Related papers: When the Zariski space is a Noetherian space

The Zariski quantization is one of the strong candidates for a quantization of the Nambu-Poisson bracket. In this paper, we apply the Zariski quantization for first quantized field theories, such as superstring and supermembrane theories,…

Orthogonal spaces are vector spaces together with a quadratic form whose associated bilinear form is non-degenerate. Over fields of characteristic two, there are many quadratic forms associated to a given bilinear form and quadratic…

I prove, answering a question of Zilber, that if $M$ is an algebraic variety of dimension strictly greater than one and $(M, \ldots)$ is a strongly minimal structure with atomic relations definable in the Zariski language on $M$, then $M$…

The schematic finite spaces are those finite ringed spaces where a theory of quasi-coherent modules can be developed with minimal natural conditions. We give various characterizations of these spaces and their natural morphisms. We show…

Let $K$ be an algebraically closed field that is complete with respect to a non-Archimedean absolute value, and let $\varphi\in K(z)$ have degree $d\geq 2$. We characterize maps for which the minimal resultant of an iterate $\varphi^n$ is…

An algebra $S$ is called a Skolem-Noether algebra (SN algebra for short) if for every central simple algebra $R$, every homomorphism $R\to R\otimes S$ extends to an inner automorphism of $R\otimes S$. One of the important properties of such…

To the best of the authors' knowledge, all previously known examples of $\mu$-, $\mu^*$-, link-, or ordinary Zariski pairs of surface singularities in $\mathbb{C}^3$ consist of (possibly weighted) L\^e-Yomdin singularities. In this paper,…

We show that a surface group contained in a reductive real algebraic group can be deformed to become Zariski dense, unless its Zariski closure acts transitively on a Hermitian symmetric space of tube type. This is a kind of converse to a…

Universal algebraic geometry is generalised from solutions of equations in a single algebra to the study of $\varphi$- or $K$-spectra, akin to the prime spectrum of a ring. We explore their basic properties and constructions, give a…

Let $F$ be a field, and let Zar$(F)$ be the space of valuation rings of $F$ with respect to the Zariski topology. We prove that if $X$ is a quasicompact set of rank one valuation rings in Zar$(F)$ whose maximal ideals do not intersect to…

A henselian valued field $K$ is called a tame field if its algebraic closure $\tilde{K}$ is a tame extension, that is, the ramification field of the normal extension $\tilde{K}|K$ is algebraically closed. Every algebraically maximal…

Let $G$ be a group with identity $e$. Let $R$ be a $G$-graded commutative ring and $M$ a graded $R$-module. A proper graded submodule $Q$ of $M$ is called a graded quasi-primary submodule if whenever $r\in h(R)$ and $m\in h(M)$ with $rm\in…

The cohomology of coherent sheaves and sheaves of Abelian groups on Noetherian schemes are interpreted in second order arithmetic by means of a finiteness theorem. This finiteness theorem provably fails for the etale topology even on…

We investigate the local properties of Berkovich spaces over Z. Using Weierstrass theorems, we prove that the local rings of those spaces are noetherian, regular in the case of affine spaces and excellent. We also show that the structure…

A finite classical polar space of rank $n$ consists of the totally isotropic subspaces of a finite vector space equipped with a nondegenerate form such that $n$ is the maximal dimension of such a subspace. A $t$-Steiner system in a finite…

A ringed finite space is a ringed space whose underlying topological space is finite. The category of ringed finite spaces contains, fully faithfully, the category of finite topological spaces and the category of affine schemes. Any ringed…

Let A be an excellent integral henselian local noetherian ring, k its residue field of characteristic p>0 and K its fraction field. Using an algebraization technique introduced by the first named author, and the one-dimension case already…

In the present survey paper, we present several new classes of Hochster's spectral spaces "occurring in nature", actually in multiplicative ideal theory, and not linked to or realized in an explicit way by prime spectra of rings. The…

An ideal $I$ in a Noetherian ring is called \textit{normal} if $I^n$ is integrally closed for all $n \geq 1$. Zariski proved that in two-dimensional regular local rings, every integrally closed ideal is normal. However, in dimension three…

Let $K$ be a field, and let $R = K[X]$ be the polynomial ring in an infinite collection $X$ of indeterminates over $K$. Let ${\mathfrak S}_{X}$ be the symmetric group of $X$. The group ${\mathfrak S}_{X}$ acts naturally on $R$, and this in…