Global Fields and Adeles Quiz

The adele ring of a global field F. It captures the information about F at all places (including archimedean places).

The ring of adeles that are integral at all places away from a finite set S of places of F.

The set of points of a scheme X with coordinates in the adele ring AF. In other words, it represents how X can be parameterized by adeles.

A continuous mapping between topological spaces that preserves their structure, with the property that a subset of both spaces is closed in one space if and only if it's closed in the other.

In this context, a mapping between schemes that preserves the underlying topological spaces and their structures. More intuitively, it preserves the 'geometry' of the schemes.

The process of combining smaller objects into one bigger object that preserves the relationships between them. It is used to create a larger structure that 'inherits' properties from its components.

The set of all points at which a function equals zero, which forms a subset of a space. It defines where a function 'vanishes' or becomes zero.

A type of topological space where any point has a neighborhood, which is both closed and open. Essentially, any point can be 'isolated' within its neighborhood.

A subscheme V is quasi-compact if it can be covered by finitely many open affine subschemes. Think of it as a 'bounded' subset of Spec(R).

The morphism ∆X/C represents the 'diagonal' of X ×C X, essentially mapping each point x ∈ X to the pair (x, x) in the product scheme.

A function that maps from the spectrum of a ring to the spectrum of another ring, representing a relationship between the algebraic structures of the rings.

A ring decomposition where the components represent different aspects of the original ring based on local rings.

A map that induces relationships between maps of affine schemes, specifically, a map between schemes that produces specific maps between their affine components.

The ability to uniquely determine an element based on its association with a particular local ring.

The ability to relate points through an open continuous map, ensuring a connection between their neighborhood structures.

A finitely presented A-scheme X can be represented as the base change of a finitely presented Ai0-scheme Xi0, where Ai0 is a suitable ring in the direct system {Ai}. This allows us to study X by analyzing its behavior over the rings in the system {Ai}.

The natural map from the limit of homomorphisms between Xi and Yi (schemes over Ai) to the homomorphisms between X and Y (schemes over A) is bijective. Essentially, this means that the homomorphisms between the schemes over A can be fully understood by studying the homomorphisms between the corresponding schemes over the rings in the system {Ai}.

A map between schemes acquires certain properties like being closed, separated, or proper upon base change to a ring Ai if and only if the induced map over the original ring A has the same property. This means that these properties are 'preserved' under descent.

A scheme X can be 'spread out' over a suitable ring, such as OF,S, where S is a finite set of places in the base field F. This means that X can be represented as the generic fiber of a scheme over OF,S, which allows us to analyze its behavior over the curve Spec OF,S.

The process of finding a scheme Xi0 over Ai0 whose base change over A is isomorphic to X is called 'descent'. This allows us to study a complex scheme by analyzing simpler schemes over rings in a direct system.

Essentially unique isomorphism means that if there are two 'descents' of a scheme X (Xi0 and Xi00), they can be made isomorphic to each other over some larger i ≥ i0. This uniqueness ensures that the different 'descents' of the same scheme essentially represent the same object.

The choice of {Ai} used in descent can have a significant impact on how we understand and study the scheme X. The most important choices are {AF,S} (with limit AF) and {OF,S} (with limit F), where F is the base field.

Given a scheme X defined over a field F, 'spreading out' X over OF,S, where S is a finite set of places of F, allows us to 'visualize' X as fibered over a curve, Spec OF,S. This approach helps us understand the behavior of X over the field F by looking at its structure over a curve.

A subset of a topological space is dense if its closure is the entire space. It means that every point in the space can be approximated by points from the dense subset.

A proper map between topological spaces is a continuous function that maps closed sets to closed sets and compact sets to compact sets. This implies that the map preserves the 'compactness' and 'closedness' properties.

A projective space Pn(F) over a field F is a set of all lines passing through the origin in an (n+1)-dimensional vector space over F. It can be thought of as a space of directions where the origin is 'removed'.

Weak approximation is a property of a global field F, where every point in a finite product of local fields can be 'approximated' by points in F. Think of F as a 'bridge' connecting all its local versions.

The adele ring AF of a global field F is a ring that combines all the local information about F. It's like a 'blueprint' containing all the local versions of F.

A compact Hausdorff space is a topological space where every point has a neighborhood, which is both closed and open. It combines the idea of 'boundedness' with 'separatability'.

A locally compact Hausdorff space is a topological space where every point has a neighborhood that is both closed and open, but the entire space itself might not be compact.

A smooth surjective map between schemes, similar to a continuous function but with extra algebraic structure, that maps points smoothly without collapsing them.

A scheme whose underlying topological space can be broken down into finitely many open sets, each of which is the spectrum of a ring. This means the scheme has a finite, well-defined structure.

The mapping that associates each point in a scheme with a specific local ring, representing the algebraic properties around that point. Think of it as a map between a scheme and its associated algebraic structures.

A point in a scheme where the corresponding local ring is a field. This point represents a 'maximal' algebraic element in the scheme.

A mapping between schemes that preserves the underlying topological structure and certain algebraic properties. This ensures a strict connection between the schemes, preserving their 'geometry' and 'algebraic relationships'.