arXiv subjects

Gabriel Angelini-Knoll

Publications and source records attributed to Gabriel Angelini-Knoll.

17 recordsLinked to original sources

arXiv2026

On the integral algebraic K-theory of Morava K-theory

We compute the cardinalities of the integral algebraic K-theory groups of connective Morava K-theory $k(n)$ in all degrees that are not congruent to $0$ or $1$ modulo $2p-2$. After base-changing the coefficients of $k(n)$ to the algebraic closure $\overline{\mathbb{F}}_p$, we determine the corresponding cardinalities in all degrees and show that the groups vanish in even degrees. Our approach uses what we call the orbit filtration on topological cyclic homology arising from the May filtration on topological Hochschild homology. Combining this with an analysis of the topological cyclic homology of formal DGAs of the form $\mathbb{F}[x_{2m}]$, where $\mathbb{F}$ is a perfect field of characteristic $p$, we prove strong cardinality results for the topological cyclic homology of $\mathbb{E}_1$-rings with homotopy $\mathbb{F}[x_{2m}]$. For Morava K-theories over finite fields, we further use the motivic filtration of Hahn--Raksit--Wilson. As an application of our methods, we compute the cardinalities of the algebraic K-theory groups of the truncation $W(\overline{\mathbb{F}}_p)/p^n$ of the $p$-typical Witt vectors of $\overline{\mathbb{F}}_p$; in particular, they vanish in even degrees.

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arXiv2026

Syntomic cohomology of truncated Brown--Peterson spectra

We compute the $\mathrm{MU}$-based syntomic cohomologies, mod $(p,v_1,\cdots,v_n)$, of all $\mathbb{E}_1$ $\mathrm{MU}$-algebra forms of the truncated Brown--Peterson spectrum $\mathrm{BP}\langle n\rangle$. As qualitative consequences, we resolve the Lichtenbaum--Quillen, telescope, and redshift questions for the algebraic K-theories of all $\mathbb{E}_{1}$ $\mathrm{MU}$-algebra forms of $\mathrm{BP} \langle n\rangle$. This extends work of the Hahn and Wilson. We also explicitly compute the algebraic K-theory of arbitrary $\mathbb{E}_{1}$ $\mathrm{MU}$-algebra forms of $\mathrm{BP}\langle 2\rangle$ at all primes $p\ge 5$ extending previous work of the author, Ausoni, Culver, Höning, and Rognes.A dditionally, we present a new computation of mod $(p, v_1, v_2, v_3)$ algebraic K-theory of arbitrary $\mathbb{E}_1$ $\mathrm{MU}$-algebra forms of $\mathrm{BP}\langle 3\rangle$ at all primes $p\ge 7$, the first explicit computation of algebraic K-theory of an $\mathbb{E}_1$-ring of height $3$.

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arXiv2026

Spoke topological Hochschild homology

Fix primes $p$ and $\ell$, and let $C_p$ be the cyclic group of order $p$. We compute the $C_p$-equivariant spoke topological Hochschild homology of $\underline{\mathbb{F}}_{\ell}$ and prove it exhibits a form of Bökstedt periodicity. Here spoke topological Hochschild homology is a variant of topological Hochschild homology where one replaces the circle in the construction with the unreduced suspension of $C_p$. As an application, we use this result to give a new proof of the Segal conjecture for the cyclic group of order an odd prime $p$.

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arXiv2026

Algebraic K-theory of real topological K-theory

We determine the A(1)-homotopy of the topological cyclic homology of the connective real K-theory spectrum ko. The answer has an associated graded that is a free F_2[v_2^4]-module of rank 52, on explicit generators in stems -1 \le * \le 30. The calculation is achieved by using prismatic and syntomic cohomology of ko as introduced by Hahn-Raksit-Wilson, extending work of Bhatt-Morrow-Scholze from the case of classical commutative rings to E_\infty rings. A new feature in our case is that there are nonzero differentials in the motivic spectral sequence from syntomic cohomology to topological cyclic homology.

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arXiv2026

Topological symmetric and braid homologies

We identify topological symmetric homology as the free $\mathbb{E}_\infty$-algebra on an $\mathbb{E}_1$-algebra and topological braid homology as the free $\mathbb{E}_2$-algebra on an $\mathbb{E}_1$-algebra. In this way, topological symmetric homology and topological braid homology can be regarded as variants of $1$-dimensional representation homology. In order to identify topological braid homology as the free $\mathbb{E}_2$-algebra on an $\mathbb{E}_1$-algebra, we prove that the $\mathbb{E}_2$-monoidal envelope of the associative operad can be identified with the braided crossed simplicial group. Using this, we also compute the topological braid homology of grouplike $\mathbb{E}_1$-spaces. Further, we develop computational tools for topological symmetric and braid homologies. These tools allow us to perform low-degree computations of topological symmetric homology and prove that it is not Morita invariant. We also compute the topological $Δ\mathbf{G}$-homology of Thom spectra in general and produce explicit formulas in the case of topological symmetric and braid homologies.

