Related papers: Algebraic valuation ring extensions as limits of c…

In this paper we provide a complete algebraic characterization of elementary equivalence of rings with a finitely generated additive group in the language of pure rings. The rings considered are arbitrary otherwise.

We study an analytically irreducible algebroid germ (X, 0) of complex singularity by considering the filtrations of its analytic algebra, and their associated graded rings, induced by the divisorial valuations associated to the irreducible…

We study projective completions of affine algebraic varieties induced by filtrations on their coordinate rings. In particular, we study the effect of the 'multiplicative' property of filtrations on the corresponding completions and…

This paper deals with properties of the algebraic variety defined as the set of zeros of a "deficient" sequence of multivariate polynomials. We consider two types of varieties: ideal-theoretic complete intersections and absolutely…

We prove that any complete regular local ring with a valuation of rank 1 can be embedded, as a valued ring, in a ring of generalized Puiseux expansions.

We construct and study a graded version of absolute perfectoidization for $G$-graded adic rings. As a main geometric application, we show that the absolute perfectoidization of the structure sheaf of a projective-type formal scheme admits…

We consider the directed union S of an infinite sequence {(R_n, m_n)} of successive local quadratic transforms of a regular local ring (R, m). If dim R = 2, Abhyankar proves that S is a valuation ring. If dim R > 2, Shannon gives necessary…

In this paper we determine the representation type of some algebras of infinite matrices continuously controlled at infinity by a compact metrizable space. We explicitly classify their finitely presented modules in the finite and tame…

Using perfectoid algebras, we introduce a mixed characteristic analog of the multiplier ideal, respectively test ideal, from characteristic zero, respectively $p > 0$, in the case of a regular ambient ring. We prove several properties about…

We classify one-dimensional restricted central extensions of the modular Witt Lie algebra in characteristic $p>3$.

This paper studies direct limits of full upper triangular matrix algebras with embeddings which are not *-extendible. A representation of the limit algebra is found so that the generated C*-algebra is the C*-envelope. Some examples are…

Let F:K be a Galois extension of number fields and Q a prime ideal of O_F lying over the prime P of O_K. By analyzing the Q-adic closure of O_K in O_F we characterize those rings of integers O_K for which every residue class ring of…

In this work we show that the classical subject of general valuation theory and Zariski-Riemann varieties has a much wider scope than commutative algebra and desingularization theory. We construct and investigate birational projective limit…

We give a new construction of free distributive p-algebras. Our construction relies on a detailed description of completely meet-irreducible congruences, so it is purely universal algebraic. It yields a normal form theorem for p-algebra…

In this paper we describe central extensions (up to isomorphism) of all complex null-filiform and filiform associative algebras.

The (full) extended plus closure was developed as a replacement for tight closure in mixed characteristic rings. Here it is shown by adapting Andr\'{e}'s perfectoid algebra techniques that, for complete local rings that have F-finite…

A finite abelian $p$-group having an automorphism $x$ such that $1+\ldots+x^{p-1}=0$, can be viewed as a module over an appropriate discrete valuation ring $\mathcal{O}$ containing $\mathbb{Z}_p$ (the ring of $p$-adic integer). This yields…

We study the algebraic and geometric properties of the integral closure of different rings of functions on a real algebraic variety : the regular functions and the continuous rational functions.

We study infinite groups interpretable in three families of valued fields: $V$-minimal, power bounded $T$-convex, and $p$-adically closed fields. We show that every such group $G$ has unbounded exponent and that if $G$ is dp-minimal then it…

Given a generically finite local extension of valuation rings $V \subset W$, the question of whether $W$ is the localization of a finitely generated $V$-algebra is significant for approaches to the problem of local uniformization of…