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On the ring of cooperations for real Hermitian K-theory

This paper provides a complete description of the ring of cooperations for the very effective cover of motivic Hermitian K-theory over the real numbers in terms of Brown–Gitler comodules by demonstrating the collapse of the motivic Adams spectral sequence and establishing a splitting result for very effective symplectic K-theory.

Original authors: Jackson Morris

Published 2026-03-27

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CC BY 4.0

Original authors: Jackson Morris

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe of mathematics as a vast, multi-layered city. In this city, there is a special neighborhood called Topology, where mathematicians study the shapes of things—not just their size or color, but how they are connected, twisted, and knotted.

For a long time, mathematicians have been trying to map the "holes" and "loops" in the most fundamental shape of all: the sphere. To do this, they use a powerful tool called a spectral sequence. Think of a spectral sequence not as a single map, but as a multi-stage telescope. You look through the first lens, see a blurry image, then adjust the focus (the next stage), and slowly the picture becomes clearer. Eventually, you hope to see the true shape of the sphere.

However, looking directly at the sphere is incredibly hard. So, mathematicians use a clever trick: they build a "proxy" object that is easier to study, hoping it reveals secrets about the real sphere.

The Main Characters

The Sphere ( $S$ ): The ultimate mystery we want to solve.
The Proxy ($kq$): A specific mathematical object called "very effective Hermitian K-theory." Think of this as a high-tech drone we send out to scout the sphere. It's a simplified version of a more complex machine (Hermitian K-theory) that is easier to fly.
The Cooperations ( $kq \otimes kq$ ): This is the tricky part. To make our drone work, we need to know how two of them interact when they crash into each other. In math, this is called the "ring of cooperations." It's like asking: "If I have two of these drones, how do their sensors talk to each other? What data do they exchange?"

The Problem: Two Different Worlds

The author, Jackson Morris, is working in a world called Motivic Homotopy Theory. This is a strange place where math mixes with geometry and algebra.

The Complex World ( $\mathbb{C}$ ): Imagine a world where you can see in full color and every direction is smooth. Mathematicians had already figured out how the drones interact here.
The Real World ( $\mathbb{R}$ ): This is our world. It's more rugged. There are "twists" and "obstacles" (mathematicians call these $\rho$ -periodic data) that don't exist in the smooth Complex world.

The big question was: Can we figure out how the drones interact in our rugged Real world, using what we know from the smooth Complex world?

The Solution: A New Kind of Lego Set

In the past, when mathematicians tried to solve similar problems in the "classical" world (just standard shapes, not this mixed math-geometry world), they used a set of building blocks called Brown-Gitler spectra. These were like pre-made Lego bricks that fit together perfectly to build the interaction map.

The Catch: In this new "Motivic" world, we don't have those physical Lego bricks yet. No one has built them!

Morris's Innovation:
Instead of trying to build the physical Lego bricks (which are hard to construct), Morris decided to build blueprints for them. He created mathematical objects called Brown-Gitler comodules.

The Analogy: Imagine you want to build a house, but you don't have the bricks. Instead, you draw a perfect, detailed blueprint of every brick and how they fit together. You don't need the physical brick to understand the structure; the blueprint is enough to do the math.

Morris used these blueprints to break down the complex interaction of the two drones ( $kq \otimes kq$ ) into smaller, manageable pieces.

The Journey: The Telescope and the Collapse

The Telescope (The Spectral Sequence): Morris set up his multi-stage telescope. He wanted to see the final picture of how the drones interact.
The First Lens (The $E_2$ -page): He calculated the first clear image. This image was made up of his blueprints (the comodules).
The Surprise (The Collapse): Usually, when you look through a telescope, the image keeps changing and shifting as you adjust the focus (differentials happen). But Morris discovered something amazing: The image stopped changing immediately.

He proved that the telescope "collapsed" on the second page. This means the blurry first guess was actually the final, perfect picture. There were no hidden surprises or shifts later on.

Why This Matters

Simplicity: By proving the telescope collapses, Morris showed that the interaction between these mathematical drones is much simpler than we thought.
A New Map: He provided a complete "instruction manual" (a full description) for how these drones talk to each other in the Real world.
Future Exploration: This map is essential for solving bigger mysteries. Just as knowing how two drones interact helps you build a fleet, knowing this "ring of cooperations" helps mathematicians solve deeper problems about the shape of the universe (the homotopy groups of spheres).

The "Symplectic" Side Quest

Along the way, Morris also solved a side mystery about a different drone called Symplectic K-theory ($ksp$). He proved that this drone is actually just a simple combination of the main drone ($kq$) and a tiny, specific piece of Lego (the first Brown-Gitler spectrum). It's like realizing that a complex machine is just a car attached to a specific wheel. This "splitting" made the calculations much easier.

