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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 ...
- 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 ...
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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mathematics. Suppose that we want to understand. xn + yn = zn: y and z 2 Z. It is well known that determining the integral solutions is very hard, and it is natural to attack such problems by considering what happens over C and also what happens when we reduce modulo p, which are both considerably easier and shed light on what happens over .
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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There are really two parts to a scheme: a topological space, and a thing called the structure sheaf which we think of as functions on the space, all of which is subject to a few conditions regarding compatibility.
Jul 5, 2020 · We give some historical background, then give the definition of a scheme and some simple examples, and finish by explaining the origin of the word "spectrum". This lecture is part of an...
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- Richard E Borcherds
