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Achim Krause

Publications and source records attributed to Achim Krause.

15 recordsLinked to original sources
arXiv2026

Prismatic cohomology relative to $δ$-rings

We develop prismatic and syntomic cohomology relative to a $δ$-ring. This simultaneously generalizes Bhatt and Scholze's absolute and relative prismatic cohomology and shows that the latter, which was defined relative to a prism, is in fact independent of the prism structure and only depends on the underlying $δ$-ring. We give several possible definitions of our new version of prismatic cohomology: a site theoretic definition, one using prismatic crystals, and a stack theoretic definition. These are equivalent under mild syntomicity hypotheses. As an application, we note how the theory of prismatic cohomology of filtered rings arises naturally in this context.

math.AG↗
arXiv2026

The de Rham and the syntomic logarithm

We define and study an integral refinement of the inverse of the Bloch-Kato exponential map which we call the de Rham logarithm. Our main tool to analyze the de Rham logarithm is the syntomic logarithm, a certain limit construction based on the theory of filtered prismatic cohomology initiated by Antieau, Krause and Nikolaus. We use the syntomic logarithm to prove a version of the Beilinson fibre square for all quasicompact, quasiseparated derived formal schemes. We also use our techniques to prove Conjecture $C_{EP}(\bq_p(n))$ of Fontaine and Perrin-Riou for all local fields $K/\bq_p$ and to compute the correction factor $C(X,n)$ introduced by Flach and Morin in their reformulation of the Bloch-Kato Tamagawa number conjecture for the Zeta function of a smooth projective scheme $X$ over a number ring.

math.NT↗
arXiv2024

Exact bounds for even vanishing of $K_* (\mathbb{Z}/p^n)$

In this note, we prove that $K_{2i} (\mathbb{Z}/p^n) \neq 0$ if and only if $p-1$ divides $i$ and $0 \leq i \leq (p-1) p^{n-2}$, refining the even vanishing theorem of Antieau, Nikolaus and the first author in this case. As a corollary of our proof, we determine that the nilpotence order of $v_1$ in $π_* K(\mathbb{Z}/p^n)/p$ is equal to $\frac{p^n-1}{p-1}$. Our proof combines the recent crystallinity result for reduced syntomic cohomology of Hahn, Levy and the second author with the explicit complex computing the syntomic cohomology of $\mathcal{O}_K /\varpi^n$ constructed by Antieau, Nikolaus and the first author.

math.KT↗
arXiv2024

On the $K$-theory of $\mathbf{Z}/p^n$

We give an explicit algebraic description, based on prismatic cohomology, of the algebraic K-groups of rings of the form $O_K/I$ where $K$ is a p-adic field and $I$ is a non-trivial ideal in the ring of integers $O_K$; this class includes the rings $\mathbf{Z}/p^n$ where $p$ is a prime. The algebraic description allows us to describe a practical algorithm to compute individual K-groups as well as to obtain several theoretical results: the vanishing of the even K-groups in high degrees, the determination of the orders of the odd K-groups in high degrees, and the degree of nilpotence of $v_1$ acting on the mod $p$ syntomic cohomology of $\mathbf{Z}/p^n$.

math.KT↗
arXiv2024

A note on quadratic forms

For a field extension $L/K$ we consider maps that are quadratic over $L$ but whose polarisation is only bilinear over $K$. Our main result is that all such are automatically quadratic forms over $L$ in the usual sense if and only if $L/K$ is formally unramified. In particular, this shows that over finite and number fields, one of the axioms in the standard definition of quadratic forms is superfluous.

math.AC↗
arXiv2023

Witt vectors with coefficients and TR

We give a new construction of $p$-typical Witt vectors with coefficients in terms of ghost maps and show that this construction is isomorphic to the one defined in terms of formal power series from the authors' previous paper. We show that our construction recovers Kaledin's polynomial Witt vectors in the case of vector spaces over a perfect field of characteristic $p$. We then identify the components of the $p$-typical TR with coefficients, originally defined by Lindenstrauss and McCarthy and later reworked by the second and third authors in joint work with McCandless, with the $p$-typical Witt vectors with coefficients. This extends a celebrated result of Hesselholt and Hesselholt-Madsen relating the components of TR with the Witt vectors. As an application, we given an algebraic description of the components of the Hill-Hopkins-Ravenel norm for cyclic $p$-groups in terms of $p$-typical Witt vectors with coefficients.

