbb{R}$-module structure on $\mathcal{A}^\mathbb{R}(1)$ can be realized as the cofiber of a map between these $\mathbb{R}$-motivic lifts of $\mathcal{Y}$. Here we show Theorem 1.4.
Theorem 1.4.
Of the self-dual $\mathcal{A}^\mathbb{R}$-module structures on $\mathcal{A}^\mathbb{R}(1)$, 8 can be realized as the cofiber of a self-map on $\mathcal{Y}^\mathbb{R}_{(2,1)}$ and 8 as the cofiber of a self-map on $\mathcal{Y}^\mathbb{R}_{(\mathsf{h},1)}$.
Notation 1.5.
In all diagrams depicting modules over the Steenrod algebra, (i.e., in Figure 3.1, Figure 4.1, and Figure 4.2), a dot • represents a rank one free module over the coefficient ring; black vertical lines indicate the action of $\operatorname {Sq}^1$; blue curved lines indicate the action of $\operatorname {Sq}^2$; and red bracket-like lines represent the action of $\operatorname {Sq}^4$. A label on an edge represents that the operation hits that multiple of the generator. For example, in Figure 3.1, $\operatorname {Sq}^2({\mathsf{x}}_{2,1})$ is ${\tau }\cdot {\mathsf{x}}_{4,1}$ and $\operatorname {Sq}^4({\mathsf{x}}_{2,1})$ is $\rho ^2\cdot {\mathsf{x}}_{4,1}$.
2.A review of the $\mathbb{R}$-motivic Steenrod algebra and its dual
In Reference V, Voevodsky defined the motivic Steenrod operations $\operatorname {Sq}^n$ for $n\geq 0$, and gave a complete description of the $\mathbb{R}$-motivic Steenrod algebra $\mathcal{A}^\mathbb{R}$. It is free as a left module over the $\mathbb{R}$-motivic homology of a point,
where the element ${\tau }$ is in bidegree $\star = (0,-1)$, and ${\rho }$ is in bidegree $\star = (-1,-1)$.
The subalgebra $\mathbb{M}_2^\mathbb{R}\subset \mathcal{A}^\mathbb{R}$ is not central, and therefore $\mathcal{A}^\mathbb{R}$ has two $\mathbb{M}_2^\mathbb{R}$-module structures, one given by left multiplication and the other by right multiplication. The $\mathbb{R}$-motivic dual Steenrod algebra $\mathcal{A}^\mathbb{R}_{\star }$ is defined to be the (left) $\mathbb{M}_2^\mathbb{R}$-linear dual of $\mathcal{A}^\mathbb{R}$; it inherits an $\mathbb{M}_2^\mathbb{R}$-module structure, which we call the left action. The right $\mathbb{M}_2^\mathbb{R}$-action on $\mathcal{A}^\mathbb{R}$ also induces an action of $\mathbb{M}_2^\mathbb{R}$ on $\mathcal{A}^\mathbb{R}_{\star }$, which we call the right action of $\mathbb{M}_2^\mathbb{R}$ on $\mathcal{A}^\mathbb{R}_{\star }$ (see Reference V, p. 48).Footnote1 These correspond to the left and the right unit
1
Since $\mathbb{M}_2^\mathbb{R}$ is commutative, there is no meaningful distinction between “left” and “right” actions. The adjectives are merely a bookkeeping device.
for all $n \in \mathbb{N}$, where $\xi _0$ is the unit 1. The conjugation map $\vcenter{\img[][74pt][11pt][{\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \setlength{\extrarowheight}{0.0pt}\begin{tikzcd} \conj\colon\cAR_\star\rar& \cAR_\star\end{tikzcd}}]{Images/img59f8e63832e445b39d658e44219c0e33.svg}}$ of the Hopf algebroid structure sends
The coproduct $\Delta$ in Equation 2.3 is an $\mathbb{M}_2^\mathbb{R}$-bimodule map.
Remark 2.5.
The conjugation is not a map of left $\mathbb{M}_2^\mathbb{R}$-modules. In fact, it interchanges the left and right $\mathbb{M}_2^\mathbb{R}$-module structures on $\mathcal{A}^\mathbb{R}_{\star }$.
2.1.Kronecker product
The $\mathbb{R}$-motivic Kronecker product is a natural pairing between $\mathbb{R}$-motivic homology and cohomology which is constructed as follows: If $\varphi : \mathrm{X} \longrightarrow \Sigma ^{\mathsf{i},{\mathsf{j}}} \mathbf{H}_\mathbb{R}\mathbb{F}_2$ represents the class $[\varphi ] \in \mathrm{H}^{\star }(\mathrm{X})$ and ${\mathsf{x}} : \Sigma ^{{\mathsf{m}},{\mathsf{n}}} \mathbb{S}_\mathbb{R}\longrightarrow \mathbf{H}_\mathbb{R}\mathbb{F}_2\wedge \mathrm{X}$ represents $[{\mathsf{x}}] \in \mathrm{H}_{{\mathsf{m}},{\mathsf{n}}} (\mathrm{X})$, then the composition
where $\mathfrak{n}(\varphi )({\mathsf{x}}) = \langle {\mathsf{x}}, \varphi \rangle$.
Remark 2.7.
When $\mathrm{H}_{\star } (\mathrm{X})$ is free and finitely generated as an $\mathbb{M}_2^\mathbb{R}$-module, the map $\mathfrak{n}$ in Equation 2.6 is an isomorphism. Consequently, elements in $\mathrm{H}^{\star }(\mathrm{X})$ can be identified with linear maps from $\mathrm{H}_{\star }(\mathrm{X})$, and the Kronecker product is simply the evaluation of functionals.
Notation 2.8.
Since both $\mathcal{A}^\mathbb{R}$ and $\mathcal{A}^\mathbb{R}_{\star }$ have a left and a right action of $\mathbb{M}_2^\mathbb{R}$, let $\mathcal{A}^\mathbb{R}\otimes _{\mathbb{M}_2^\mathbb{R}}^{\mathrm{left}} \mathcal{A}^\mathbb{R}_\star$ (likewise $\mathcal{A}^\mathbb{R}\otimes _{\mathbb{M}_2^\mathbb{R}}^{\mathrm{right}} \mathcal{A}^\mathbb{R}_\star$) denote the tensor product of left (likewise right) $\mathbb{M}_2^\mathbb{R}$-modules.
Remark 2.9.
When $\mathrm{X}$ is $\mathbf{H}_\mathbb{R}\mathbb{F}_2$, the Kronecker product is a map of left $\mathbb{M}_2^\mathbb{R}$-modules$\mathcal{A}^\mathbb{R}_{\star } \otimes _{\mathbb{M}_2^\mathbb{R}}^{\mathrm{left}} \mathcal{A}^\mathbb{R}\to \mathbb{M}_2^\mathbb{R}$.
