Abstract
The objective of the chapter is to introduce the notion of schemes and their morphisms, in particular, to explain the construction of affine schemes. If A is a ring, the set of its prime ideals is viewed as a point set, called the prime spectrum of A; the latter is denoted by Spec A. Any element f∈A can be viewed as a function on Spec A. Just let \(f(\mathfrak{p})\), for any prime ideal \(\mathfrak{p} \subset A\), be the residue class of f in \(A/\mathfrak{p}\). Furthermore, write D(f) for the subset of all points \(\mathfrak{p} \in \mathrm{Spec}\,A\) such that \(f(\mathfrak{p})\) is not the zero class. Then D(f) is called a basic open subset in Spec A, while the topology generated by all these sets is referred to as the Zariski topology on Spec A. There is a remarkable fact to be discovered: for any element f∈A, the point set D(f)⊂Spec A can canonically be identified with the prime spectrum Spec A f of the localization of A by f. In particular, the natural localization morphism A⟶A f can be interpreted as the process of restricting functions living on Spec A and given by elements in A, to functions on the basic open subset D(f). In order to handle local functions on Spec A in convenient terms, the notion of sheaf is introduced. This way Spec A is viewed as a ringed space, i.e. as a topological space together with a sheaf of functions on it. Such a pair is called an affine scheme. More general schemes occur as ringed spaces that look locally like affine schemes.
There is additional material in the chapter. It exposes quasi-coherent modules on schemes, as well as the basics on direct and inverse images of module sheaves with respect to scheme morphisms.
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© 2013 Springer-Verlag London
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Bosch, S. (2013). Affine Schemes and Basic Constructions. In: Algebraic Geometry and Commutative Algebra. Universitext. Springer, London. https://doi.org/10.1007/978-1-4471-4829-6_6
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DOI: https://doi.org/10.1007/978-1-4471-4829-6_6
Publisher Name: Springer, London
Print ISBN: 978-1-4471-4828-9
Online ISBN: 978-1-4471-4829-6
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