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Morten Lüders

Publications and source records attributed to Morten Lüders.

15 recordsLinked to original sources
arXiv2026

The Gersten conjecture for $p$-adic étale Tate twists and the $p$-adic cycle class map

We prove the Gersten conjecture for $p$-adic étale Tate twists for a smooth scheme $X$ in mixed characteristic in the Nisnevich topology. Our main observation is that, while $p$-adic étale Tate twists are not $\mathbb A^1$-invariant, for the proof of the Gersten conjecture it suffices that they satisfy the $\mathbb P^1$-bundle formula. This fits nicely with the emphasis on the projective bundle formula in non $\mathbb A^1$-invariant motivic cohomology recently developed by Elmanto-Morrow and Annala-Hoyois-Iwasa. Furthermore, identifying $p$-adic étale Tate twists with the syntomic cohomology defined by Bhatt-Morrow-Scholze, the result generalises the Gersten conjecture for logarithmic deRham-Witt sheaves due to Gros-Suwa to arbitrary characteristic. We also extend work of Schmidt-Strunk on the Gersten conjecture for presheaves of spectra to the non-$\mathbb A^1$-invariant situation. In the second part of the article, we revisit the cycle class map from thickened zero-cycles on the special fiber of $X$ to étale cohomology with coefficients in $p$-adic étale Tate twists previously studied in [26]. This cycle class map is important in the study of zero-cycles on smooth projective varieties over local fields and the approach to the cycle class map which we use in this article is more conceptual and, in contrast to the approach in loc. cit., works for arbitrary finite residue fields.

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arXiv2026

Higher dimensional local class field theory

In this largely expository article we give an introduction to higher dimensional local class field theory, more precisely class field theory for smooth projective schemes over local fields. We begin with a quick summary of some of the main results of classical local class field theory. Then we explain the main definitions and statements in the higher dimensional setting. We give a new and simple proof of the $\ell$-part, which is originally due to Jannsen--Saito and Forré. Inspired by this proof, we propose a conjecture which would imply the remaining open cases with $p$-coefficients and which improves our understanding of the $p$-part. A particular goal of the exposition is to outline analogies between the classical and the higher dimensional theory and on how the unit filtration and ramification, i.e. the $p$- and the $\ell$-part, can be interpreted in the higher dimensional setting.

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arXiv2026

On the diagonal of low bidegree hypersurfaces

We study the existence of a decomposition of the diagonal for bidegree hypersurfaces in a product of projective spaces. Using a cycle theoretic degeneration technique due to Lange, Pavic and Schreieder, we develop an inductive procedure that allows one to raise the degree and dimension starting from the quadric surface bundle of Hassett, Pirutka and Tschinkel. Furthermore, we are able to raise the dimension without raising the degree in a special case, showing that a very general $(3,2)$ complete intersection in $\mathbb P^4\times \mathbb P^3$ does not admit a decomposition of the diagonal. As a corollary of these theorems, we show that in a certain range, bidegree hypersurfaces which were previously only known to be stably irrational over fields of characteristic zero by results of Moe, Nicaise and Ottem, are not retract rational over fields of characteristic different from two.

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arXiv2025

On the diagonal of quartic hypersurfaces and $(2,3)$-complete intersection $n$-folds

We study the question of the existence of a decomposition of the diagonal for very general quartic and $(2,3)$-complete intersection $n$-folds. Using cycle-theoretic techniques of Lange, Pavic and Schreieder we reduce the question via a degeneration argument to the existence of such a decomposition for $n-1$-dimensional cubic hypersurfaces and their essential dimension. A result of Voisin on the essential dimension of complex cubic hypersurfaces of odd dimension (and of dimension four) then yields conditional statements that extend results of Nicaise and Ottem from stable rationality to the existence of a decomposition of the diagonal. As an application, we use a recent result of Engel, de Gaay Fortman and Schreieder on the decomposition of the diagonal for cubic threefolds to give a new proof of the non-retract rationality of a very general complex quartic $4$-fold, originally due to Totaro, and of a very general complex $(2,3)$-complete intersection $4$-fold, originally due to Skauli.

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arXiv2024

On analogues of the Kato conjectures and proper base change for $1$-cycles on rationally connected varieties

In 1986, Kato set up a framework of conjectures relating (higher) $0$-cycles and étale cohomology for smooth projective schemes over finite fields or rings of integers in local fields through the homology of so-called Kato complexes. In analogy, we develop a framework of conjectures for $1$-cycles on smooth projective rationally connected varieties over algebraically closed fields and for families of such varieties over henselian discrete valuation rings with algebraically closed fields. This is partly motivated by results of Colliot-Thélène-Voisin [3] in dimension $3$. We prove some special cases building on recent results of Kollár-Tian [16].

