Question on equivalence of definitions of quasi-affine varieties

I'm hoping to introduce some basic concepts of classical algebraic geometry to some students in a seminar soon. To avoid the usual ambiguity and confusion around the terms in this picture, I was hoping to present things in a more consistent, if atypical, fashion. Since I have a tendency to overlook some subtleties, I'd like to know if there are any issues with the following outline:

1. Fix a field $$k=\overline{k}$$ and define $$\mathbb{A}^n$$
2. Define the Zariski topology on $$\mathbb{A}^n$$ and use it to distinguish closed and irreducible closed subsets.
3. Define a morphism $$f:X \to Y$$ between subsets $$X \subseteq \mathbb{A}^n$$ and $$Y \subseteq \mathbb{A}^m$$ to be a set-theoretic map $$f =(f_1/g_1, \ldots, f_m/g_m)$$, where $$f_i,g_i \in k[x_1,\ldots,x_n]$$ and $$\forall x \in X, g_i \neq 0$$ (a regular map). Isomorphisms are biregular bijections.
4. Give the following extrinsic definitions: $$X \subseteq \mathbb{A}^n$$ is a quasi-affine variety if it's isomorphic to a closed subset of some $$\mathbb{A}^m$$, and an affine variety if it's isomorphic to an irreducible closed subset of some $$\mathbb{A}^m$$.

I think this captures the picture with some consistency, but I'm not sure if the definition of quasi-affine above is equivalent to the usual definition as locally closed subset. If $$X$$ is locally closed then I believe it has a closed image in some $$\mathbb{A}^m$$ (if it's an open subset of a closed set $$X \subset \mathbb{A}^n$$, then it can be written as a finite intersection of closed sets in $$\mathbb{A}^{n+1}$$). I'm not sure about the converse, but the idea that quasi-affine means we have a solution of a polynomial system minus some other solution set should mean it's a union of solutions somewhere.

• i've had some teaching success doing everything extrinsically in a course for non-specialists. are you going to use the fact that e.g. Aˆ1 \setminus 0 "is" a variety in some sense? or can you just work with closed things? – hunter Jan 8 at 21:17
• Qiaochu's already given you a good answer to the question you're asking below, but I feel compelled to advise against trying to give too simple of a definition/explanation here. Unless these students will never again see any algebraic geometry, I think it's important to let them know that this definition gets upgraded substantially as they move through the subject – KReiser Jan 8 at 21:38
• My goal was to kick things off by focusing on affine subsets of interest (solutions to systems of polys), and to (mistakenly) demonstrate a symmetry in definition between affine and quasi-affine sets. Mainly because I've had some students get bewildered by the sudden enlargement of the category from aff ("If we care about solutions, why are we adding in 'local' solutions now?"). Now that I think about it, I like the idea of justifying "quasi-" by saying it's natural to subtract some solutions out of a solution set, and noting that such things can be written as unions of other solution sets. – SBRJCT Jan 8 at 22:38

1 Answer

$$X \subseteq \mathbb{A}^n$$ is a quasi-affine variety if it's isomorphic to a closed subset of some $$\mathbb{A}^m$$

This is not equivalent to the usual definition of quasi-affine; for example, $$\mathbb{A}^2 \setminus \{ 0 \}$$ is quasi-affine in the usual sense but not in this sense. (Personally I would call this "affine.")

• I had a feeling I did something wrong. I'm not sure how I suddenly convinced myself that I'd get a bijection between an open set covered by principal opens and the union of the closed images of the principal opens. D'oh! – SBRJCT Jan 8 at 22:20