Affine Geometry – What is Affine Space?

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In my recent reading of various books and notes on algebraic geometry and scheme theory, I have come across three definitions of affine $n$-space over a field $k$:

  1. $\mathbb{A}_k^n$ is $k^n$ 'without an origin';

  2. $\mathbb{A}_k^n$ is simply $k^n$ with the Zariski topology (i.e. zero loci of sets of polynomials are closed);

  3. $\mathbb{A}_k^n=\operatorname{Spec}k[x_1,\ldots,x_n]$.

I know that things like the Nullstellensatz give us a nice way of going from one point of view to another when it comes to polynomial rings and affine varieties, but here I'm having some confusion.
For $k=\mathbb{C}$ (which is algebraically closed, so things should all act reasonably nicely) we have that the points of $\operatorname{Spec}\mathbb{C}[x]$ are in bijective correspondence with points of $\mathbb{C}$, apart from the general point $[(0)]$.
So it seems to me that these definitions can't all be the same, but searching for affine space on Google or in books leads to any one of the above definitions, depending on the level of the text.

Am I just making a silly mistake, or is there a difference in these definitions?
If so, which one is the 'right' one?

Best Answer

So it seems to me that these definitions can't all be the same, but searching for affine space on Google or in books leads to any one of the above definitions, depending on the level of the text.

Right, they're not all the same. They define objects in three different categories, which are respectively

  1. The category of affine spaces over $k$. This requires a bit of elaboration. One way to define a vector space is as a set equipped with an $n$-ary operation for every $n$-tuple $(r_1, r_2, \dots r_n)$ of elements of $k$, corresponding to the linear combination operation $(x_1, x_2, \dots x_n) \mapsto \sum r_i x_i$, together with various axioms relating these. An affine space is a slightly restricted version of this where you only allow operations where $\sum r_i = 1$ ("affine linear combinations"). In particular you can't multiply by zero, so you don't know where your origin is. In this category, endomorphisms $k^n \to k^n$ correspond to maps of the form $v \mapsto Av + w$ where $A$ is a matrix and $w$ is a vector.

  2. The category of what I'll call "naive affine varieties" over $k$. This is the category whose objects are Zariski-closed subsets of $k^n$ for some $n$ and whose morphisms are polynomial functions over $k$. The big difference between this option and #1 is that polynomial functions of degree greater than $1$ are allowed. I don't want to call these things varieties because if $k$ is not algebraically closed this is the wrong category to work in. In this category, endomorphisms $k^n \to k^n$ correspond to $n$-tuples of polynomials of $n$ variables over $k$.

  3. The category of varieties over $k$, which itself has several definitions depending on your level of sophistication. The big difference between this option and #2 is that given a variety $X$ over a field $k$ it makes sense to talk about the points $X(L)$ of that variety over any field extension $L$ of $k$ (and in fact much more generally than this, but let's stick to field extensions for simplicity). For example, if $k = \mathbb{R}$ then $\{ x^2 + y^2 = -1 \}$ defines the empty set in option #2 but it defines an interesting variety (with interesting points over $\mathbb{C}$) in this option. In this category, endomorphisms $k^n \to k^n$ look the same as above, but by looking at field extensions you can get a more interesting endomorphism object involving polynomials over field extensions of $k$.

The relationship between these categories is that there are functors from #1 to #3 to #2. The tendency to give an object the same name in different categories where it appears when they're related by a functor is pretty common in mathematics (for example, think of how many categories have an object called $\mathbb{R}$) and it's good to get used to it sooner rather than later.

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