2.2.The Milnor basis
The dual Steenrod algebra $\mathrm{H}_\star (\mathbf{H}_\mathbb{R}\mathbb{F}_2) \cong \mathcal{A}^\mathbb{R}_{\star }$ is free and degreewise finitely generated as an $\mathbb{M}_2^\mathbb{R}$-module. Consequently, the natural map of Equation 2.6 gives an isomorphism
of left $\mathbb{M}_2^\mathbb{R}$-modules. Taking advantage of the above isomorphism, Voevodsky Reference V, $\mathsection$13 defines the Milnor basis of the $\mathbb{R}$-motivic Steenrod algebra using the monomial basis of the dual Steenrod algebra Equation 2.2.
For finite sequences $\mathrm{E} = ({\mathsf{e}}_0, {\mathsf{e}}_1, \dots , {\mathsf{e}}_m )$ and $\mathrm{R} =({\mathsf{r}}_1, \dots , {\mathsf{r}}_n)$ of nonnegative integers, let $\muprho (\mathrm{E}, \mathrm{R})$ denote the element in $\mathcal{A}^\mathbb{R}$ dual to the monomial
in $\mathcal{A}^\mathbb{R}_{\star }$. It is standard practice to set $\mathcal{P}^{\mathrm{R}} \coloneq \muprho ( \mathbf{0}, \mathrm{R})$ and $\mathcal{Q}^{\mathrm{E}} \coloneq \muprho (\mathrm{E}, \mathbf{0})$. Moreover, $\mathcal{Q}_i$ is shorthand for the dual to $\muptau _i$.
In Table 2.1, we record, for each monomial $\muptau (\mathrm{E}) \mupxi (\mathrm{R})\in \mathcal{A}^\mathbb{R}_{\star }$ in low degree, its image under the conjugation $\mathsf{c}$ and its dual element in $\mathcal{A}^\mathbb{R}$, both in terms of the Milnor basis as well as in terms of the generators $\mathcal{G}\coloneq \{ \operatorname {Sq}^{2^k}: k \geq 1 \}$. The latter description will be used in Section 3.3 and Section 4.
A number of these descriptions in terms of $\mathcal{G}$ can be found in Reference V. For example, see Reference V, Lemma 13.1 and Lemma 13.6. The Adem relations (see Reference BGL, Appendix A) are another useful tool. For example, the Adem relation $\operatorname {Sq}^2 \operatorname {Sq}^4 = \operatorname {Sq}^6 + {\tau }\operatorname {Sq}^5 \operatorname {Sq}^1$ leads to the description for $P^3=\operatorname {Sq}^6$. The formula for $\mathcal{P}^{(0,1)}$ follows from Reference K, (6). Finally, the formula for $\mathcal{P}^{(1,1)}$ can be deduced from expressing $\operatorname {Sq}^6 \operatorname {Sq}^2$ in terms of the Milnor basis. This can be done by evaluating the formula Reference V, (12.9)
at $\varphi =\operatorname {Sq}^6$,$\psi =\operatorname {Sq}^2$, and $\mathsf{x}$ monomials in low degree. This shows that $\operatorname {Sq}^6 \operatorname {Sq}^2$ is the sum $\mathcal{P}^{(1,1)} + \tau \mathcal{Q}_0 \mathcal{Q}_1 \mathcal{P}^2$.
3.Dualizing $\mathcal{A}^\mathbb{R}$-modules
For any $\mathbb{R}$-motivic spectrum $\mathrm{X}$, its Spanier–Whitehead dual is the function spectrum $\mathrm{D}\mathrm{X}\coloneq \mathrm{F} (\mathrm{X}, \mathbb{S}_{\mathbb{R}})$. The goal of this section is to identify the $\mathcal{A}^\mathbb{R}$-module structure $\mathrm{H}^{\star }(\mathrm{DX})$ given the $\mathcal{A}^\mathbb{R}$-module structure on $\mathrm{H}^{\star }(\mathrm{X})$ under the following assumption.
Assumption 3.1.
Let $\mathrm{X}$ be a finite $\mathbb{R}$-motivic spectrum such that its homology $\mathrm{H}_{\star }(\mathrm{X})$ is free over $\mathbb{M}_2^\mathbb{R}$.
Notation 3.2.
For an $\mathbb{M}_2^\mathbb{R}$-module$\mathbf{N}$ let
be the set of $\mathbb{M}_2^\mathbb{R}$-linear functionals.
3.1.From $\muppsi _{\mathrm{L}}$ to $\mupphi _{\mathrm{L}}'$
Recall that $\mathrm{H}^\star (\mathrm{X})$ is naturally a left $\mathcal{A}^\mathbb{R}$-module. We will also use an $\mathcal{A}^\mathbb{R}_\star$-comodule structure on $\mathrm{H}^{\star }(\mathrm{X})$
First, note that $\mathcal{A}^\mathbb{R}_\star$ is free as a right $\mathbb{M}_2^\mathbb{R}$-module with basis $\mathcal{B}$ given by the conjugate of any left $\mathbb{M}_2^\mathbb{R}$-module basis. Then we have a splitting
where $\mupiota$ is the unit map of $\mathbf{H}_\mathbb{R}\mathbb{F}_2$. For any finite motivic spectrum, the map $\muppsi$ induces the map $\muppsi _{\mathrm{L}}$ (see Reference B, Theorem 2.9(b)), giving $\mathrm{H}^\star (\mathrm{X})$ the structure of an $\mathcal{A}^\mathbb{R}_\star$-comodule as explained in Reference B, Section 6. Further, Boardman showed that we get Proposition 3.4.
for $\varphi \in \mathcal{A}^\mathbb{R}$,$\muplambda \in \mathbf{N}^\vee$, and $n \in \mathbf{N}$.
Remark 3.6.
If $\muppsi _{\mathrm{L}}(n) = \sum _i a_i \otimes n_i$, for $a_i \in \mathcal{A}^\mathbb{R}_\star$ and $n_i \in \mathbf{N}$, then Equation 3.5 can be rewritten as
Combining Proposition 3.4 with the following result, one can deduce the left $\mathcal{A}^\mathbb{R}$-module structure on $\mathrm{H}^\star (\mathrm{DX})$($\mupphi _{\mathrm{L}}'$ in Figure 1.1) from the left $\mathcal{A}^\mathbb{R}_{\star }$-comodule structure on $\mathrm{H}^\star (\mathrm{X})$($\muppsi _{\mathrm{L}}$ in Figure 1.1).
Proposition 3.8.
