Related papers: Algebraic geometry in mixed characteristic
We discuss recent developments in $p$-adic geometry, ranging from foundational results such as the degeneration of the Hodge-to-de Rham spectral sequence for "compact $p$-adic manifolds" over new period maps on moduli spaces of abelian…
A survey on recent developments in (algebraic) integral geometry is given. The main focus lies on algebraic structures on the space of translation invariant valuations and applications in integral geometry.
We develop the basic theory of geometrically closed rings as a generalisation of algebraically closed fields, on the grounds of notions coming from positive model theory and affine algebraic geometry. For this purpose we consider several…
In this paper we introduce elements of algebraic geometry over an arbitrary algebraic structure. We prove Unification Theorems which gather the description of coordinate algebras by several ways.
We aim at studying collections of algebraic structures defined over a commutative ring and investigating the complexity of significant constructions carried out on these objects. The assignment of measures of size, via a multiplicity…
We develop basic notions and methods of algebraic geometry over the algebraic objects called hyperrings. Roughly speaking, hyperrings generalize rings in such a way that an addition is `multi-valued'. This paper largely consisits of two…
This article describes recent applications of algebraic geometry to noncommutative algebra. These techniques have been particularly successful in describing graded algebras of small dimension.
We discuss a selection of recent developments in arithmetic combinatorics having to do with ``approximate algebraic structure'' together with some of their applications.
We formulate a number of new results in Algebraic Geometry and outline their derivation from Theorem 2.12 which belongs to Algebraic Combinatorics.
Over the past few years there has been considerable progress in the structural understanding of special Colombeau algebras. We present some of the main trends in this development: non-smooth differential geometry, locally convex theory of…
This is a survey of some recent developments concerning the p-adic cohomology of algebraic varieties over fields of positive characteristic and local fields of mixed characteristic, plus some related areas like p-adic Hodge theory.
We develop algebraic geometry for general Segal's Gamma-rings and show that this new theory unifies two approaches we had considered earlier on (for a geometry under Spec Z). The starting observation is that the category obtained by gluing…
We show that an algebraic immediate valuation ring extension of characteristic $p>0$ is a filtered union of complete intersection algebras of finite type.
Superstring compactifications have been vigorously studied for over four decades, and have flourished involving an active iterative feedback between physics and (complex) algebraic geometry. This led to an unprecedented wealth of…
In this paper, for a given finitely generated algebra (an algebraic structure with arbitrary operations and no predicates) A we study finitely generated limit algebras of A, approaching them via model theory and algebraic geometry. Along…
Some basic properties of the ring of integers $\mathbb{Z}$ are extended to entire rings. In particular, arithmetic in entire principal rings is very similar than arithmetic in the ring of integers $\mathbb{Z}$. These arithmetic properties…
We expand our previously founded basic theory of equiresidual algebraic geometry over an arbitrary commutative field, to a well-behaved theory of (equiresidual) algebraic varieties over a commutative field, thanks to the generalisation of…
We introduce the first geometric construction of codes in the sum-rank metric, which we called linearized Algebraic Geometry codes, using quotients of the ring of Ore polynomials with coefficients in the function field of an algebraic…
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…
In this note we introduce generalised pairs from the perspective of the evolution of the notion of space in birational algebraic geometry. We describe some applications of generalised pairs in recent years and then mention a few open…