Preprints
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The invariant cell number property for E_4-rings (joint with Chedalavada)
[arxiv link]
For any non-zero E_4-ring R, we prove that an n-fold direct sum of suspensions of R cannot admit an exhaustive filtration by fewer than n cells. We do this by showing that certain integral representations of the symmetric group lift to Schur functors on additively E_3-monoidal ∞-categories. We then reduce the statement to a vanishing fact for rational Schur functors due to Deligne.
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Nilpotence phenomena over E_∞-rings in positive characteristic
[arxiv link] (submitted)
We study the behavior of nilpotent classes in E_∞-rings over finite fields. In characteristic p=2, we show that any nilpotent class can be killed by a nilpotence detecting map of E_∞-rings. As a consequence, we prove the existence of "residue fields", namely that any non-zero E_∞-ring over F_2 admits a map to a separably closed graded field. Conversely, for any odd prime p, we construct non-zero E_∞-rings over F_p which admit nilpotent classes that cannot be killed by any non-zero map of E_∞-rings, a phenomenon that does not occur classically or at any finite height.
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On the K-Theory of algebraic tori (joint with Bai, Carmeli and Juran; [img]) [arxiv link] (submitted)
Given an algebraic torus T over a field F, its lattice of characters 𝛬 gives rise to a topological torus B𝛬(T) with a continuous action of the absolute Galois group G. We construct a natural equivalence between the algebraic K-theory of T and the equivariant homology of B𝛬(T) with coefficients in the G-equivariant K-theory of F. We obtain this equivalence by analyzing the KGL-motive of T in the stable motivic category SH(F) of Voevodsky and Morel. We construct a natural comparison map as a special case of a "motivic Fourier transform" and prove that it is an equivalence by using a motivic Eilenberg--Moore formula for classifying spaces of tori.
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On the deformation theory of E_∞-coalgebras
[updated version]
[arxiv link] (submitted)
We show that, for any prime p and any space X, the F_p-chains of X admit a unique and functorial lift to an E_∞-coalgebra in the ∞-category of p-complete connective spectra, given by the p-completed suspension spectrum of X. We do this by introducing a notion of formally étale coalgebras and prove that the coalgebraic right adjoint to base change along the map S_p -> F_p is fully faithful when restricted to formally étale coalgebras. We conclude by showing that the F_p-chains are formally étale, by reducing to the case of an Eilenberg--MacLane space X=K(F_p,n) and using an explicit presentation of the F_p-chains via power operations.
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