Suppose $\mathrm{X}$ satisfies Assumption 3.1. There are isomorphisms of left $\mathcal{A}^\mathbb{R}$-modules$\mathrm{H}^\star (\mathrm{DX}) \cong ( \mathrm{H}_\star ( \mathrm{DX }) )^\vee \cong ( \mathrm{H}^\star (\mathrm{X}) )^\vee$.
Proof.
Under Assumption 3.1 the map $\mathfrak{n}:\mathrm{H}^\star (\mathrm{DX}) \longrightarrow (\mathrm{H}_\star (\mathrm{DX}))^\vee$ defined in Equation 2.6, is not just an isomorphism of $\mathbb{M}_2^\mathbb{R}$-modules (see Remark 2.7), but also an isomorphism of left $\mathcal{A}^\mathbb{R}$-modules according to Reference B, Lemma 6.2.
For the second isomorphism, first note that Assumption 3.1 implies that there exists an isomorphism
of $\mathbb{M}_2^\mathbb{R}$-modules. By Proposition 3.4, it is enough to lift Equation 3.9 to an isomorphism of $\mathcal{A}^\mathbb{R}_\star$-comodules. To this end, we first observe that the comodule structure on $\mathrm{H}^\star (\mathrm{X})$ is induced by the map
where the horizontal maps are evaluation at $\mathrm{X}$.
■
3.2.From $\mupphi _{\mathrm{L}}$ to $\muppsi _{\mathrm{L}}$
For any $\varphi \in \mathcal{A}^\mathbb{R}\cong \mathrm{Hom}_{\mathbb{M}_2^\mathbb{R}}(\mathcal{A}^\mathbb{R}_{\star }, \mathbb{M}_2^\mathbb{R})$, let $\varphi \mathsf{c}$ denote the composition
which is a right $\mathbb{M}_2^\mathbb{R}$-module map as the conjugation $\mathsf{c}$ is an isomorphism from the right $\mathbb{M}_2^\mathbb{R}$-module structure to the left $\mathbb{M}_2^\mathbb{R}$-module structure of $\mathcal{A}^\mathbb{R}_{\star }$.
Proposition 3.10.
Let $\mathbf{N}$ be a left $\mathcal{A}^\mathbb{R}_\star$-comodule with coproduct $\muppsi _{\mathrm{L}}$. Then, for $n\in \mathbf{N}$ and $\varphi \in \mathcal{A}^\mathbb{R}$, the formula
for $\varphi , \psi \in \mathcal{A}^\mathbb{R}$ and $a \in \mathcal{A}^\mathbb{R}_\star$. Formula Equation 3.11 follows from combining Reference B, Lemma 3.3(a) with $\mathsf{c}\circ \mupeta _{\mathrm{L}} = \mupeta _{\mathrm{R}}$ and
The right $\mathbb{M}_2^\mathbb{R}$-module structure on $\mathcal{A}^\mathbb{R}_\star$ is defined Reference V, Section 12 such that
$$\begin{equation*} a \cdot \mupeta _{\mathrm{R}}(m)(\varphi ) = a(\varphi \cdot m) \end{equation*}$$
for $m \in \mathbb{M}_2^\mathbb{R}$,$a\in \mathcal{A}^\mathbb{R}_\star$ and $\varphi \in \mathcal{A}^\mathbb{R}$. This shows that the evaluation pairing defines a map
of $\mathbb{M}_2^\mathbb{R}$-bimodules, where the left $\mathbb{M}_2^\mathbb{R}$-module structure on $\mathcal{A}^\mathbb{R}\otimes _{\mathbb{M}_2^\mathbb{R}}^{\mathrm{right}} \mathcal{A}^\mathbb{R}_\star$ is obtained via the left action on $\mathcal{A}^\mathbb{R}$, and the right $\mathbb{M}_2^\mathbb{R}$-module structure via the left action on $\mathcal{A}^\mathbb{R}_\star$. Consequently, the left action constructed in Proposition 3.10 can be described as the composition $\mupphi _{\mathrm{L}}$ in the diagram
If we set $\mathbf{N}= \mathrm{H}^\star (\mathrm{X})$, i.e., the cohomology of a finite spectrum $\mathrm{X}$ with the $\mathcal{A}_{\star }$-comodule structure of Equation 3.3, Proposition 3.10 recovers the usual $\mathcal{A}^\mathbb{R}$-module structure on $\mathrm{H}^\star (\mathrm{X})$ (see Reference B, Lemma 6.3). Our next result reverse engineers Proposition 3.10 to obtain a formula that calculates the $\mathcal{A}^\mathbb{R}_\star$-comodule on $\mathrm{H}^{\star }(\mathrm{X})$($\muppsi _{\mathrm{L}}$ in Figure 1.1) from the $\mathcal{A}^\mathbb{R}$-module on $\mathrm{H}^{\star }(\mathrm{X})$($\mupphi _{\mathrm{L}}$ in Figure 1.1).
Let $\mathscr{B}$ be the monomial basis of the left $\mathbb{M}_2^\mathbb{R}$-module structure on $\mathcal{A}^\mathbb{R}_{\star }$ (as in Section 2.2). For simplicity, let $\mathbf{b}_i$ denote the elements of $\mathscr{B}$, and let $\mathbf{B}^i \in \mathcal{A}^\mathbb{R}$ be the dual basis in the following result.
Proposition 3.13.
Let $\mathbf{N}$ be a left $\mathcal{A}^\mathbb{R}_\star$-comodule with coaction map $\muppsi _{\mathrm{L}}$. Then $\muppsi _{\mathrm{L}}$ is related to $\mupphi _{\mathrm{L}}$ using the formula
where $\cdot$ is the action of $\mathcal{A}^\mathbb{R}$ on $\mathbf{N}$ constructed using Proposition 3.10.
Proof.
Since $\{c(\mathbf{b}_i)\}$ is a basis for $\mathcal{A}^\mathbb{R}_\star$ as a free right $\mathbb{M}_2^\mathbb{R}$-module, it follows that there is a unique expression $\muppsi _{\mathrm{L}}(n) = \sum _i c(\mathbf{b}_i) \otimes n_i$ for appropriate elements $n_i$. On the other hand,
We now demonstrate the usefulness of Proposition 3.4, Proposition 3.10, and Proposition 3.13 by identifying the $\mathcal{A}^\mathbb{R}$-module structure on $\mathrm{H}^{\star }(\mathrm{DX})$, for a few well-known finite $\mathbb{R}$-motivic finite complexes $\mathrm{X}$.
Notation 3.14.
