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my question is basically already in the title. I just started to read the Algebraic Geometry book by Hartshorne where he defines an algebraic set but in some propositions a bit later talks about affine algebraic sets. So does he mean the same thing because i didn't see him define affine algebraic sets?

Thank you!

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  • $\begingroup$ Is he saying affine algebraic set or affine algebraic variety. He defines the latter as irreducible algebraic sets. $\endgroup$ Dec 24, 2019 at 14:41
  • $\begingroup$ Saw him in Proposition 1.7 using affine algebraic set. He is using it there as his definition of algebraic set. $\endgroup$ Dec 24, 2019 at 14:43

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an algebraic set (or variety) is the zero set of a polynomial whose coefficients belong to a given field k, considered as a subset of k^n.

if k contains R k^n may be considered affine space, in the sense that it is closed under u,v-->au+(1-a)v for all a in R, and then the algebraic set may be called affine.

some authors use variety only to mean na irreducible algebraic set.

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  • $\begingroup$ usually, defined as a zero set of a FINITE SYSTEM of polynomials. but once you multiply them all, all you need is just one polynomial. $\endgroup$
    – Nir Cohen
    Jan 3, 2020 at 15:31

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