Related papers: A Berkovich Approach to Perfectoid Spaces
A uniform space is a topological space together with some additional structure which allows one to make sense of uniform properties such as completeness or uniform convergence. Motivated by previous work of J. Rivera-Letelier, we give a new…
This text contributes to the foundations of the theory of global Berkovich spaces, that is to say Berkovich spaces over Banach rings with nice properties such as $\mathbf{Z}$, rings of integers of number fields, discrete valuation rings,…
An exploration into the uses of the Fourier transform in the areas of algebraic and arithmetic geometry. In particular this treats the topics of Banach-Colmez spaces, for which an introduction to the theory of perfectoid spaces is given.…
Perfectoid spaces have become a crucial tool in $p$-adic geometry, serving as a bridge between adic spaces in characteristic $0$ and those in characteristic $p$. In this article, we develop a systematic way to study the structure of…
To understand the structure of an algebraic variety we often embed it in various projective spaces. This develops the notion of projective geometry which has been an invaluable tool in algebraic geometry. We develop a perfectoid analog of…
We develop a theory of perfect algebraic spaces that extend the so-called perfect schemes to the setting of algebraic spaces. We prove several desired properties of perfect algebraic spaces. This extends some previous results of perfect…
In this thesis we develop the foundations for a theory of analytic geometry over a valued field, uniformly encompassing the case when the base field is equipped with a non-archimedean valuation and the case when it has an archimedean one.…
In this paper, we first study the local rings of a Berkovich analytic space from the point of view of commutative algebra. We show that those rings are excellent ; we introduce the notion of a an analytically separable extension of…
We investigate affine Berkovich spaces over maximally complete fields and prove that they may be approximated by simpler spaces when the only functions we need to evaluate are polynomials of bounded degree. We derive applications to…
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…
Let $K$ be a perfectoid field. We describe all quotient fields of the perfectoid Tate algebra\begin{equation*}T_{n,K}^{\text{perfd}}=K\langle X_{1}^{1/p^{\infty}},\dots, X_{n}^{1/p^{\infty}}\rangle\end{equation*}in any number $n\geq1$ of…
We construct a continuum of mutually non-isomorphic separable Banach spaces which are complemented in each other. Consequently, the Schroeder-Bernstein Index of any of these spaces is $2^{\aleph_0}$. Our construction is based on a Banach…
We define a notion of global analytic space with overconvergent structure sheaf. This gives an analog on a general base Banach ring of Grosse-Kloenne's overconvergent p-adic spaces and of Bambozzi's generalized affinoid varieties over R.…
We develop some of the foundations of affinoid pre-adic spaces without Noetherian or finiteness hypotheses. We give some explicit examples of non-adic affinoid pre-adic spaces (including a locally perfectoid one). On the positive side, we…
This is a survey of recent advances in commutative algebra, especially in mixed characteristic, obtained by using the theory of perfectoid spaces. An explanation of these techniques and a short account of the author's proof of the direct…
A new category of algebro-geometric objects is defined. This construction is a vast generalization of existing F1-theories, as it contains the the theory of monoid schemes on the one hand and classical algebraic theory, e.g. Grothendieck…
Using arithmetic jet spaces, we attach perfectoid spaces to smooth schemes and to $\delta$-morphisms of smooth schemes. We also study perfectoid spaces attached to arithmetic differential equations defined by some of the remarkable…
We show that Berkovich analytic geometry can be viewed as relative algebraic geometry in the sense of To\"{e}n--Vaqui\'{e}--Vezzosi over the category of non-Archimedean Banach spaces. For any closed symmetric monoidal quasi-abelian category…
This paper is a subsequent paper of arXiv:2303.07672. We will continue our research on the subject of perfect algebraic spaces that is developed in arXiv:2303.07672. By means of algebraic Frobenius morphisms, we define the perfection of…
The Grassmannian of affine subspaces is a natural generalization of both the Euclidean space, points being zero-dimensional affine subspaces, and the usual Grassmannian, linear subspaces being special cases of affine subspaces. We show…