相关论文: On the integrality of \'etale extensions of polyno…

We study integral points on varieties with infinite \'etale fundamental groups. More precisely, for a number field $F$ and $X/F$ a smooth projective variety, we prove that for any geometrically Galois cover $\varphi\colon Y \to X$ of degree…

This paper is inspired by Michael Artin's paper "On The Join of Hensel Rings". In his paper, Artin proves that in an absolutely integrally closed ring the sum of two prime ideals is either prime or the whole ring. A more elementary proof of…

We study projective completions of affine algebraic varieties which are given by filtrations, or equivalently, 'degree like functions' on their rings of regular functions. For a quasifinite polynomial map P (i.e. with all fibers finite) of…

By a theorem of Roberts, the integral closure of a regular local ring in a finite abelian extension of its fraction field is Cohen-Macaulay, provided that the degree of the extension is coprime to the characteristic of the residue field. We…

We show that by working over the absolute base $\mathbb S$ (the categorical version of the sphere spectrum) instead of $\mathbb S[\pm 1]$ improves our previous Riemann-Roch formula for $\overline{{\rm Spec\,}\mathbb Z}$. The formula equates…

Given $p$ polynomials with coefficients in a commutative unitary integral ring $\mathcal{C}$ containing $\mathbb{Q}$, we define the notion of a generic Bernstein-Sato polynomial on an irreducible affine scheme $V \subset…

A ring is *unit-additive* if a sum of units is always either a unit or nilpotent. For example, $k[X]$ and $k[X]/(X^2)$ are unit-additive, but $\mathbb Z$ is not. We prove a wide-ranging theorem about unit-additivity in semigroup rings,…

In the note we construct a family of \'etale coverings of the affine line. More specifically, let $F$ be a finite field of characteristic $p$ and suppose that the cardinality of $F$ is at least 4. Let $A = F[T]$ be the polynomial ring in…

Let $A\subset B$ be an integral ring extension of integral domains with fields of fractions $K$ and $L$, respectively. The integral degree of $A\subset B$, denoted by ${\rm d}_A(B)$, is defined as the supremum of the degrees of minimal…

We prove Dirichlet's theorem for polynomial rings: Let F be a pseudo algebraically closed field. Then for all relatively prime polynomials a(X), b(X)\in F[X] and for every sufficiently large positive integer n there exist infinitely many…

Let $D$ be an integrally closed local Noetherian domain of Krull dimension 2, and let $f$ be a nonzero element of $D$ such that $fD$ has prime radical. We consider when an integrally closed ring $H$ between $D$ and $D_f$ is determined…

For an extension A/B of neither necessarily associative nor necessarily unital rings, we investigate the connection between simplicity of A with a property that we call A-simplicity of B. By this we mean that there is no non-trivial ideal I…

The classical Hilbert specialization property is a field-theoretic tool ensuring that polynomial irreducibility over a field is preserved under specialization of some of the variables. We develop an integral counterpart by introducing the…

Let $D$ be an integral domain with quotient field $K$ and $\Omega$ a finite subset of $D$. McQuillan proved that the ring ${\rm Int}(\Omega,D)$ of polynomials in $K[X]$ which are integer-valued over $\Omega$, that is, $f\in K[X]$ such that…

We prove that, for any $g \geq 2$, the \'etale double cover $\rho_g:\mathcal{E}_{g} \to \widehat{\mathcal{E}}_{g}$ from the moduli space $\mathcal{E}_{g}$ of complex polarized genus $g$ Enriques surfaces to the moduli space…

We revisit the concept of special algebras, also known as \textit{purely inseparable ring extensions}. This concept extends the notion of purely inseparable field extensions to the more general context of extensions of commutative rings. We…

Let $R$ be a complete discrete valuation ring with fraction field $K$ and with algebraically closed residue field. Let $X$ be a faithfully flat $R$-scheme of finite type of relative dimension 1 and $G$ be any affine $K$-group scheme of…

We explore an analogy between, on one hand, the notions of integral closure of ideals and Rees valuations in commutative algebra and, on the other hand, the notions of spectral seminorm and Shilov boundary in nonarchimedean geometry. For…

Let $A$ be an abelian scheme of dimension at least four over a $\mathbb{Z}$-finitely generated integral domain $R$ of characteristic zero, and let $L$ be an ample line bundle on $A$. We prove that the set of smooth hypersurfaces $D$ in $A$…

We develop a class of integrals on a manifold M called exponential iterated integrals, an extension of K. T. Chen's iterated integrals. It is shown that the matrix entries of any upper triangular representation of the fundamental group of M…