Scheme theory

Scheme theory is a branch of algebraic geometry that explores the properties of schemes, which are the fundamental objects of study in this field. Developed in the 1960s by mathematicians such as Alexander Grothendieck, scheme theory provides a unifying framework for various concepts in geometry and algebra. A **scheme** is locally defined by the spectra of rings, specifically the spectrum of a commutative ring, which can be thought of as a space of prime ideals.

The Chevalley scheme is a concept from algebraic geometry and is primarily related to the study of algebraic groups and their representations. It is named after the mathematician Claude Chevalley, who made significant contributions to the theory of algebraic groups. In basic terms, a Chevalley scheme is a certain type of scheme that comes from a connected linear algebraic group defined over a field.

Equivariant sheaves are a concept in algebraic geometry and representation theory that involve the notion of symmetry with respect to a group action. To understand equivariant sheaves, it's useful to break down the terminology: 1. **Sheaf**: A sheaf is a mathematical tool that captures local data of a space in a consistent global manner.

The fiber product of schemes is a fundamental construction in algebraic geometry, analogous to the notion of the fiber product in category theory. It allows us to "pull back" schemes along morphisms, producing a new scheme that encodes information from each of the original schemes.

In scheme theory, a branch of algebraic geometry, the concept of a "function field" is a fundamental idea associated with the geometrical notion of a variety. Let's break it down: ### Definition 1. **Function Field**: The function field of a variety (or scheme) can be informally thought of as the "field of rational functions" on that variety.

In the context of algebraic geometry, "geometrically" often refers to concepts, properties, or constructions that have a clear geometric interpretation or manifestation. Algebraic geometry itself is the study of geometric objects that can be defined by polynomial equations. The relationship between algebraic equations and geometric objects is a fundamental aspect of the field.

Gluing schemes is a concept in algebraic geometry and more broadly in the study of schemes, which are the fundamental objects of interest in modern geometric contexts. The idea of gluing schemes typically arises in the context of constructing new schemes from given ones, often using the tools of categorical theory and sheaf theory. ### Overview of Gluing 1.

The Picard group, often denoted as \(\text{Pic}(X)\), is an important concept in algebraic geometry and algebraic topology. It is primarily associated with the study of line bundles (or invertible sheaves) over a scheme or a topological space \(X\). ### Definition 1. **Line Bundles:** A line bundle over a space \(X\) is a locally free sheaf of rank 1.

Proj construction is likely a reference to "projection construction," which is used in various fields, including computer graphics, engineering, and project management. However, since "Proj construction" could refer to different concepts depending on context, let me outline a few possible interpretations: 1. **Projection Construction in Mathematics/Geometry**: This refers to methods of creating projections of geometric shapes, often to simplify complex visuals or to work within different dimensions.

In mathematics, particularly in set theory and algebra, the concept of a quotient by an equivalence relation is an important one. When you have a set \( S \) and an equivalence relation \( \sim \) defined on that set, you can partition the set into disjoint subsets, known as equivalence classes.

In mathematics, particularly in the field of algebraic geometry, a **Scheme** is a fundamental concept that generalizes the notion of algebraic varieties. Introduced by Alexander Grothendieck in the 1960s, schemes provide a framework for working with solutions to polynomial equations, accommodating both classical geometric objects and more abstract algebraic structures.

The term "Smooth scheme" typically refers to a concept in algebraic geometry. In this context, a scheme is a fundamental object that generalizes concepts of varieties and allows for a more flexible treatment of geometric objects. A smooth scheme is important because it captures certain desirable properties, particularly around points in the scheme.

The Séminaire de Géométrie Algébrique du Bois Marie (SGABM) is a mathematical seminar focused on algebraic geometry. It is held at the Institut des Hautes Études Scientifiques (IHES), which is located in Bures-sur-Yvette, France, near Paris. The seminar typically features talks on various topics in algebraic geometry, presented by both leading researchers and emerging scholars in the field.