In the following examples, the $\mathbb{R}$-motivic spectrum $\mathrm{X}$ will satisfy Assumption 3.1. In particular, $\mathrm{H}^{\star }(\mathrm{X})$ will be a free $\mathbb{M}_2^\mathbb{R}$-module. By ${\mathsf{x}}_{\mathsf{i},j}$, we will denote an element of its $\mathbb{M}_2^\mathbb{R}$-basis which lives in cohomological bidegree $(\mathsf{i}, {\mathsf{j}})$. By $\hat{\mathsf{x}}_{\mathsf{i}, {\mathsf{j}}}$, we will denote an element of $(\mathrm{H}^{\star }(\mathrm{X}))^{\vee }$ dual to ${\mathsf{x}}_{\mathsf{i},j}$. Note that the bidegree of $\hat{\mathsf{x}}_{\mathsf{i},j}$ is $(-\mathsf{i}, -{\mathsf{j}})$ under the isomorphism $(\mathrm{H}^{\star }(\mathrm{X}))^{\vee } \cong \mathrm{H}^{\star }(\mathrm{DX})$.
Example 3.15(The $\mathbb{R}$-motivic mod $2$ Moore spectrum).
As an $\mathbb{M}_2^\mathbb{R}$-module,$\mathrm{H}^\star (\mathbb{S}_\mathbb{R}/2)$ has generators ${\mathsf{x}}_{0,0}$ and ${\mathsf{x}}_{1,0}$. The $\mathcal{A}^\mathbb{R}$-module structure is then determined by the relations
which determines the $\mathcal{A}^\mathbb{R}_\star$-comodule structure on $\mathrm{H}^{\star }(\mathbb{S}_\mathbb{R}/2)$. Then we apply Proposition 3.4, in particular, formula Equation 3.7, to obtain
as $\mathcal{A}^\mathbb{R}$-modules. This aligns with the fact that $D(\mathbb{S}_\mathbb{R}/2)$ is equivalent to $\Sigma ^{-1} \mathbb{S}_\mathbb{R}/2$.
Example 3.16($\mathbb{R}$-motivic mod ${\mathsf{h}}$ Moore spectrum).
As a graded $\mathbb{M}_2^\mathbb{R}$-module,$\mathrm{H}^{\star }(\mathbb{S}/\mathsf{h})$ is isomorphic to $\mathrm{H}^{\star }(\mathbb{S}/2)$. However, they differ in their $\mathcal{A}^\mathbb{R}$-module structures in that
and using Equation 3.7 we see that $\Big ( \mathrm{H}^\star (\mathbb{S}_\mathbb{R}/\mathsf{h}) \Big )^\vee \cong \Sigma ^{-1} \mathrm{H}^\star (\mathbb{ S}_\mathbb{R}/\mathsf{h})$. This aligns with the fact that $D(\mathbb{S}_\mathbb{R}/\mathsf{h})$ is equivalent to $\Sigma ^{-1} \mathbb{S}_\mathbb{R}/\mathsf{h}$.
Example 3.17(The $\mathbb{R}$-motivic$\mathfrak{J}$oker).
The $\mathcal{A}^\mathbb{R}(1)$-module of the $\mathbb{R}$-motivic$\mathfrak{J}$oker$\mathcal{J}_\mathbb{R}$ (discussed in Reference GL) is the quotient $\mathcal{A}^\mathbb{R}(1)/\operatorname {Sq}^3$. In Figure 3.1, we have displayed a particular $\mathcal{A}^\mathbb{R}$-module extension of $\mathcal{A}^\mathbb{R}(1)/\operatorname {Sq}^3$ obtained using Theorem 4.1. Using Proposition 3.13, in conjunction with Table 2.1, we notice that
determines the $\mathcal{A}^\mathbb{R}_{\star }$-comodule structure of $\mathrm{H}^{\star }(\mathcal{J}_{\mathbb{R}})$. Then Equation 3.7 produces the $\mathcal{A}^\mathbb{R}$-module structure on the dual displayed in Figure 3.1.
4.Self-dual $\mathcal{A}^\mathbb{R}$-module structures on $\mathcal{A}^\mathbb{R}(1)$
Let ${\mathsf{x}}_{\mathsf{i},j}$ and ${\mathsf{y}}_{\mathsf{i},j}$ denote the elements of the $\mathbb{M}_2^\mathbb{R}$-basis of $\mathcal{A}^\mathbb{R}(1)$ introduced in Reference BGL, Notation 1.5 in bidegree $(\mathsf{i},j)$.
there exists a unique isomorphism class of $\mathcal{A}^\mathbb{R}$-module structures on $\mathcal{A}^\mathbb{R}(1)$, which we denote by $\mathcal{A}^\mathbb{R}_{ \overline{\mathrm{v}}}(1)$, determined by the formulas
where $\mathscr{j}_{24}= \beta _{03} \gamma _{36} + \alpha _{03}(\beta _{25} + \beta _{26})$. Further, any $\mathcal{A}^\mathbb{R}$-module whose underlying $\mathcal{A}^\mathbb{R}(1)$-module is free on one generator is isomorphic to one listed above.
Using Proposition 3.13, we calculate the $\mathcal{A}^\mathbb{R}_\star$-comodule structure $\muppsi _{\mathrm{L}}$ on $\mathcal{A}^\mathbb{R}_{\overline{\mathrm{v}}}(1)$:
The $\mathcal{A}^\mathbb{R}(1)$-module structure on the dual $(\mathcal{A}^\mathbb{R}_{ \overline{\mathrm{v}}}(1))^\vee$ is as displayed in the right of Figure 4.1. Moreover, its $\mathcal{A}^\mathbb{R}$-module structure is determined by
where $\mupdelta (\overline{\mathrm{v}}) = (\gamma _{36},\beta _{25}+\beta _{26},\beta _{25},\mathscr{j}_{24}+\beta _{06},\beta _{14},\beta _{03} + \beta _{14},\alpha _{03})$. Thus, $\mathcal{A}^\mathbb{R}_{ \overline{\mathrm{v}}}(1)$ is self dual if and only if
(1)
$\alpha _{03} = \gamma _{36}$,
(2)
$\beta _{03} = \beta _{25} + \beta _{26}$, and
(3)
$\beta _{14} = \beta _{25}$.
Remark 4.4.
The constant $\mathscr{j}_{24}$ has a geometric significance noted in Reference BGL, Remark 1.21. It follows from Corollary 4.3 that $\mathscr{j}_{24}=0$ whenever $\mathcal{A}^\mathbb{R}_{ \overline{\mathrm{v}}}(1)$ is self-dual.
Remark 4.5.
The underlying classical $\mathcal{A}$-module structure on $\mathcal{A}(1)$ is self-dual if and only if $\beta _{26} = \beta _{03} + \beta _{14}$. In the presence of (3), this is equivalent to (2). Thus the conditions of Corollary 4.3 can be thought of as the classical condition, plus conditions (1) and (3).
