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In mathematics, specifically algebraic geometry, a scheme is a structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x = 0 and x 2 = 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).
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, ... Basic Algebraic Geometry II Schemes and complex manifolds (2nd ed.).
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.
algebraic geometry Algebraic geometry is a branch of mathematics that studies solutions to algebraic equations. algebraic geometry over the field with one element One goal is to prove the Riemann hypothesis. [2] See also the field with one element and Peña, Javier López; Lorscheid, Oliver (2009-08-31). "Mapping F_1-land:An overview of ...
An abstract algebraic variety is a particular kind of scheme; the generalization to schemes on the geometric side enables an extension of the correspondence described above to a wider class of rings. A scheme is a locally ringed space such that every point has a neighbourhood that, as a locally ringed space, is isomorphic to a spectrum of a ring .
In algebraic geometry, a new scheme (e.g. an algebraic variety) can be obtained by gluing existing schemes through gluing maps. Statement
In mathematics, specifically in algebraic geometry, the fiber product of schemes is a fundamental construction. It has many interpretations and special cases. For example, the fiber product describes how an algebraic variety over one field determines a variety over a bigger field, or the pullback of a family of varieties, or a fiber of a family of varieties.
Linear algebraic group. Additive group; Multiplicative group; Algebraic torus; Reductive group; Borel subgroup; Radical of an algebraic group; Unipotent radical; Lie-Kolchin theorem; Haboush's theorem (also known as the Mumford conjecture) Group scheme; Abelian variety. Theta function; Grassmannian; Flag manifold; Weil restriction; Differential ...