Related papers: $\mathbf{A}_{\text{inf}}$ is infinite dimensional

Assume that $K$ is an algebraically closed field, $R$ a locally support-finite locally bounded $K$-category, $G$ a torsion-free admissible group of $K$-linear automorphisms of $R$ and $A=R/G$. We show that the Krull-Gabriel dimension…

Consider a complete discrete valuation ring $\mathcal{O}$ with quotient field $F$ and finite residue field. Then the inclusion map $\mathcal{O} \hookrightarrow F$ induces a map $\hat{\mathrm{K}}^\mathrm{M}_*\mathcal{O} \to…

We construct an example of an $A_{\infty}$ algebra structure defined over a finite dimensional graded vector space.

Let $K$ be a large field such that $K[\sqrt{-1}]$ is not algebraically closed and $F/K$ a function field in one variable. Extending techniques and results from earlier work with Becher and Dittmann, we show that every valuation ring on $F$…

For a finitely generated algebra over a field, the transcendence degree is known to be equal to the Krull dimension. The aim of this paper is to generalize this result to algebras over rings. A new definition of the transcendence degree of…

We construct a nil algebra over a countable field which has finite but non-zero Gelfand-Kirillov dimension.

We define radically finite rings and show that finite dimensional radically finite rings are Noetherian, and that if either R is a finite character Hilbert domain that contains a field of characteristic zero or a finite dimensional Prufer…

Let $K$ be a complete discrete valued field of characteristic zero with residue field $k_K$ of characteristic $p > 0$. Let $L/K$ be a finite Galois extension with the Galois group $G$ and suppose that the induced extension of residue fields…

Suppose that $R$ is a ring essentially of finite type over a perfect field of characteristic $p > 0$ and that $a \subseteq R$ is an ideal. We prove that the set of $F$-jumping numbers of $\tau_b(R; a^t)$ has no limit points under the…

We establish that all rings of $S$-integers are universally definable in function fields in one variable over certain ground fields including global and non-archimedean local fields. That is, we show that the complement of such a ring of…

We show that for a global field $K$, every ring of $S$-integers has a universal first-order definition in $K$ with $10$ quantifiers. We also give a proof that every finite intersection of valuation rings of $K$ has an existential…

We prove the explicit characterization of the so-called "best f" for degree $p$ Artin-Schreier and degree $p$ Kummer extensions of Henselian valuation rings in residue characteristic $p$. This characterization is mentioned briefly in [Th16,…

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…

Let $K$ be a complete discrete valuation field of characteristic $0$ with perfect residue field of characteristic $p>0$. Let $\mathbb{A}_K$ denote the imperfect coefficient ring of $(\varphi,\Gamma)$-modules defined by Jean-Marc Fontaine.…

In this article, we give a full description of the topology of the one dimensional affine analytic space $\mathbb{A}_R^1$ over a complete valuation ring $R$ (i.e. a valuation ring with "real valued valuation" which is complete under the…

The small finitistic dimension $\fPD(R)$ of a ring $R$ is defined to be the supremum of projective dimensions of $R$-modules with finite projective resolutions. In this paper, we investigate the small finitistic dimensions of four types of…

In \cite{SSZ}, the authors proved that an Artin algebra $A$ with infinite global dimension has an indecomposable module with infinite projective and infinite injective dimension, giving a new characterisation of algebras with finite global…

We develop a new technique for studying monomial ideals in the standard polynomial rings $A[X_1,\ldots,X_d]$ where $A$ is a commutative ring with identity. The main idea is to consider induced ideals in the semigroup ring…

We show that a valuation ring containing its residue field of characteristic $p>0$ is a filtered direct limit of complete intersection ${\bf F}_p$-algebras.

For a finite totally ramified extension $L$ of a complete discrete valuation field $K$ with the perfect residue field of characteristic $p>0$, it is known that $L/K$ is an abelian extension if the upper ramification breaks are integers and…