arXiv subjects

Tess Bouis

Publications and source records attributed to Tess Bouis.

7 recordsLinked to original sources

arXiv2026

On the motivic cohomology of some singular rings

Using non-$\mathbb{A}^1$-invariant motivic cohomology, we prove motivic refinements of certain known computations of the algebraic $K$-theory of singular rings, such as rings of the form $\mathbb{Z}/p^n$, $\mathbb{Z}[x]/(x^e)$, and $\mathscr{C}(X;\mathbb{C})$ for $X$ a compact Hausdorff space. These refinements are made possible by the use of integral $p$-adic Hodge theory, as a replacement for the standard use of trace methods in $K$-theory.

math.AG↗

arXiv2026

The cyclosyntomic regulator of a number field

We construct a q-deformation of the p-adic regulator of a number field, called the cyclosyntomic regulator, building on the Habiro ring of Garoufalidis-Scholze-Wheeler-Zagier. The key new ingredient in our construction is a refinement of Sulyma's norm maps in prismatic cohomology, which interpolate between classical powers and Frobenius maps at various prime numbers p. Furthermore, we compute the values of the cyclosyntomic regulator at units of the form $1-\zeta$, where $\zeta$ is a root of unity.

math.NT↗

arXiv2025

$\mathbb{A}^1$-connectivity of motivic spaces

We prove a version of Morel's unstable $\mathbb{A}^1$-connectivity theorem over arbitrary base schemes. In the stable setting, this recovers (and simplifies the proof of) the known connectivity bounds due to Morel, Schmidt--Strunk, Deshmukh--Hogadi--Kulkarni--Yadav, and Druzhinin, and extends them to possibly non-noetherian schemes. Using the recent work of Bachmann--Elmanto--Morrow, this also implies that the slice filtration on homotopy $K$-theory is convergent for qcqs schemes of finite valuative dimension.

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arXiv2025

Weibel vanishing and the projective bundle formula for mixed characteristic motivic cohomology

We prove that the motivic cohomology of mixed characteristic schemes, introduced in our previous work, satisfies various expected properties of motivic cohomology, including a motivic refinement of Weibel's vanishing in algebraic $K$-theory, the projective bundle formula, a comparison to Milnor $K$-theory, and a universal characterisation in terms of pro cdh descent. These results extend those of Elmanto--Morrow to schemes which are not necessarily defined over a field.

math.AG↗

arXiv2025

Beilinson--Lichtenbaum phenomenon for motivic cohomology

The goal of this paper is to study non-$\mathbb{A}^1$-invariant motivic cohomology, recently defined by Elmanto, Morrow, and the first-named author, for smooth schemes over possibly non-discrete valuation rings. We establish that the cycle class map from $p$-adic motivic cohomology to a suitable truncation of Bhatt--Lurie's syntomic cohomology is an isomorphism, thereby verifying the Beilinson--Lichtenbaum conjecture in this generality. As a first consequence, we prove that this motivic cohomology integrally recovers the classical definition of motivic cohomology in terms of Bloch's cycle complexes, whenever the latter is defined. As a second consequence, we show a purity theorem for this cohomology theory over perfectoid rings, thus motivically refining a result of Nizio\l{} in algebraic $K$-theory. The key ingredient in our approach is a version of Gabber's presentation lemma applicable in mixed characteristic, non-noetherian settings.

math.AG↗

arXiv2024

Motivic cohomology of mixed characteristic schemes

We introduce a theory of motivic cohomology for quasi-compact quasi-separated schemes, which generalises the construction of Elmanto--Morrow in the case of schemes over a field. Our construction is non-$\mathbb{A}^1$-invariant in general, but it uses the classical $\mathbb{A}^1$-invariant motivic cohomology of smooth $\mathbb{Z}$-schemes as an input. The main new input of our construction is a global filtration on topological cyclic homology, whose graded pieces provide an integral refinement of derived de Rham cohomology and Bhatt--Morrow--Scholze's syntomic cohomology. Our theory satisfies various expected properties of motivic cohomology, including relations to \'etale cohomology and to non-connective algebraic $K$-theory.

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arXiv2023

Cartier smoothness in prismatic cohomology

We introduce the notion of a $p$-Cartier smooth algebra. It generalises that of a smooth algebra and includes valuation rings over a perfectoid base. We give several characterisations of $p$-Cartier smoothness in terms of prismatic cohomology, and deduce a comparison theorem between syntomic and étale cohomologies under this hypothesis.

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