Abstract
The chapter introduces schemes as the new foundational object in algebraic geometry, with an extensive discussion of the ideas underlying this new notion. The prime spectrum SpecA of an arbitrary commutative ring with a 1 is defined as the set of prime ideals of A. It has a Zariski topology and a structure sheaf, a sheaf of rings with stalk at a point \({\frak{p}}\) the local ring \(A_{{\frak{p}}}\). Several examples are discussed, along with foundation notions, such as the dimension and product of affine schemes. General schemes are defined, together with the notion of product of schemes and the separatedness axiom in terms of closure of the diagonal.
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Notes
- 1.
Appendix refers to the Algebraic Appendix at the end of Book 1.
References
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Shafarevich, I.R. (2013). Schemes. In: Basic Algebraic Geometry 2. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38010-5_1
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DOI: https://doi.org/10.1007/978-3-642-38010-5_1
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-38009-9
Online ISBN: 978-3-642-38010-5
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