In Reference BGL, it was shown that the $\mathcal{A}^\mathbb{R}$-modules$\mathcal{A}^\mathbb{R}_{ \overline{\mathrm{v}}}(1)$ can be realized as the cohomology of an $\mathbb{R}$-motivic spectrum for all values of $\overline{\mathrm{v}}$.
Corollary 4.6.
Suppose $\mathcal{A}^\mathbb{R}_1[\overline{\mathrm{v}}]$ is an $\mathbb{R}$-motivic spectrum realizing $\mathcal{A}^\mathbb{R}_{ \overline{\mathrm{v}}}(1)$, and suppose that $\mathcal{A}^\mathbb{R}_{ \overline{\mathrm{v}}}(1)$ is a self-dual $\mathcal{A}^\mathbb{R}$-module. Then $\mathcal{A}^\mathbb{R}_1[\overline{\mathrm{v}}]$ is the cofiber of a $v_1$-self-map on either $\mathcal{Y}^\mathbb{R}_{2,1}$ or $\mathcal{Y}^\mathbb{R}_{\mathsf{h},1}$.
Proof.
By Reference BGL, Theorem 1.8, the $\mathbb{R}$-motivic spectrum $\mathcal{A}^\mathbb{R}_1[\overline{\mathrm{v}}]$ is the cofiber of a $v_1$-self map on $\mathcal{Y}^\mathbb{R}_{2,1}$ if $\beta _{25}+\beta _{26}+\gamma _{36}=1$ and $\alpha _{03}+\beta _{03}=1$, whereas it is the cofiber of a $v_1$-self-map on $\mathcal{Y}^\mathbb{R}_{\mathsf{h},1}$ if $\beta _{25}+\beta _{26}+\gamma _{36}=0$ and $\alpha _{03}+\beta _{03}=0$. But conditions (1) and (2) of Corollary 4.3 imply that $\beta _{25}+\beta _{26}+\gamma _{36}$ is equal to $\alpha _{03}+\beta _{03}$.
■
Our main results Theorem 1.2 and Theorem 1.4 follow from Corollary 4.3 and Corollary 4.6, respectively.
Remark 4.7.
Using the Betti realization functor, Reference BGL produced $\mathrm{C}_2$-equivariant realizations of analogous $\mathcal{A}^{\mathrm{C}_2}$-modules$\mathcal{A}_{\overline{\mathrm{v}}}^{\mathrm{C}_2} (1)$. Using the comparison result Reference BGL, Theorem 1.19, the $\mathcal{A}$-module structures on $\Phi (\mathcal{A}_{1}^{\mathrm{C}_2} [\overline{\mathrm{v}}])$, the geometric fixed points of $\mathcal{A}_{1}^{\mathrm{C}_2} [\overline{\mathrm{v}}]$, was identified in Reference BGL, Figure 4.12. In Figure 4.2, we record the $\mathcal{A}$-module structure on the geometric fixed points of a self-dual $\mathcal{A}_{1}^{\mathrm{C}_2} [\overline{\mathrm{v}}]$.
Appendix A.On the antiautomorphism of $\mathcal{A}^\mathbb{R}$
Although Boardman Reference B, $\mathsection$6 pointed out that the set of $\mathrm{E}$-cohomology operations $[\mathrm{E}, \mathrm{E}]^{\ast }$ may not necessarily have an antiautomorphism for a cohomology theory $\mathrm{E}$, we find the case of $\mathrm{E} = \mathbf{H}_\mathbb{R}\mathbb{F}_2$ a rather curious one.
The case of $\mathrm{E} = \mathbf{H}\mathbb{F}_2$ is exceptional; the Steenrod algebra $\mathcal{A}\coloneq [\mathbf{H}\mathbb{F}_2, \mathbf{H}\mathbb{F}_2]_*$ is well known to be a Hopf algebra and, therefore, equipped with an antiautomorphism $\vcenter{\img[][69pt][10pt][{\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \setlength{\extrarowheight}{0.0pt}\begin{tikzcd} \chi:\cA\ar[r] & \cA. \end{tikzcd}}]{Images/imgcc5eb664780ef97e6d9f60cebf07bfbc.svg}}$ The composition of extension of scalars and Betti realization,
as the $\mathbb{C}$-motivic Steenrod algebra is a connected bialgebra. However, lifting $\chi ^{\mathbb{C}}$ along $\muppi _1$ is less straightforward. The dual $\mathbb{R}$-motivic Steenrod algebra $\mathcal{A}^\mathbb{R}_\star$ is a Hopf algebroid, rather than a Hopf algebra, so that its dual is not a Hopf algebra.
One feature that distinguishes $\mathcal{A}^\mathbb{R}$ from $\mathcal{A}^\mathbb{C}$ is the fact that ${\tau }$ is not central in $\mathcal{A}^\mathbb{R}$. In the following result, we use the commutators $[{\tau }, \operatorname {Sq}^{2^n}]$ in $\mathcal{A}^\mathbb{R}$ (computed using the Cartan formula Reference V, Proposition 9.7) to compute the values of a hypothetical antiautomorphism in low degrees.
Proposition A.1.
Suppose that $\chi ^\mathbb{R}\colon \mathcal{A}^\mathbb{R}\longrightarrow \mathcal{A}^\mathbb{R}$ is a ring antihomomorphism and an involution. Then
in characteristic $2$. Since $\tau$ and $\operatorname {Sq}^1$ are unique $\mathbb{F}_2$-generators in their bidegree and $\chi ^\mathbb{R}$ is an automorphism, it follows that
For degree reasons, $\chi ^\mathbb{R}(\operatorname {Sq}^2)$ must be $\operatorname {Sq}^2 + \varepsilon {\rho }\operatorname {Sq}^1$, where $\varepsilon$ is either 0 or 1. But the commutator $[{\tau },\operatorname {Sq}^2]$ is equal to ${\rho }{\tau }\operatorname {Sq}^1$. Applying Equation A.2, we see that
and therefore, $\delta =0$,$\varepsilon =1$, and $\lambda =1$ as desired.
■
Proposition A.1 suggests there might be an $\mathbb{R}$-motivic antiautomorphism on the subalgebra $\mathcal{A}^\mathbb{R}(2): = \mathbb{M}_2^\mathbb{R}\langle \operatorname {Sq}^1, \operatorname {Sq}^2, \operatorname {Sq}^4 \rangle \subset \mathcal{A}^\mathbb{R}$. It seems likely that the method above can be extended to produce an antiautomorphism on all of $\mathcal{A}^\mathbb{R}$. However, we leave open the question of whether or not this is possible.
On the other hand, Remark A.3 shows that an antihomomorphism on $\mathcal{A}^\mathbb{R}$ may not be directly of use in dualizing $\mathcal{A}^\mathbb{R}$-modules.
Remark A.3.
