Related papers: Hyperfields, truncated DVRs and valued fields

Let $A$ be a discrete valuation ring with field of fractions $F$ and (sufficiently large) residue field $k$. We prove that there is a natural exact sequence $H_3(\mathrm{SL}_2(A),\mathbb{Z}[\frac{1}{2}]) \to…

We consider the algebraic K-theory of a truncated polynomial algebra in several commuting variables, K(k[x_1, ..., x_n]/(x_1^a_1, ..., x_n^a_n)). This naturally leads to a new generalization of the big Witt vectors. If k is a perfect field…

Let $T$ be a polynomially bounded o-minimal theory extending the theory of real closed ordered fields. Let $K$ be a model of $T$ equipped with a $T$-convex valuation ring and a $T$-derivation. If this derivation is continuous with respect…

We prove that the $p$-adically completed periodic topological cyclic homology of a DG category over a perfect field $k$ of characteristic $p>2$ is isomorphic to the ($p$-adically completed) periodic cyclic homology of a lifting of the DG…

Let K be a valued field of characteristic p>0 with non-p-divisible value group. We show that every finite embedding problem for K whose kernel is a p-group is properly solvable.

Let $k$ be a perfect field of characteristic $p > 2$, and let $K$ be a finite totally ramified extension over $W(k)[\frac{1}{p}]$ of ramification degree $e$. Let $R_0$ be a relative base ring over $W(k)\langle t_1^{\pm 1}, \ldots, t_m^{\pm…

Let $K$ be a number field, let $v$ be a finite place of $K$, let $f\in K[z]$ be a degree $d\geqslant2$ polynomial with $v|d$, and let $a\in K$. We show that if $f$ is postcritically bounded and has potential good reduction with respect to…

Let k be a field of characteristic zero, let X be a geometrically integral k-variety of dimension n and let K be its field of fractions. Under the assumption that K contains all r-th roots of unity for an integer r, we prove that, given an…

For a large class of cohomology theories, we prove that refined unramified cohomology is canonically isomorphic to the hypercohomology of a natural truncated complex of Zariski sheaves. This generalizes a classical result of Bloch and Ogus…

An $H$-field is a type of ordered valued differential field with a natural interaction between ordering, valuation, and derivation. The main examples are Hardy fields and fields of transseries. Aschenbrenner and van den Dries proved…

Let k be an algebraically closed field of characteristic p>0. Let H be a supersingular p-divisible group over k of height 2d. We show that H is uniquely determined up to isomorphism by its truncation of level d (i.e., by H[p^d]). This…

Let $K$ be a local field with finite residue field, we define a normal form for Eisenstein polynomials depending on the choice of a uniformizer $\pi_K$ and of residue representatives. The isomorphism classes of extensions generated by the…

Consider a system of polynomials in many variables over the ring of integers of a number field $K$. We prove an asymptotic formula for the number of integral zeros of this system in homogeneously expanding boxes. As a consequence, any…

Suppose $M^{n+1}$ is a complex manifold and K is a hypersurface with isolated singularities. Let $\omega$ be a holomorphic form on $M\setminus K$ with the first order pole on K. The Leray residue of such form gives an element in the n-th…

In different areas of discrete mathematics, a certain type of polynomials, having coefficients in a field K of finite characteristic, has been considered. The form and the degree of these polynomials, here called projective, are simply…

Let $V$ be a valuation ring of a global field $K$. We show that for all positive integers $k$ and $1 < n_1 \leq \ldots \leq n_k$ there exists an integer-valued polynomial on $V$, that is, an element of $\text{Int}(V) = \{ f \in K[X] \mid…

We investigate fields of characteristic 0 and dp-rank 2. While we do not obtain a classification, we prove that any unstable field of characteristic 0 and dp-rank 2 admits a unique definable V-topology. If this statement could be…

A quadratic form over a Henselian-valued field of arbitrary residue characteristic is tame if it becomes hyperbolic over a tamely ramified extension. The Witt group of tame quadratic forms is shown to be canonically isomorphic to the Witt…

We propose topological Hochschild homology as a tool for measuring ramification of maps of structured ring spectra. We determine second order topological Hochschild homology of the $p$-local integers. For the tamely ramified extension of…

Local fields, and fields complete with respect to a discrete valuation, are essential objects in commutative algebra, with applications to number theory and algebraic geometry. We formalize in Lean the basic theory of discretely valued…