Summary

Jackson Morris took a very difficult, abstract problem about how mathematical shapes interact in a "real" geometric world. Since the physical building blocks for this world didn't exist, he invented a new way to draw the blueprints. Using these blueprints, he showed that the complex interaction simplifies instantly, giving mathematicians a clear, stable map to navigate the future of this field.

In short: He built a blueprint for a missing Lego set, used it to map a complex interaction, and discovered that the map was surprisingly simple and didn't change as you looked closer.

1. Problem Statement and Motivation

The paper addresses a fundamental problem in stable motivic homotopy theory: computing the ring of cooperations $\pi_{**}^{\mathbb{R}}(kq \otimes kq)$ , where $kq$ is the very effective cover of the Hermitian K-theory spectrum $KQ$ over the real numbers $\mathbb{R}$ .

Context: In classical stable homotopy theory, the computation of the homotopy groups of spheres often relies on the Adams Spectral Sequence (ASS). A crucial step in this process is understanding the smash powers of the spectrum used for the resolution. For the real K-theory spectrum $bo$, Mahowald famously utilized a splitting $bo \otimes bo \simeq \bigvee \Sigma^{4k} bo \otimes H\mathbb{Z}_k$ involving integral Brown–Gitler spectra to compute the $bo$-resolution.
Motivic Analogue: In motivic homotopy theory, the $kq$-resolution (analogous to the $bo$-resolution) is a powerful tool for studying $v_1$ -periodic phenomena and the motivic sphere spectrum. However, unlike the classical case, there are no known constructions of motivic integral Brown–Gitler spectra over general base fields (except in very specific cases). This absence prevents the direct application of Mahowald's topological splitting arguments.
Goal: The author aims to compute the $E_2$ -page of the motivic Adams spectral sequence (mASS) for $kq \otimes kq$ over $\mathbb{R}$ , describe its structure as a module over $\pi_{**}^{\mathbb{R}}(kq)$ , and prove that the spectral sequence collapses, thereby determining the ring of cooperations.

2. Methodology

The paper employs a sophisticated algebraic approach, adapting and extending methods used by Culver–Quigley (for the complex case $\mathbb{C}$ ) and Behrens–Ormsby–Stapleton–Stojanoska (for classical $tmf$). The methodology proceeds in three main stages:

A. Decomposition via Brown–Gitler Comodules

Since geometric Brown–Gitler spectra do not exist in the motivic setting, the author works entirely with Brown–Gitler comodules.

The motivic homology $H_{**}(kq)$ is identified with the dual of the subalgebra $(A // A(1))^\vee_{\mathbb{R}}$ .
Using the Mahowald weight filtration, this homology decomposes into a direct sum of integral motivic Brown–Gitler comodules $B^{\mathbb{R}}_0(k)$ :
$H_{**}(kq) \cong \bigoplus_{k \geq 0} \Sigma^{4k, 2k} B^{\mathbb{R}}_0(k)$
By the Künneth isomorphism, the homology of the smash product $kq \otimes kq$ splits similarly, allowing the $E_2$ -page of the mASS to be expressed as a sum of Ext groups over the dual Steenrod subalgebra $A(1)^\vee_{\mathbb{R}}$ :
$E_2^{s,f,w} \cong \bigoplus_{k \geq 0} \text{Ext}^{s,f,w}_{A(1)^\vee_{\mathbb{R}}}(\Sigma^{4k, 2k} B^{\mathbb{R}}_0(k))$

B. Inductive Computation via Algebraic Atiyah–Hirzebruch Spectral Sequences (aAHSS)

To compute the Ext groups $\text{Ext}(B^{\mathbb{R}}_0(k))$ , the author uses an inductive process based on short exact sequences relating the comodules $B^{\mathbb{R}}_0(k)$ .

Base Case: The computation begins with $B^{\mathbb{R}}_0(1)$ . The author constructs an algebraic Atiyah–Hirzebruch spectral sequence (aAHSS) filtering $B^{\mathbb{R}}_0(1)$ by topological degree.
Differentials: The differentials in the aAHSS are determined by the attaching maps in the algebraic cell complex of the comodule. Crucially, the $d_3$ -differential in the $\mathbb{R}$ -motivic case is non-trivial and depends on the field (specifically involving the element $\rho$ ), unlike the $\mathbb{C}$ -motivic or classical cases where it vanishes for degree reasons.
Induction: For $k > 1$ , the author uses short exact sequences (e.g., $0 \to \Sigma B_0(k/2) \to B_0(k) \to \dots \to 0$ ) to relate $\text{Ext}(B_0(k))$ to $\text{Ext}(B_0(k/2))$ and $\text{Ext}(B_1(\dots))$ . This reduces the problem to computing Ext groups of tensor products of lower-order comodules.

C. Handling the $\mathbb{R}$ -Motivic Complexity

The $\mathbb{R}$ -motivic dual Steenrod algebra $A^\vee_{\mathbb{R}}$ is a Hopf algebroid (not a Hopf algebra) due to the presence of the class $\rho$ (representing $-1$).

