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Håkon Kolderup

Publications and source records attributed to Håkon Kolderup.

8 recordsLinked to original sources
arXiv2025

Hermitian $K$-theory and Milnor-Witt motivic cohomology over $\mathbb Z$

The theme of this paper is to compute hermitian $K$-groups in terms of the recently developed theory of Milnor-Witt motivic cohomology. Our approach makes use of the very effective slice spectral sequence within the motivic stable homotopy category, which we analyze in detail for base schemes of arithmetic interest. We show a Grothendieck-Riemann-Roch theorem, determine the map between Milnor-Witt and hermitian $K$-theory up to degree five for all fields, and compute the hermitian $K$-groups and the higher Witt-groups of the ring of integers.

math.AG↗
arXiv2025

A symmetry approach to number tricks

We generalize the classical "1089-number trick", which states that a certain combination of addition, subtraction and swapping the digits of a three-digit number will always output 1089. More precisely, we show that any pair of zero divisors $fg=0$ in the group ring ${\mathbb Z}[\Sigma_n]$ on the n-th symmetric group gives rise to a partition of the set of n-digit numbers into subsets $U_{\mathbf e}$ defined by linear inequalities, such that the zero divisors act constantly on each $U_{\mathbf e}$ and hence define a number trick.

math.GM↗
arXiv2024

Cousin complexes in motivic homotopy theory

We investigate Cousin (bi-)complexes in the setting of motives. Over essentially smooth local schemes, the columns of the Cousin bicomplex with coefficients in any stable motivic homotopy type are shown to be acyclic. On the other hand, we also construct a family of non-acyclic Cousin complexes over any positive dimensional base scheme. Our method of proof employs the notion of extended compactified framed correspondences. Three major motivations for this study are to further our understanding of strict homotopy invariance, motivic infinite loop spaces, and connectivity in stable motivic homotopy theory. As applications of our main results on motivic Cousin complexes, we generalize several fundamental results in these topics to finite dimensional base schemes.

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arXiv2020

The trivial fiber topology and framed motives over the integers

This paper introduces the trivial fiber topology on schemes. For one-dimensional base schemes, we use it to describe fibrant replacements in the stable motivic homotopy category and motivic infinite loop spaces. We also extend the Garkusha-Panin and Voevodsky strict $\mathbb{A}^{1}$-invariance theorems to one-dimensional base schemes. The trivial fiber topology plays a central role in the proof of refined localization results for motivic homotopy categories. Moreover, we extend Morel's $\mathbb{A}^{1}$-connectivity theorem on Nisnevich sheaves of stable motivic homotopy groups. These results open new vistas for computations of motivic invariants over deeper base schemes of arithmetic interest.

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arXiv2020

Cohomological correspondence categories

We prove that homotopy invariance and cancellation properties are satisfied by any linear category of correspondences that is defined, via Calmès and Fasel's construction, by an underlying cohomology theory. In particular, this includes any category of correspondences arising from the cohomology theory defined by an MSL-algebra.

math.AG↗
arXiv2019

Homotopy invariance of Nisnevich sheaves with Milnor-Witt transfers

The category of finite Milnor-Witt correspondences, introduced by Calmès and Fasel, provides a new type of correspondences closer to the motivic homotopy theoretic framework than Suslin-Voevodsky's correspondences. A fundamental result of the theory of ordinary correspondences concerns homotopy invariance of sheaves with transfers, and in the present paper we address this question in the setting of Milnor-Witt correspondences. Employing techniques due to Druzhinin, Fasel-Østvær and Garkusha-Panin, we show that homotopy invariance of presheaves with Milnor-Witt transfers is preserved under Nisnevich sheafification.

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arXiv2019

On Modules Over Motivic Ring Spectra

In this note, we provide an axiomatic framework that characterizes the stable $\infty$-categories that are module categories over a motivic spectrum. This is done by invoking Lurie's $\infty$-categorical version of the Barr--Beck theorem. As an application, this gives an alternative approach to Röndigs and Østvær's theorem relating Voevodsky's motives with modules over motivic cohomology, and to Garkusha's extension of Röndigs and Østvær's result to general correspondence categories, including the category of Milnor-Witt correspondences in the sense of Calmès and Fasel. We also extend these comparison results to regular Noetherian schemes over a field (after inverting the residue characteristic), following the methods of Cisinski and Déglise.

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