Concept

Formal scheme

In mathematics, specifically in algebraic geometry, a formal scheme is a type of space which includes data about its surroundings. Unlike an ordinary scheme, a formal scheme includes infinitesimal data that, in effect, points in a direction off of the scheme. For this reason, formal schemes frequently appear in topics such as deformation theory. But the concept is also used to prove a theorem such as the theorem on formal functions, which is used to deduce theorems of interest for usual schemes. A locally Noetherian scheme is a locally Noetherian formal scheme in the canonical way: the formal completion along itself. In other words, the category of locally Noetherian formal schemes contains all locally Noetherian schemes. Formal schemes were motivated by and generalize Zariski's theory of formal holomorphic functions. Algebraic geometry based on formal schemes is called formal algebraic geometry. Formal schemes are usually defined only in the Noetherian case. While there have been several definitions of non-Noetherian formal schemes, these encounter technical problems. Consequently, we will only define locally noetherian formal schemes. All rings will be assumed to be commutative and with unit. Let A be a (Noetherian) topological ring, that is, a ring A which is a topological space such that the operations of addition and multiplication are continuous. A is linearly topologized if zero has a base consisting of ideals. An ideal of definition for a linearly topologized ring is an open ideal such that for every open neighborhood V of 0, there exists a positive integer n such that . A linearly topologized ring is preadmissible if it admits an ideal of definition, and it is admissible if it is also complete. (In the terminology of Bourbaki, this is "complete and separated".) Assume that A is admissible, and let be an ideal of definition. A prime ideal is open if and only if it contains . The set of open prime ideals of A, or equivalently the set of prime ideals of , is the underlying topological space of the formal spectrum of A, denoted Spf A.

About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Related courses (7)
MATH-657: Deformation Theory
We will study classical and modern deformation theory of schemes and coherent sheaves. Participants should have a solid background in scheme-theory, for example being familiar with the first 3 chapter
MATH-662: Perfectoid spaces
The course is about defining perfectoid spaces, and possibly presenting some applications.
MATH-489: Number theory II.c - Cryptography
The goal of the course is to introduce basic notions from public key cryptography (PKC) as well as basic number-theoretic methods and algorithms for cryptanalysis of protocols and schemes based on PKC
Show more
Related lectures (10)
Algebraic Geometry: Rings and Bodies
Explores algebraic geometry, focusing on rings, bodies, quotient rings, and irreducible polynomials.
Finite Difference Grids
Explains finite difference grids for computing solutions of elastic membranes using Laplace's equation and numerical methods.
Show more
Related publications (102)

On the (Im)possibility of Commitment over Gaussian Unfair Noisy Channels

Commitment is a key primitive which resides at the heart of several cryptographic protocols. Noisy channels can help realize information-theoretically secure commitment schemes; however, their imprecise statistical characterization can severely impair such ...
2023

Unified theory of atom-centered representations and message-passing machine-learning schemes

Michele Ceriotti, Guillaume André Jean Fraux, Sergey Pozdnyakov, Jigyasa Nigam

Data-driven schemes that associate molecular and crystal structures with their microscopic properties share the need for a concise, effective description of the arrangement of their atomic constituents. Many types of models rely on descriptions of atom-cen ...
AIP Publishing2022

Budget-Bounded Incentives for Federated Learning

Boi Faltings, Aris Filos Ratsikas, Adam Julian Richardson

We consider federated learning settings with independent, self-interested participants. As all contributions are made privately, participants may be tempted to free-ride and provide redundant or low-quality data while still enjoying the benefits of the FL ...
Springer Nature Switzerland AG 20202022
Show more
Related concepts (2)
Scheme (mathematics)
In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x = 0 and x2 = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers). Scheme theory was introduced by Alexander Grothendieck in 1960 in his treatise "Éléments de géométrie algébrique"; one of its aims was developing the formalism needed to solve deep problems of algebraic geometry, such as the Weil conjectures (the last of which was proved by Pierre Deligne).
Moduli space
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spaces frequently arise as solutions to classification problems: If one can show that a collection of interesting objects (e.g., the smooth algebraic curves of a fixed genus) can be given the structure of a geometric space, then one can parametrize such objects by introducing coordinates on the resulting space.

Graph Chatbot

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.