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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 ...
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.
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 .
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 11.4 A structure result for ...
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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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In particular, for any index i, the complex Caug(Xi; R) is quasi-isomorphic to H1(Caug(Xi; R))[ 1], since Caug(Xi; R) is a two term complex. Since Caug((Xi2I); R) is also a complex of flat R-modules, Lemma 3.10 implies that. Hk(Caug((Xi2I); R)) = Hk(C aug(X1; R) Caug((Xi2I; i6=1); R)) ' Hk(H1(C aug(X1; R))[.
Basically, any serious study of varieties in families (whether arithmetic families, i.e. schemes over Z, or geometric families, i.e. parameterized families of varieties) requires scheme-theoretic techniques and the consideration of non-closed points.