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arXiv2026

Topological Hochschild homology of truncated Brown-Peterson spectra II

We compute topological Hochschild homology of $\mathbb{E}_3$-MU-algebra forms of the second truncated Brown-Peterson spectrum with Adams summand coefficients at $p=2$ and conditionally at arbitrary primes. We also provide a new computational tool, a variant of the Brun spectral sequence, for computing topological Hochschild homology of truncated Brown-Peterson spectra with certain coefficients. As a consequence, we show that $\mathbb{E}_3$-MU-algebra forms of truncated Brown-Peterson spectra are not Thom spectra $\mathrm{BP}\langle n\rangle$ at the prime $p=2$ for any $n\ge 2$.

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arXiv2026

Topological $ΔG$ homology of rings with twisted $G$-action

We construct topological $ΔG$-homology for rings with twisted $G$-action. Here a ring with twisted $G$-action is a common generalization of a ring with anti-involution and a ring with $G$-action. This construction recovers as special cases topological Hochschild homology (THH) of rings, with its $S^1$-action, and Real topological Hochschild homology (THR) of rings with anti-involution, with its $O(2)$-action. A new example of this construction is quaternionic topological Hochschild homology (THQ) of rings with twisted $C_4$-action, which carries a $Pin(2)$-action. We prove that THQ of a loop space with twisted $C_4$-action can be $Pin(2)$-equivariantly identified with a twisted free loop space. Other new examples of interest are topological symmetric homology and topological hyperoctrahedral homology and more generally topological twisted symmetric homology. We prove a homotopical version of results of Fiedorowicz, Ault, and Graves computing these new topological homology theories on loop spaces with twisted $G$-action. A key step of independent interest in this program is the construction of a new family of crossed simplicial groups, which correspond to operads that encode the structure of rings with twisted $G$-action.

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arXiv2025

Real topological Hochschild homology via the norm and Real Witt vectors

We prove that Real topological Hochschild homology can be characterized as the norm from the cyclic group of order $2$ to the orthogonal group $O(2)$. From this perspective, we then prove a multiplicative double coset formula for the restriction of this norm to dihedral groups of order $2m$. This informs our new definition of Real Hochschild homology of rings with anti-involution, which we show is the algebraic analogue of Real topological Hochschild homology. Using extra structure on Real Hochschild homology, we define a new theory of $p$-typical Witt vectors of rings with anti-involution. We end with an explicit computation of the degree zero $D_{2m}$-Mackey functor homotopy groups of $\operatorname{THR}(\underline{\mathbb{Z}})$ for $m$ odd. This uses a Tambara reciprocity formula for sums for general finite groups, which may be of independent interest.

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arXiv2025

Real syntomic cohomology

We introduce a theory of syntomic cohomology for ring spectra with involution, which we call Real syntomic cohomology. We show that our construction extends the theory of syntomic cohomology for rings with involution due to Park. Our construction also refines syntomic cohomology as developed by Bhatt--Morrow--Scholze, Morin, Bhatt--Lurie, and Hahn--Raksit--Wilson. We compute the Real syntomic cohomology of Real topological K-theory and topological modular forms with level structure.

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arXiv2025

A deformation of Borel equivariant homotopy

We describe a deformation of the $\infty$-category of Borel $G$-spectra for a finite group $G$. This provides a new presentation of the $a$-complete real Artin--Tate motivic stable homotopy category when $G=C_2$ and gives a new interpretation of the $a$-completed $C_2$-effective slice spectral sequence. As a new computational tool, we present a modified Adams--Novikov spectral sequence which computes the $RO(G)$-graded Mackey functor valued homotopy of Borel $G$-spectra.

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arXiv2025

Commuting unbounded homotopy limits with Morava K-theory

This paper provides conditions for Morava $K$-theory to commute with certain homotopy limits. These conditions extend previous work on this question by allowing for homotopy limits of sequences of spectra that are not uniformly bounded below. As an application, we prove the $K(n)$-local triviality (for sufficiently large $n$) of the algebraic $K$-theory of algebras over truncated Brown--Peterson spectra, building on work of Bruner--Rognes and extending a classical theorem of Mitchell on $K(n)$-local triviality of the algebraic K-theory spectrum of the integers for large enough $n$.

