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Is it possible to define the category of affine algebraic varieties over a general (not necessarily algebraically closed) field $k$, by using the language of schemes, perhaps in a categorical way?

This should be standard I think, but I can not find an account of it anywhere. I know how to get affine space, by looking at the equivalence between affine schemes over $k$ and commutative $k$-algebras, but then how does one define an algebraic variety in this way?

Would be grateful for any comments.

John M.
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Yes. The category of affine $k$-varieties for any field $k$ (more generally, any base ring) is the 'image' of the subcategory of affine $k$-algebras under the functor $\operatorname{Spec}: \mathrm{CRing} \to \mathrm{Schemes}$.

After the edit to the question: Every affine variety can be embedded in affine space in the following manner. Let $(X,\mathcal{O}_X)$ be our affine variety. Since $\mathcal{O}_X(X)$ is finitely generated over $k$, pick some $x_1,\cdots,x_n$ which generated $\mathcal{O}_X(X)$. This gives a surjection $k[y_1,\cdots,y_n]\to \mathcal{O}_X(X)$ by sending $y_i\mapsto x_i$. This map is surjective with kernel $I$, and taking Specs of both sides exhibits $X$ as a subset of $\mathbb{A}^n$ with defining ideal $I$.

You can go back, too: given a closed subset of affine space, you can ask for the ideal defining it, and this will give you $I$ from above which lets you reconstruct the rest of the data.

(I'm a little unclear on what you mean by 'define an algebraic variety', but I hope this suffices. If you have something more specific in mind, please let me know in a comment.)

KReiser
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  • Thanks @Kreiser, I have in fact just edited my question. What do you mean exactly by image here? And what is an affine $k$-algebra? You mean finitely generated? – John M. May 05 '17 at 06:57
  • Nominally set-theoretic concepts like images can be a bit tricky to get right in the case of categories and functors between them, sometimes. I was hasty and included the quotes to ignore this subtlety. I guess I mean the following: affine schemes are recognizable amongst all schemes by the property that $X\cong \operatorname{Spec} \mathcal{O}_X(X)$. The 'image' is all such affine schemes where $\mathcal{O}_X(X)$ is a $k$-algebra. I'll take a look at your edit and attempt to add to my answer. – KReiser May 05 '17 at 07:02
  • thanks. In your last comment, you mean $\mathcal{O}_X(X)$ is an affine $k$-algebra, right? Also, your spectrum functor should be defined on commutative algebras over $k$, not just commutative rings, right? – John M. May 05 '17 at 07:09
  • Yes, you're right, $\mathcal{O}_X(X)$ should be an affine $k$-algebra. As for Spec, either way works. I can define my functor on some category containing affine $k$-algebras and then restrict it, or I can just define my functor on that category. – KReiser May 05 '17 at 07:12
  • thanks for all this. One last thing, regarding affine $k$-algebras. You mean the kind of commutative $k$-algebras that arise as coordinate rings of affine schemes, right? I think such things have a classification as finitely generated noetherian associative commutative unital $k$-algebras without nilpotent elements. Is this what you have in mind? And also, what are the morphisms between such specs supposed to be, in order to get a category of affine algebraic varieties over $k$? Do we take everything? – John M. May 05 '17 at 07:16
  • Yes, it is. That classification is correct. – KReiser May 05 '17 at 07:18
  • Ah just added an edit in my question...You were quicker. – John M. May 05 '17 at 07:19