Note that if $\mathbf{N}$ is an $\mathcal{A}^\mathbb{R}$-module, then the action of $\mathcal{A}^\mathbb{R}$ on $\mathbf{N}$ is not $\mathbb{M}_2^\mathbb{R}$-linear, so that, in contrast to the classical case, it does not induce a right $\mathcal{A}^\mathbb{R}$-action on the dual $\mathbf{N}^{\vee }$. Even if $\mathcal{A}^\mathbb{R}$ were to be hypothetically equipped with an antiautomorphism $\chi ^\mathbb{R}$, this may not be so useful for the purpose of dualization. The reason is that the classical formula Equation 1.1 does not work in this setting. More precisely, let $\mathbf{N}$ be an $\mathcal{A}^\mathbb{R}$-module, let $\lambda \in \mathbf{N}^{\vee }$,$\varphi \in \mathcal{A}^\mathbb{R}$, and $n \in \mathbf{N}$. Then defining a new action $\varphi \odot \lambda$ by
from Example 3.16. Then $(\operatorname {Sq}^2\odot \hat{\mathsf{x}}_{1,0})( {\tau }{\mathsf{x}}_{0,0})$ vanishes, whereas $(\operatorname {Sq}^2\odot \hat{\mathsf{x}}_{1,0})( {\mathsf{x}}_{0,0})$ is equal to ${\rho }$. It follows that the formula for $\operatorname {Sq}^2 \odot \hat{\mathsf{x}}_{1,0}$ is not $\mathbb{M}_2^\mathbb{R}$-linear and is therefore not a valid element of $\mathbf{N}^\vee$.
Acknowledgements
We thank Agnès Beaudry, Mike Hill, Clover May, Sarah Petersen, Liz Tatum, and Doug Ravenel for a stimulating conversation at the conference, Homotopy Theory in honor of Paul Goerss, held at Northwestern University in March 2023. We also thank William Balderrama for an illuminating conversation.
Among the 128 different $\mathcal{A}^\mathbb{R}$-module structures on $\mathcal{A}^\mathbb{R}(1)$, only 16 are self-dual.
Theorem 1.4.
Of the self-dual $\mathcal{A}^\mathbb{R}$-module structures on $\mathcal{A}^\mathbb{R}(1)$, 8 can be realized as the cofiber of a self-map on $\mathcal{Y}^\mathbb{R}_{(2,1)}$ and 8 as the cofiber of a self-map on $\mathcal{Y}^\mathbb{R}_{(\mathsf{h},1)}$.
When $\mathrm{H}_{\star } (\mathrm{X})$ is free and finitely generated as an $\mathbb{M}_2^\mathbb{R}$-module, the map $\mathfrak{n}$ in Equation 2.6 is an isomorphism. Consequently, elements in $\mathrm{H}^{\star }(\mathrm{X})$ can be identified with linear maps from $\mathrm{H}_{\star }(\mathrm{X})$, and the Kronecker product is simply the evaluation of functionals.
Assumption 3.1.
Let $\mathrm{X}$ be a finite $\mathbb{R}$-motivic spectrum such that its homology $\mathrm{H}_{\star }(\mathrm{X})$ is free over $\mathbb{M}_2^\mathbb{R}$.
for $\varphi \in \mathcal{A}^\mathbb{R}$,$\muplambda \in \mathbf{N}^\vee$, and $n \in \mathbf{N}$.
Remark 3.6.
If $\muppsi _{\mathrm{L}}(n) = \sum _i a_i \otimes n_i$, for $a_i \in \mathcal{A}^\mathbb{R}_\star$ and $n_i \in \mathbf{N}$, then Equation 3.5 can be rewritten as
Let $\mathbf{N}$ be a left $\mathcal{A}^\mathbb{R}_\star$-comodule with coproduct $\muppsi _{\mathrm{L}}$. Then, for $n\in \mathbf{N}$ and $\varphi \in \mathcal{A}^\mathbb{R}$, the formula
Let $\mathbf{N}$ be a left $\mathcal{A}^\mathbb{R}_\star$-comodule with coaction map $\muppsi _{\mathrm{L}}$. Then $\muppsi _{\mathrm{L}}$ is related to $\mupphi _{\mathrm{L}}$ using the formula
where $\cdot$ is the action of $\mathcal{A}^\mathbb{R}$ on $\mathbf{N}$ constructed using Proposition 3.10.
Example 3.16($\mathbb{R}$-motivic mod ${\mathsf{h}}$ Moore spectrum).
As a graded $\mathbb{M}_2^\mathbb{R}$-module,$\mathrm{H}^{\star }(\mathbb{S}/\mathsf{h})$ is isomorphic to $\mathrm{H}^{\star }(\mathbb{S}/2)$. However, they differ in their $\mathcal{A}^\mathbb{R}$-module structures in that
and using Equation 3.7 we see that $\Big ( \mathrm{H}^\star (\mathbb{S}_\mathbb{R}/\mathsf{h}) \Big )^\vee \cong \Sigma ^{-1} \mathrm{H}^\star (\mathbb{ S}_\mathbb{R}/\mathsf{h})$. This aligns with the fact that $D(\mathbb{S}_\mathbb{R}/\mathsf{h})$ is equivalent to $\Sigma ^{-1} \mathbb{S}_\mathbb{R}/\mathsf{h}$.
there exists a unique isomorphism class of $\mathcal{A}^\mathbb{R}$-module structures on $\mathcal{A}^\mathbb{R}(1)$, which we denote by $\mathcal{A}^\mathbb{R}_{ \overline{\mathrm{v}}}(1)$, determined by the formulas
where $\mathscr{j}_{24}= \beta _{03} \gamma _{36} + \alpha _{03}(\beta _{25} + \beta _{26})$. Further, any $\mathcal{A}^\mathbb{R}$-module whose underlying $\mathcal{A}^\mathbb{R}(1)$-module is free on one generator is isomorphic to one listed above.
Corollary 4.3.
For the $\mathcal{A}^\mathbb{R}$-module$\mathcal{A}^\mathbb{R}_{ \overline{\mathrm{v}}}(1)$, its (regraded) dual is isomorphic to
where $\mupdelta (\overline{\mathrm{v}}) = (\gamma _{36},\beta _{25}+\beta _{26},\beta _{25},\mathscr{j}_{24}+\beta _{06},\beta _{14},\beta _{03} + \beta _{14},\alpha _{03})$. Thus, $\mathcal{A}^\mathbb{R}_{ \overline{\mathrm{v}}}(1)$ is self dual if and only if
(1)
$\alpha _{03} = \gamma _{36}$,
(2)
$\beta _{03} = \beta _{25} + \beta _{26}$, and
(3)
$\beta _{14} = \beta _{25}$.
Corollary 4.6.