math.AT↗
arXiv2023

Polygonic spectra and TR with coefficients

We introduce the notion of a polygonic spectrum which is designed to axiomatize the structure on topological Hochschild homology $\mathrm{THH}(R,M)$ of an $\mathbb{E}_1$-ring $R$ with coefficients in an $R$-bimodule $M$. For every polygonic spectrum $X$, we define a spectrum $\mathrm{TR}(X)$ as the mapping spectrum from the polygonic version of the sphere spectrum $\mathbb{S}$ to $X$. In particular if applied to $X = \mathrm{THH}(R,M)$ this gives a conceptual definition of $\mathrm{TR}(R,M)$. Every cyclotomic spectrum gives rise to a polygonic spectrum and we prove that TR agrees with the classical definition of TR in this case. We construct Frobenius and Verschiebung maps on $\mathrm{TR}(X)$ by exhibiting $\mathrm{TR}(X)$ as the $\mathbb{Z}$-fixedpoints of a quasifinitely genuine $\mathbb{Z}$-spectrum. The notion of quasifinitely genuine $\mathbb{Z}$-spectra is a new notion that we introduce and discuss inspired by a similar notion over $\mathbb{Z}$ introduced by Kaledin. Besides the usual coherences for genuine spectra, this notion additionally encodes that $\mathrm{TR}(X)$ admits certain infinite sums of Verschiebung maps.

math.AT↗
arXiv2022

On the K-theory of $\mathbb{Z}/p^n$ -- announcement

We announce new methods for using prismatic cohomology to compute the K-groups of $\mathbb{Z}/p^n$ and related rings. We use computer algebra methods to compute these K-groups through a large range in specific cases and also obtain explicit formulas for their orders in large degrees.

math.KT↗
arXiv2020

The Picard group in equivariant homotopy theory via stable module categories

We develop a mechanism of "isotropy separation for compact objects" that explicitly describes an invertible $G$-spectrum through its collection of geometric fixed points and gluing data located in certain variants of the stable module category. As an application, we carry out a complete analysis of invertible G-spectra in the case $G=A_5$. A further application is given by showing that the Picard groups of $\mathrm{Sp}^G$ and a category of derived Mackey functors agree.

math.AT↗
arXiv2020

Witt vectors with coefficients and characteristic polynomials over non-commutative rings

For a not-necessarily commutative ring R we define an abelian group W(R;M) of Witt vectors with coefficients in an R-bimodule M. These groups generalize the usual big Witt vectors of commutative rings and we prove that they have analogous formal properties and structure. One main result is that W(R) := W(R;R) is Morita invariant in R. For an R-linear endomorphism f of a finitely generated projective R-module we define a characteristic element $χ_f \in W(R)$. This element is a non-commutative analogue of the classical characteristic polynomial and we show that it has similar properties. The assignment $f \mapsto χ_f$ induces an isomorphism between a suitable completion of cyclic K-theory and W(R).

math.KT↗
arXiv2019

Bökstedt periodicity and quotients of DVRs

In this note we compute the topological Hochschild homology of quotients of DVRs. Along the way we give a short argument for Bökstedt periodicity and generalizations over various other bases. Our strategy also gives a very efficient way to redo the computations of THH (resp. logarithmic THH) of complete DVRs originally due to Lindenstrauss-Madsen (resp. Hesselholt-Madsen).

math.AT↗
arXiv2018

C-motivic modular forms

We construct a topological model for cellular, 2-complete, stable C-motivic homotopy theory that uses no algebro-geometric foundations. We compute the Steenrod algebra in this context, and we construct a "motivic modular forms" spectrum over C.

math.AT↗
arXiv2018

Vanishing lines for modules over the motivic Steenrod algebra

We study criteria for freeness and for the existence of a vanishing line for modules over certain Hopf subalgebras of the motivic Steenrod algebra over $\mathrm{Spec}(\mathbb{C})$ at the prime 2. These turn out to be determined by the vanishing of certain Margolis homology groups in the quotient Hopf algebra $\mathcal{A}/τ$.

math.AT↗
arXiv2014

Bianchi's classification of 3-dimensional Lie algebras revisited

We present Bianchi's proof on the classification of real (and complex) $3$-dimensional Lie algebras in a coordinate free version from a strictly representation theoretic point of view. Nearby we also compute the automorphism groups and from this the orbit dimensions of the corresponding orbits in the algebraic variety $X\subseteqΛ^2V^*\otimes V$ describing all Lie brackets on a fixed vector space $V$ of dimension $3$. Moreover we clarify which orbits lie in the closure of a given orbit and therefore the topology on the orbit space $X/G$ with $G=\mathrm{Aut}(V)$.

math.RT↗