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Schemes

In: Basic Algebraic Geometry 2

Author

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  • Igor R. Shafarevich

    (Steklov Mathematical Institute of the Russian Academy of Sciences, Algebra Section)

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.

Suggested Citation

  • Igor R. Shafarevich, 2013. "Schemes," Springer Books, in: Basic Algebraic Geometry 2, edition 3, chapter 0, pages 3-47, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-38010-5_1
    DOI: 10.1007/978-3-642-38010-5_1
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