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arXiv2024

Zero-cycles in families of rationally connected varieties

We study zero-cycles in families of rationally connected varieties. We show that for a smooth projective scheme over a henselian discrete valuation ring the restriction of relative zero cycles to the special fiber induces an isomorphism on Chow groups if the special fiber is separably rationally connected. We further extend this result to certain higher Chow groups and develop conjectures in the non-smooth case. Our main results generalise a result of Kollár [31].

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arXiv2024

$p$-adic tame Tate twists

Recently, Hübner-Schmidt defined the tame site of a scheme. We define $p$-adic tame Tate twists in the tame topology and prove some first properties. We establish a framework analogous to the Beilinson-Lichtenbaum conjectures in the tame topology for $p$-adic tame Tate twists and tame logarithmic deRham-Witt sheaves. Both only differ from their étale counterpart in cohomological degrees above the weight. These cohomology groups can be analysed using the Gersten conjecture which, at least conjecturally, has a nice shape in the tame topology. We prove the Gersten conjecture for tame logarithmic deRham-Witt sheaves for curves in positive characteristic and note that the conjecture in arbitrary dimension would follow from strict $\mathbb{A}^1$-invariance.

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arXiv2024

On the relative Gersten conjecture for Milnor K-theory in the smooth case

We show that the Gersten complex for the (improved) Milnor K-sheaf on a smooth scheme over an excellent discrete valuation ring is exact except at the first place and that exactness at the first place may be checked at the discrete valuation ring associated to the the generic point of the special fiber. This complements results of Gillet and Levine for K-theory, Geisser for motivic cohomology and Schmidt and Strunk and the author for étale cohomology.

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arXiv2024

Bloch-Ogus theory for smooth and semi-stable schemes in mixed characteristic

We study Bloch-Ogus theory and the Gersten conjecture for homology theories with duality satisfying certain properties, in particular for étale cohomology with finite coefficients coprime to the residue characteristic of the base, for smooth and semi-stable schemes in mixed characteristic. We prove the Gersten conjecture in the smooth case and prove a special case in the semi-stable situation. As a corollary of the smooth case we obtain the surjectivity of the Galois symbol map for arbitrary local rings over an excellent discrete valuation ring.

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arXiv2021

Milnor $K$-theory of $p$-adic rings

We study the mod $p^r$ Milnor $K$-groups of $p$-adically complete and $p$-henselian rings, establishing in particular a Nesterenko-Suslin style description in terms of the Milnor range of syntomic cohomology. In the case of smooth schemes over complete discrete valuation rings we prove the mod $p^r$ Gersten conjecture for Milnor $K$-theory locally in the Nisnevich topology. In characteristic $p$ we show that the Bloch-Kato-Gabber theorem remains true for valuation rings, and for regular formal schemes in a pro sense.

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arXiv2020

Deformation theory of the Chow group of zero-cycles

We study the deformations of the Chow group of zero-cycles of the special fibre of a smooth scheme over a henselian discrete valuation ring. Our main tools are Bloch's formula and differential forms. As a corollary we get an algebraization theorem for thickened zero-cycles previously obtained using idelic techniques. In the course of the proof we develop moving lemmata and Lefschetz theorems for cohomology groups with coefficients in differential forms.

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arXiv2019

Algebraization for zero-cycles and the $p$-adic cycle class map

Using an idelic argument and assuming the Gersten conjecture for Milnor K-theory, we show that the restriction map from one-cycles on a smooth projective scheme over a henselian local ring to a pro-system of thickened zero-cycles is surjective. We relate this restriction map to the $p$-adic cycle class map.

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arXiv2019

A local to global principle for higher zero-cycles

We study a local to global principle for certain higher zero-cycles over global fields. We thereby verify a conjecture of Colliot-Thélène for these cycles. Our main tool are the Kato conjectures proved by Jannsen, Kerz and Saito. Our approach also allows to reprove the ramified global class field theory of Kato and Saito. Finally, we apply the Kato conjectures to study the $p$-adic cycle class map over henselian discrete valuation rings of mixed characteristic and to deduce finiteness theorems for arithmetic schemes in low degree.

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arXiv2017

On a base change conjecture for higher zero-cycles

We show the surjectivity of a specialisation map on higher $(0,1)$-cycles for a smooth projective scheme over an excellent henselian discrete valuation ring. This gives evidence for a conjecture stated in an article of Kerz, Esnault and Wittenberg saying that base change holds for such schemes in general for motivic cohomology in degrees $(i,d)$ for fixed $d$ being the relative dimension over the base. Furthermore, the specialisation map we study is related to a finiteness conjecture for the $n$-torsion of $CH_0(X)$, where $X$ is a variety over a $p$-adic field.

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