Konstantin Emming | Searcharxiv

arXiv2026

Filtrations in $\mathbb{C}$-motivic stable homotopy theory

We study the effective, connective, and very effective filtrations in the $\mathbb{C}$-motivic, $2$-complete, cellular, stable homotopy category. We do so by using the filtered spectrum model for this category due to Gheorghe-Isaksen-Krause-Ricka, and in particular the motivic analogue functor $\Gamma_\star$. Then we can express the covers making up the respective filtrations of a nice motivic analogue $\Gamma_\star(X)$ via filtered spectra, and use these to compute the slices. Applying this in the case of $X$ being the sphere spectrum, $\text{MU}$, $\text{ku}$, or an Eilenberg-MacLane spectrum recovers a number of conjectures due to Voevodsky. Applying it to $\text{ko}$ recovers a computation of Ananyevskiy-R\"ondigs-{\O}stv{\ae}r. We can also apply it to $\text{tmf}$ and compute the effective slices of the motivic modular forms spectrum $\text{mmf}$. We also study the effective slice spectral sequence for $\Gamma_\star(X)$, which turns out to contain the same information as the classical Adams-Novikov spectral sequence for $X$.

math.AT↗

arXiv2025

The cohomology of $\mathbb{R}$-motivic $\mathcal{A}(2)$

We compute the cohomology of the quotient algebra $\mathcal{A}(2)$ of the $\mathbb{R}$-motivic dual Steenrod algebra. We do so by running a $\rho$-Bockstein spectral sequence whose input is the cohomology of $\mathbb{C}$-motivic $\mathcal{A}(2)$. The purpose of our computation is that the cohomology of $\mathcal{A}(2)$ is the input to an Adams spectral sequence of a hypothetical $\mathbb{R}$-motivic modular forms spectrum. This Adams spectral sequence computes the homotopy groups of such an $\mathbb{R}$-motivic modular forms spectrum, which in turn can be used to make inferences about the homotopy groups of the $\mathbb{R}$-motivic sphere spectrum and eventually about the classical stable stems.

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