Coweight Organization: To manage the complexity, the author organizes computations by coweight ($cw = s - w$) modulo 4.
Module Classification: The author introduces a family of specific Ext-modules (e.g., "big diamonds" $D$ , "big staircases" $S$ , "big flags" $F$ , and various towers $H, P, J$ ) to describe the structure of the $E_\infty$ -pages of the spectral sequences.
Comparison with $\mathbb{C}$ : The author utilizes the base change functor $-\otimes \mathbb{C}: SH(\mathbb{R}) \to SH(\mathbb{C})$ and the $\rho$ -Bockstein spectral sequence to lift results from the complex case and resolve hidden extensions.

3. Key Contributions and Results

A. The Ring of Cooperations (Theorem A & Theorem 8.4)

The paper provides a full description of the $E_2$ -page of the mASS for $kq \otimes kq$ over $\mathbb{R}$ .

Structure: The $E_2$ -page is isomorphic to a direct sum of shifted copies of specific modules $Z^{\mathbb{R}}_i$ (defined via the "big flag" and "staircase" modules) and $h_0$ -towers.
$E_2 \cong \bigoplus_{k \geq 0} \left( \Sigma^{4k-4, 2k-2} Z^{\mathbb{R}}_{\alpha(k)} \oplus \bigoplus_{j} (\Sigma^{4j, 2j} H \oplus \Sigma^{4j, 2j-2} H) \right)$
where $\alpha(k)$ is the number of 1s in the dyadic expansion of $k$ .
Collapse: The spectral sequence collapses at the $E_2$ -page. This is proven by showing that every class on the $E_2$ -page is $\tau^4$ -periodic, and base change to $\mathbb{C}$ implies no differentials can exist between $\tau$ -free classes.
Significance: This establishes the ring of cooperations $\pi_{**}^{\mathbb{R}}(kq \otimes kq)$ modulo $v_1$ -torsion, providing the necessary input to study the $kq$-resolution over $\mathbb{R}$ .

B. Splitting of Symplectic K-Theory (Theorem D & Theorem 3.3)

The author proves a novel splitting result for the very effective cover of symplectic K-theory, $ksp$:
$ksp \simeq kq \otimes H\mathbb{Z}^{\mathbb{R}}_1$
where $H\mathbb{Z}^{\mathbb{R}}_1$ is the first integral motivic Brown–Gitler spectrum (realized as a finite complex). This result is derived using the very effective slice filtration and mirrors the classical splitting $bsp \simeq bo \otimes H\mathbb{Z}_1$ .

C. Generalization to Higher Smash Powers (Theorem B)

The methodology is extended to compute the cooperations for $n$ -fold smash powers $kq^{\otimes n}$ . The $E_2$ -page is described as a sum over tuples $K = (k_1, \dots, k_n)$ involving the tensor products of Brown–Gitler comodules $B^{\mathbb{R}}_0(K)$ .

D. Correction of Previous Literature

The paper identifies and corrects an error in the work of Culver–Quigley regarding the structure of $\text{Ext}(B_0(k))$ over $\mathbb{C}$ . The author demonstrates that the previous formula was incorrect for $k \geq 2$ and provides the corrected decomposition involving shifts of $Z_i$ and $h_0$ -towers.

4. Significance and Future Directions

Foundational for $v_1$ -Periodicity: The $kq$-resolution is the primary tool for detecting $v_1$ -periodic phenomena in motivic homotopy theory. By computing the cooperations and proving the collapse of the spectral sequence, this paper lays the groundwork for calculating the $v_1$ -torsion-free component of the motivic sphere spectrum over $\mathbb{R}$ .
Bridging Fields: The work connects motivic homotopy theory over $\mathbb{R}$ with $C_2$ -equivariant homotopy theory. The author notes that the Ext computations here are isomorphic to a summand of the $C_2$ -equivariant Ext groups for $ko_{C_2}$ , suggesting a path to solving the splitting problem for $ko_{C_2} \otimes ko_{C_2}$ .
Arithmetic Applications: The results, combined with pullback squares involving finite fields and $\mathbb{C}$ , allow for the computation of arithmetic invariants over $\mathbb{Z}[1/2]$ , bypassing the lack of a description for Ext over the integers.
Future Work: The author plans to use these results to analyze the differentials in the $kq$-resolution over $\mathbb{R}$ , utilizing $C_2$ -equivariant Betti realization and the $ko_{C_2}$ -resolution to handle $\rho$ -periodic data.

In summary, this paper successfully adapts classical topological techniques to the motivic setting over the reals, overcoming the lack of geometric Brown–Gitler spectra by developing a robust algebraic framework based on comodules and spectral sequences. It provides a complete description of the $kq$-cooperations and sets the stage for deeper investigations into motivic periodicity.