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I'm used to the definition of Affine Variety as $$V(f_1, ..., f_s)=\{a=(a_1, ..., a_n) \in k^n \hspace{2mm}| \hspace{2mm} f_i(a)=0 \hspace{3mm} \text{for every} \hspace{3mm} 1 \leq i \leq s\}$$ for some field $k$. However, I've read recently (on a French book) the definition of an Affine Variety $V$ as a subset of some vector space such that, for every $x, y \in V$, the line that intersects $x$ and $y$ are entirely contained in $V$. Formally speaking, an affine variety $V$ is a set such that $$ x, y \in V, t \in \mathbb{R} \implies (1-t)x+ty \in V.$$ These two definitions, of course, are different, but could someone explain me why there is this conflict in definitions? It seems to me that in French Affine Varieties are called Algebraic Affine Varieties, and what they call by Affine Variety is what I just wrote here. What would be the equivalent definition in english?

Thanks.

J. W. Tanner
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Will199
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    Your second definition isn't an affine variety in the sense of algebraic geometry, it's an affine space in the sense of "vector space without an origin". – KReiser Dec 02 '20 at 02:34
  • Also affine variety needs that the $f_i$ generate a prime ideal, and it is not only the set but also all the algebraic structure in particular the coordinate ring $k[x_1,\ldots,x_n]/(f_1,\ldots,f_s)$, its field of fractions, maximal ideals, localizations.. That we can identity with the vanishing set of the ideal in $\overline{k}^n$. – reuns Dec 02 '20 at 02:44
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    Some authors don't require that affine = irreducible, so the $f_i$ don't always need to generate a prime ideal (although usually the convention is to have them generate a prime ideal). – mi.f.zh Dec 02 '20 at 03:07

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Let me upgrade my comment in to an answer. Your first definition is one of many for affine varieties. The second definition is for an "affine space", which is a vector space where we forget where $0$ is - that means we're allowed to add any two elements of our space as long as the coefficients sum to one, but we don't have scalar multiplication, a choice of a $0$ vector, etc.

One thing that may be contributing to your confusion is that the French word "variété" has multiple meanings - this has been discussed here on MSE in the past.

KReiser
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  • Thank you very much for the explanation, and, yes, "variété" seems to be used interchangeably with other concepts, such as manifold. I believe this also happens in Spanish and Portuguese. – Will199 Dec 02 '20 at 16:14