Related papers: One-dimensional bad Noetherian domains

Let $R$ be a complete equicharacteristic noetherian local domain with an algebraically closed residue field $k$. Let $\nu$ be a zero dimensional valuation of rank one centered in $R$ with value group $\Phi$. We show that there is a…

In this paper the question of which semigroups are realizable as the semigroup of values attained on a Noetherian local ring which is dominated by a valuation is considered. We give some striking examples, indicating that there may be no…

Let $(R,\m,k)$ be a local (Noetherian) ring of positive prime characteristic $p$ and dimension $d$. Let $G_\dt$ be a minimal resolution of the residue field $k$, and for each $i\ge 0$, let $\gothic t_i(R) = \lim_{e\to \8}…

We explicitly describe the divisor class groups and semidualizing modules for ladder determinantal rings with coefficients in an arbitrary normal domain for arbitrary ladders, not necessarily connected, and all sizes of minors.

In this paper, an algebraic theory for local rings of finite embedding dimension is developed. Several extensions of (Krull) dimension are proposed, which are then used to generalize singularity notions from commutative algebra. Finally,…

We show that there exists a complete local Noetherian normal domain of prime characteristic whose perfection is a non-coherent GCD domain, answering a question of Patankar in the negative concerning characterizations of $F$-coherent rings.…

The aim of this article is to prove some results on the existence of an integral extension domain of a complete local Noetherian domain in mixed characteristic $p>0$ having certain distinguished properties with respect to the Frobenius map.…

We classify all non-degenerate skew-hermitian forms defined over certain local rings, not necessarily commutative, and study some of the fundamental properties of the associated unitary groups, including their orders when the ring in…

We establish a new fundamental class of varieties in nonnoetherian algebraic geometry related to the central geometry of dimer algebras. Specifically, given an affine algebraic variety $X$ and a finite collection of non-intersecting…

The focus of this note is on the Chow group problem over ramified regular local rings $(R, m)$. Our goal is threefold: i) to introduce a characterization of a ramified regular local ring essentially of finite type over a dvr, ii) to address…

Let X be a singular affine normal variety with coordinate ring R and assume that there is an R-order admitting a stability structure such that the scheme of relevant semistable representations is smooth, then we construct a partial…

We describe new classes of noetherian local rings $R$ whose finitely generated modules $M$ have the property that $Tor_i^R(M,M)=0$ for $i\gg 0$ implies that $M$ has finite projective dimension, or $Ext^i_R(M,M)=0$ for $i\gg 0$ implies that…

In this paper we consider birational properties of ramification in excellent local rings. We extend earlier results of the author with Olivier Piltant to show that strong local monomialization is true along a valuation dominating a…

We prove that every local derivation on a finite-dimensional semisimple Lie algebra over an algebraically closed field of characteristic zero is a derivation. We also give examples of finite-dimensional nilpotent Lie algebras $\mathcal{L}$…

In this paper we consider a family of non local functionals of convolution-type depending on a small parameter $\varepsilon>0$ and $\Gamma$-converging to local functionals defined on Sobolev spaces as $\varepsilon\to 0$. We study the…

We give an introduction to the valuation theoretical phenomenon of "defect", also known as "ramification deficiency". We describe the role it plays in deep open problems in positive characteristic: local uniformization (the local form of…

Given a noetherian local domain $R$ and a valuation $\nu$ of its field of fractions which is non negative on $R$, we derive some very general bounds on the growth of the number of distinct valuation ideals of $R$ corresponding to values…

This article discusses ramification and the structure of relative K\"ahler differentials of extensions of valued fields. We begin by surveying the theory developed in recent work with Franz-Viktor Kuhlmann and Anna Rzepka constructing the…

In this paper we consider birational properties of ramification in excellent local rings. We give an example showing that local monomialization (and weak local monomialization) can fail for extensions of algebraic local rings in algebraic…

Assume that $(L,v)$ is a finite Galois extension of a valued field $(K,v)$. We give an explicit construction of the valuation ring $\mathcal O_L$ of $L$ as an $\mathcal O_K$-algebra, and an explicit description of the module of relative…