Suppose $\mathcal{A}^\mathbb{R}_1[\overline{\mathrm{v}}]$ is an $\mathbb{R}$-motivic spectrum realizing $\mathcal{A}^\mathbb{R}_{ \overline{\mathrm{v}}}(1)$, and suppose that $\mathcal{A}^\mathbb{R}_{ \overline{\mathrm{v}}}(1)$ is a self-dual $\mathcal{A}^\mathbb{R}$-module. Then $\mathcal{A}^\mathbb{R}_1[\overline{\mathrm{v}}]$ is the cofiber of a $v_1$-self-map on either $\mathcal{Y}^\mathbb{R}_{2,1}$ or $\mathcal{Y}^\mathbb{R}_{\mathsf{h},1}$.
Proposition A.1.
Suppose that $\chi ^\mathbb{R}\colon \mathcal{A}^\mathbb{R}\longrightarrow \mathcal{A}^\mathbb{R}$ is a ring antihomomorphism and an involution. Then
Note that if $\mathbf{N}$ is an $\mathcal{A}^\mathbb{R}$-module, then the action of $\mathcal{A}^\mathbb{R}$ on $\mathbf{N}$ is not $\mathbb{M}_2^\mathbb{R}$-linear, so that, in contrast to the classical case, it does not induce a right $\mathcal{A}^\mathbb{R}$-action on the dual $\mathbf{N}^{\vee }$. Even if $\mathcal{A}^\mathbb{R}$ were to be hypothetically equipped with an antiautomorphism $\chi ^\mathbb{R}$, this may not be so useful for the purpose of dualization. The reason is that the classical formula Equation 1.1 does not work in this setting. More precisely, let $\mathbf{N}$ be an $\mathcal{A}^\mathbb{R}$-module, let $\lambda \in \mathbf{N}^{\vee }$,$\varphi \in \mathcal{A}^\mathbb{R}$, and $n \in \mathbf{N}$. Then defining a new action $\varphi \odot \lambda$ by
from Example 3.16. Then $(\operatorname {Sq}^2\odot \hat{\mathsf{x}}_{1,0})( {\tau }{\mathsf{x}}_{0,0})$ vanishes, whereas $(\operatorname {Sq}^2\odot \hat{\mathsf{x}}_{1,0})( {\mathsf{x}}_{0,0})$ is equal to ${\rho }$. It follows that the formula for $\operatorname {Sq}^2 \odot \hat{\mathsf{x}}_{1,0}$ is not $\mathbb{M}_2^\mathbb{R}$-linear and is therefore not a valid element of $\mathbf{N}^\vee$.
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![\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \setlength{\extrarowheight}{0.0pt}\begin{tikzpicture} \begin{scope}[ thick, every node/.style={sloped,allow upside down}, scale=0.8] \draw(0,0) node[inner sep=0] (v0) {} -- (0,1) node[inner sep=0] (v1) {}; \draw(-0.5,2) node[inner sep=0] (v2) {}; \draw(0,3) node[inner sep=0] (v3) {} -- (0,4) node[inner sep=0] (v4) {}; \draw[color=blue] (v1) to [bend right=50] (v3); \draw[color=blue] (v0) to [bend left=50] (v2); \draw[color=blue] (v2) to [bend left=50] node[pos=0.3,above={4pt},rotate=-95] {$\mtau$} (v4); \draw[red] (v2) to (-1.5,2) to node[pos=0.3,above={5pt},rotate=-90] {$\mrho^2$} (-1.5,4) to (v4); \draw[red] (v0) to (1.15,0) to node[pos=0.6,below={5pt},rotate=-90] {$\mtau$} (1.15,4) to (v4); \filldraw(v0) circle (2.5pt); \filldraw(v1) circle (2.5pt); \filldraw(v2) circle (2.5pt); \filldraw(v3) circle (2.5pt); \filldraw(v4) circle (2.5pt); \draw(0,0) node[below]{$ {\mathsf{x}}_{0,0}$} (0,1) node[right]{$\ {\mathsf{x}}_{1,0} $} (-0.5,2) node[below right={-4pt},yshift={-1pt}]{$\ {\mathsf{x}}_{2,1} $}; \draw(0,4) node[above]{${\mathsf{x}}_{4,1} $} (0,3) node[right]{$\ {\mathsf{x}}_{3,1}$}; \draw(0,-1.5) node { $\mr{H}^{\star}(\mathcal{J}_{\R})$}; \end{scope} \end{tikzpicture}](Images/img80862914b3c7933a97bf0b0b08902e7b.svg)
![\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \setlength{\extrarowheight}{0.0pt}\begin{tikzpicture} \begin{scope}[ thick, every node/.style={sloped,allow upside down}, scale=0.8] \draw(0,0) node[inner sep=0] (v0) {} -- (0,1) node[inner sep=0] (v1) {}; \draw(-0.5,2) node[inner sep=0] (v2) {}; \draw(0,3) node[inner sep=0] (v3) {} -- (0,4) node[inner sep=0] (v4) {}; \draw[red] (v0) to (-1.5,0) to node[pos=0.25,above={5pt},rotate=-90] {$\mrho^2$} (-1.5,2) to (v2); \draw[color=blue] (v1) to [bend right=50] (v3); \draw[color=blue] (v0) to [bend left=50] node[pos=0.4,above={5pt},rotate=-110] {$\mtau$} (v2); \draw[color=blue] (v2) to [bend left=50] (v4); \filldraw(v0) circle (2.5pt); \filldraw(v1) circle (2.5pt); \filldraw(v2) circle (2.5pt); \filldraw(v3) circle (2.5pt); \filldraw(v4) circle (2.5pt); \draw(0,0) node[below ]{$ \hat{\mathsf{x}}_{4,1}$} (0,1) node[ right]{$\ \hat{\mathsf{x}}_{3,1} $} (-0.5,2) node[below right={-4.5pt},yshift={1pt}]{$\ \hat{\mathsf{x}}_{2,1} $}; \draw(0,4) node[above]{$\hat{\mathsf{x}}_{0,0} $} (0,3) node[right]{$\ \hat{\mathsf{x}}_{1,0}$}; \draw(0,-1.5) node {$\mr{H}^{\star}(\mr{D}\mathcal{J}_{\R})$}; \end{scope} \end{tikzpicture}](Images/imgbff89fd59a091171217a4c87f87382a5.svg)