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arXiv2025

A homological approach to chromatic complexity of algebraic K-theory

The family of Thom spectra $y(n)$ interpolates between the sphere spectrum and the mod two Eilenberg--MacLane spectrum. Computations of Mahowald, Ravenel, Shick, and the authors show that the associative ring spectrum $y(n)$ has type $n$. Using trace methods, we give evidence that algebraic K-theory preserves this chromatic complexity. Our approach sheds light on the chromatic complexity of topological negative cyclic homology and topological periodic cyclic homology, which approximate algebraic K-theory and are of independent interest. Our main contribution is a homological approach that can be applied in great generality, such as to associative ring spectra $R$ without additional structure whose coefficient rings are not completely understood.

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arXiv2025

Syntomic cohomology of Morava K-theory

We compute the MU-based syntomic cohomologies, mod $(p,v_1,\cdots,v_{n+1})$, of all $\mathbb{E}_1$-MU-algebra forms of connective Morava K-theory k(n). As qualitative consequences, we deduce the Lichtenbaum--Quillen conjecture, telescope conjecture, and redshift conjecture for the algebraic K-theories of all $\mathbb{E}_{1}$-$\mathbb{S}$-algebra forms of $(2p^n-2)$-periodic Morava K-theory. Notably, the motivic spectral sequence computing $π_*TC(k(n))_p$ is concentrated on at most three lines, independently of $n$.

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arXiv2023

Algebraic K-theory of elliptic cohomology

We calculate the mod (p, v_1, v_2) homotopy V(2)_* TC(BP<2>) of the topological cyclic homology of the truncated Brown--Peterson spectrum BP<2>, at all primes p\ge7, and show that it is a finitely generated and free F_p[v_3]-module on 12p+4 generators in explicit degrees within the range -1 \le * \le 2p^3+2p^2+2p-3. At these primes BP<2> is a form of elliptic cohomology, and our result also determines the mod (p, v_1, v_2) homotopy of its algebraic K-theory. Our computation is the first that exhibits chromatic redshift from pure v_2-periodicity to pure v_3-periodicity in a precise quantitative manner.

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arXiv2023

Topological Hochschild homology of truncated Brown-Peterson spectra I

We compute topological Hochschild homology of sufficiently structured forms of truncated Brown--Peterson spectra with coefficients. In particular, we compute $\mathrm{THH}_*(B\langle n\rangle ;H\mathbb{Z}_{(p)})$ for all $n$ and $\mathrm{THH}_*(B\langle 2\rangle ;M)$ for $M\in \{ k(1),k(2)\}$ where $B\langle n\rangle$ is an $E_3$ form of $BP\langle n\rangle$ for certain primes $p$. For example, this gives a computation of $\mathrm{THH}(\mathrm{taf}^{D};M)$ for $M\in \{H\mathbb{Z}_{(3)},k(1),k(1))$ where $\mathrm{taf}^{D}$ is the $E_{\infty}$ form of $BP\langle 2\rangle$ constructed by Hill--Lawson.

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arXiv2022

Detecting $β$ elements in iterated algebraic K-theory

The Lichtenbaum--Quillen conjecture (LQC) relates special values of zeta functions to algebraic K-theory groups. The Ausoni--Rognes red-shift conjectures generalize the LQC to higher chromatic heights in a precise sense. In this paper, we propose an alternate generalization of the LQC to higher chromatic heights and give evidence for it at height two. In particular, if the $n$-th Greek letter family is detected by a commutative ring spectrum $R$, then we conjecture that the $n+1$-st Greek letter family will be detected by the algebraic K-theory of $R$. We prove this in the case $n=1$ for $R=\text{K}(\mathbb{F}_q)$ modulo $(p,v_1)$ where $p\ge 5$ and $q=\ell^k$ is a prime power generator of the units in $\mathbb{Z}/p^2\mathbb{Z}$. In particular, we prove that the commutative ring spectrum $\text{K}(\text{K}(\mathbb{F}_q))$ detects the part of the $p$-primary $β$-family that survives mod $(p,v_1)$. The method of proof also implies that these $β$ elements are detected in iterated algebraic K-theory of the integers. Consequently, one may relate iterated algebraic K-theory groups of the integers to integral modular forms satisfying certain congruences.

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arXiv2022

Complex orientations and $\text{TP}$ of complete DVRS

Let $L$ be finite extension of $\mathbb{Q}_p$ with ring of integers $\mathcal{O}_L$. I show that periodic topological cyclic homology of $\mathcal{O}_L$, over the base $\mathbb{E}_{\infty}$-ring $\mathbb{S}_{W(\mathbb{F}_q)}[z]$ carries a $p$-height one formal group law mod $p$ that depends on the Eisenstein polynomial of $L$ over $\mathbb{Q}_p$ for a choice of uniformizer $\varpi\in \mathcal{O}_L$.

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