# What is a "closed subvariety"?

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?

• 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
Commented Jun 27, 2018 at 9:23
• @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? Commented Jun 27, 2018 at 9:26
• (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
Commented Jun 27, 2018 at 9:35
• 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 $n$. Commented Jun 28, 2018 at 22:13