![\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \setlength{\extrarowheight}{0.0pt}\begin{tikzpicture} \begin{scope}[thick, every node/.style={sloped,allow upside down}, scale=0.7] \draw(0,0) node[inner sep=0] (v00) {} -- (0,1) node[inner sep=0] (v01) {}; \draw(0,2) node[inner sep=0] (v11) {} -- (0,3) node[inner sep=0] (v12) {}; \draw(1,3) node[inner sep=0] (v22) {} -- (1,4) node[inner sep=0] (v23) {}; \draw(1,5) node[inner sep=0] (v33) {} -- (1,6) node[inner sep=0] (v34) {}; \draw[color=blue] (v00) to [out=150,in=-150] (v11); \draw[color=blue] (v01) to [out=15,in=-90] (v22); \draw[color=blue] (v12) to [out=90,in=-165] (v33); \draw[color=blue] (v23) to [out=30,in=-30] (v34); \draw[color=blue] (v11) to [out=15,in=-165] node[pos=0.6,above={3pt},rotate=-85] {{$\mtau$}} (v23); \filldraw(v00) circle (2.5pt); \filldraw(v01) circle (2.5pt); \filldraw(v11) circle (2.5pt); \filldraw(v12) circle (2.5pt); \filldraw(v22) circle (2.5pt); \filldraw(v23) circle (2.5pt); \filldraw(v33) circle (2.5pt); \filldraw(v34) circle (2.5pt); \draw(0,0) node[right]{$ {\mathsf{x}}_{0,0}$} (0,1) node[right]{ \ \ ${\mathsf{x}}_{1,0}$} (0,2) node[left,xshift={1pt}]{${\mathsf{x}}_{2,1} \ $ } (0,3) node[left]{${\mathsf{x}}_{3,1} $ }; \draw(1,6) node[right]{$\ {\mathsf{y}}_{6,2}$} (1.3,5.3) node[left]{${\mathsf{y}}_{5,2} \ $} (1,4) node[right]{$ \ {\mathsf{y}}_{4,1} $} (1,3) node[right]{$ {\mathsf{y}}_{3,1}$}; \end{scope} \end{tikzpicture}](Images/img9767635e8aa603748a39c612ba9be961.svg)
![\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \setlength{\extrarowheight}{0.0pt}\begin{tikzpicture} \begin{scope}[thick, every node/.style={sloped,allow upside down}, scale=0.7] \draw(0,0) node[inner sep=0] (v00) {} -- (0,1) node[inner sep=0] (v01) {}; \draw(0,2) node[inner sep=0] (v11) {} -- (0,3) node[inner sep=0] (v12) {}; \draw(1,3) node[inner sep=0] (v22) {} -- (1,4) node[inner sep=0] (v23) {}; \draw(1,5) node[inner sep=0] (v33) {} -- (1,6) node[inner sep=0] (v34) {}; \draw[color=blue] (v00) to [out=150,in=-150] (v11); \draw[color=blue] (v01) to [out=15,in=-90] (v22); \draw[color=blue] (v12) to [out=90,in=-165] (v33); \draw[color=blue] (v23) to [out=30,in=-30] (v34); \draw[color=blue] (v11) to [out=15,in=-165] node[pos=0.6,above={3pt},rotate=-85] {{$\mtau$}} (v23); \filldraw(v00) circle (2.5pt); \filldraw(v01) circle (2.5pt); \filldraw(v11) circle (2.5pt); \filldraw(v12) circle (2.5pt); \filldraw(v22) circle (2.5pt); \filldraw(v23) circle (2.5pt); \filldraw(v33) circle (2.5pt); \filldraw(v34) circle (2.5pt); \draw(0,0) node[right]{$ \hat{\mathsf{y}}_{6,2}$} (0,1) node[right]{ \ \ $\hat{\mathsf{y}}_{5,2}$} (0,2) node[left,xshift={1pt}]{$\hat{\mathsf{y}}_{4,1} \ $ } (0,3) node[left]{$\hat{\mathsf{y}}_{3,1} $ }; \draw(1,6) node[right]{$\ \hat{\mathsf{x}}_{0,0}$} (1.3,5.3) node[left]{$\hat{\mathsf{x}}_{1,0} \ $} (1,4) node[right]{$ \ \hat{\mathsf{x}}_{2,1} $} (1,3) node[right]{$ \hat{\mathsf{x}}_{3,1}$}; \end{scope} \end{tikzpicture}](Images/imge1667db48af2905ef8e4d92c2e460c32.svg)
![\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \setlength{\extrarowheight}{0.0pt}\begin{tikzpicture}\begin{scope}[ thick, every node/.style={sloped,allow upside down}, scale=0.8] \draw(0,0) node[inner sep=0] (s0) {} -- (0,1) node[inner sep=0] (s1a) {}; \draw(1.5,2) node[inner sep=0] (t2) {} -- (1.5,3) node[inner sep=0] (t3a) {}; \draw(3,1) node[inner sep=0] (s1b) {} -- (3,2) node[inner sep=0] (s2) {}; \draw(4.5,3) node[inner sep=0] (t3b) {} -- (4.5,4) node[inner sep=0] (t4) {}; \draw[color=blue,bend right] (s0) to node[pos=0.5,above] {$\alpha_{03}$} (t2); \draw[color=blue,out=20,in=-160] (s0) to node[pos=0.5,below] {$\beta_{03}$} (s2); \draw[color=blue,bend left=20] (s1a) to node[pos=0.4,above={-1pt}] {$\beta_{14}$} (t3a); \draw[color=blue,out=150,in=-30] (s1b) to (t3a); \draw[color=blue,bend right] (s1b) to node[pos=0.5,below] {$\beta_{14}$}(t3b); \draw[color=blue,out=30,in=210] (s2) to node[pos=0.5,below] {$\alpha_{03}$} (t4); \draw[color=blue,out=15,in=195] (t2) to node[pos=0.7,above] {$\beta_{03}$} (t4); \draw[color = red] (s0) to (5.5,0) to (5.5,4) to (t4); \filldraw(s0) circle (2.5pt); \filldraw(s1a) circle (2.5pt); \filldraw(t2) circle (2.5pt); \filldraw(t3a) circle (2.5pt); \filldraw(s1b) circle (2.5pt); \filldraw(s2) circle (2.5pt); \filldraw(t3b) circle (2.5pt); \filldraw(t4) circle (2.5pt); \draw(s0) node[left]{$ {\mathsf{s}}_0$} (0,1) node[left]{$ {\mathsf{s}}_{1a}$} (s1b) node[below]{${\mathsf{s}}_{1b} $} (s2) node[right]{$\ {\mathsf{s}}_2 $}; \draw(t2) node[left]{$ {\mathsf{t}}_2$} (t3a) node[above]{$ {\mathsf{t}}_{3a}$} (t3b) node[below right={-2pt}]{${\mathsf{t}}_{3b} $} (t4) node[above]{${\mathsf{t}}_4 $}; \draw(5.5,2) node[right]{$\beta_{06}$}; \end{scope}\end{tikzpicture}](Images/imgbf35357dd466c054aab6ea4ea91c064d.svg)