Affine Varieties
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Chapter 1 Affine Varieties In this chapter, we will assume that k is infinite, since when k is finite the only irreducible algebraic sets in An (k) are singletons. 1.0.1 Definition. An irreducible affine algebraic set is called an affine (alge- braic) variety, or simply a variety if the context is clear. Any variety X is endowed with the induced (Zariski) topology, whose open sets are of the form X U for some open subset U ⊆ An . T 1.1 Coordinate Rings Since affine varieties are defined by polynomials over a field, the most natural functions to consider on an affine variety are…
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