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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 x2 = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat c...
1. Why schemes? Schemes were introduced by Grothendieck more than fty years ago into the world of algebraic geometry. In much the same way as measure theory, nearly everyone in the eld almost immediately adopted the new de nitions. But like measure theory for someone on the outside the whole theory seems remarkably abstract and hard to absorb. For
10.1 Separated schemes 184 10.2 Properties of separated schemes 186 10.3 Proper morphisms 190 10.4 The valuative criteria* 194 10.5 Exercises 195 11 Schemes of finite type over a field 199 11.1 The formal definition of a variety 199 11.2 Schemes of finite type over a field 200 11.3 Dimension theory for schemes of finite type over a field 201
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- Basic Concepts and Properties.
- Cohomology of schemes.
- Construction of schemes.
Let $(X,\cO_X)$ be a scheme. For every point $x\in X$, the stalk $\cO_{X,x}$ at $x$ of the sheaf is alocal ring; the residue field of this ring is denoted by $k(x)$ and is called the residue field of the point $X$. As the topological properties of the scheme the properties of the underlying space $x$ are considered (for example, quasi-compactness, ...
Studies of schemes and related algebraic-geometric objects can often be divided into two problems — local and global. Local problems are usually linearized and their data are described by somecoherent sheaf or by sheaf complexes. For example, in the study of the local structure of a morphism $X\to S$, the sheaves $\def\O{\Omega}\O_{X/S}P$ of relati...
In the construction of a concrete scheme one most frequently uses the concepts of an affine or projective spectrum (seeAffine morphism;Projective scheme), including the definition of a subscheme by a sheaf of ideals. The construction of a projective spectrum makes it possible, in particular, to construct a monoidal transformation of schemes. Fibre ...
To take a simple example, consider the a ne plane A2. Over a eld k , the a ne plane can be thought of as having the ring of regular functions k [x; y], where x is the function that assigns a point to its x-coordinate, and y is the function that assigns a point to its y-coordinate.
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To help students become more proficient at solving word problems, teachers can help students recognize the problem schema, which refers to the underlying structure of the problem or the problem type (e.g., adding or combining two or more sets, finding the difference between two sets).
Grothendieck saw how to convert a ring into a space with a topology, a so-called affine scheme, and then defined schemes to be the things you can get by gluing together affine schemes.
