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I'm studying algebraic geometry from the classical viewpoint in which the Zariski topology takes center stage and schemes have yet to be invented. I sometimes see the term "closed subvariety" thrown around, but I can't find a proper definition for this. For example on p.43 of J.S. Milne's notes we find:

PROPOSITION 2.27. Let $V$ be an irreducible variety such that $k[V]$ is a unique factorization domain. If $W\subseteq V$ is a closed subvariety of dimension $\mathrm{dim}(V)-1$, then $I(W)$ is a principal ideal.

(In this context $I(W)$ means the set of all polynomials that vanish on $W$.)

I'm not quite sure what a closed subvariety is. Isn't every variety automatically closed?

goblin GONE
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    Well - for example, you might consider $\mathbb{A}^1$ and its open subset $\mathbb{A}^1-{0}$. This is not closed in $\mathbb{A}^1$, but is still a variety (it is isomorphic to the zero locus $xy-1=0$ in $\mathbb{A}^2$. – loch Jun 27 '18 at 09:23
  • @loch, I thought it wouldn't be considered a variety, because (i) a variety is an irreducible algebraic set, and (ii) an algebraic set is automatically closed. Am I wrong here? – goblin GONE Jun 27 '18 at 09:26
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    (i) and (ii) are not wrong - but what I'm basically saying is that an open set in one space can be isomorphic to a closed set in another. – loch Jun 27 '18 at 09:35
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    Often in this context, "variety" in general will mean a quasiprojective variety, i.e. an open subset of a closed subset of $\mathbb{P}^n$ for some $n$. – Daniel Schepler Jun 28 '18 at 22:13

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Maybe your confusion lies in the difference between algebraic sets, affine algebraic varieties and algebraic varieties. By algebraic sets I mean the zero sets of polynomials whereas (affine) varieties are defined as ringed spaces. But depending on the author these can mean different things.

It seems like your are using an old version of the notes by Milne. I would recommend you to switch to the newest version (6.02) of the notes. There he starts by defining algebraic sets and later moves to (affine) algebraic varieties. On page 68 is a definition of closed subvarities.

Florian
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