arXiv subjects

Arpon Raksit

Publications and source records attributed to Arpon Raksit.

5 recordsLinked to original sources

arXiv2026

THH(Z) and the image of J

Let $p$ be an odd prime number and $\mathrm{j}_p$ the $p$-complete connective image of J spectrum. We establish an equivalence of cyclotomic $\mathbb{E}_\infty$-rings $\mathrm{THH}(\mathbb{Z})^{\wedge}_p \simeq \mathrm{sh}(\mathrm{j}_p^{\mathrm{triv}})$ and an equivalence of $\mathbb{E}_\infty$-rings $\mathrm{TP}(\mathbb{Z})^{\wedge}_p \simeq \mathrm{j}_p^{\mathrm{t}\mathrm{S}^1}$. We also record a few applications of this: a new perspective, with some new information, on the description of $\mathrm{TC}(\mathbb{Z})^{\wedge}_p$ as a spectrum; height $1$ analogues of the fiber squares of Antieau-Mathew-Morrow-Nikolaus, resulting in new calculations in $\mathrm{K}(1)$-localized algebraic K-theory; and a proof of a slight refinement of the noncommutative crystalline-de Rham comparison result of Petrov-Vologodsky.

math.AT↗

arXiv2026

Notes on Tate cohomology

We formulate a definition of Tate cohomology in the context of three functor formalisms, and we establish basic monoidality and functoriality properties of it in this context. Our approach to these properties is based on the treatment of Nikolaus-Scholze in the setting of local systems of spectra on spaces. We discuss a couple of other specific settings of interest that are accommodated by our generalization.

math.AT↗

arXiv2026

Hochschild homology and the derived de Rham complex revisited

We characterize two objects by universal property: the derived de Rham complex and Hochschild homology together with its Hochschild-Kostant-Rosenberg (HKR) filtration. This involves endowing these objects with extra structure, built on notions of "homotopy-coherent cochain complex" and "filtered circle action" that we study here. We use these universal properties to give a conceptual proof that the associated graded of the HKR filtration identifies with the derived de Rham complex, as well as to give a new construction of the filtrations on cyclic, negative cyclic, and periodic cyclic homology that relate these invariants to derived de Rham cohomology.

math.AG↗

arXiv2025

A motivic filtration on the topological cyclic homology of commutative ring spectra

For a prime number $p$ and a $p$-quasisyntomic commutative ring $R$, Bhatt--Morrow--Scholze defined motivic filtrations on the $p$-completions of $\mathrm{THH}(R), \mathrm{TC}^{-}(R), \mathrm{TP}(R),$ and $\mathrm{TC}(R)$, with the associated graded objects for $\mathrm{TP}(R)$ and $\mathrm{TC}(R)$ recovering the prismatic and syntomic cohomology of $R$, respectively. We give an alternate construction of these filtrations that applies also when $R$ is a well-behaved commutative ring spectrum; for example, we can take $R$ to be $\mathbb{S}$, $\mathrm{MU}$, $\mathrm{ku}$, $\mathrm{ko}$, or $\mathrm{tmf}$. We compute the mod $(p,v_1)$ syntomic cohomology of the Adams summand $\ell$ and observe that, when $p \ge 3$, the motivic spectral sequence for $V(1)_*\mathrm{TC}(\ell)$ collapses at the $\mathrm{E}_2$-page.

math.KT↗

arXiv2020

Motivic Gauß-Bonnet formulas

The apparatus of motivic stable homotopy theory provides a notion of Euler characteristic for smooth projective varieties, valued in the Grothendieck-Witt ring of the base field. Previous work of the first author and recent work of Déglise-Jin-Khan establishes a "Gauß-Bonnet formula" relating this Euler characteristic to pushforwards of Euler classes in motivic cohomology theories. In this paper, we apply this formula to SL-oriented motivic cohomology theories to obtain explicit characterizations of this Euler characteristic. The main new input is a unicity result for pushforward maps in SL-oriented theories, identifying these maps concretely in examples of interest.

math.AG↗