Exploring Affine n-space in Algebraic Geometry

[Bleys]

I need some help about the definition of this. I cannot find it properly anywhere. This concerns algebraic geometry.

What is an affine n-space?
In some books, k is a field and it says this is just the point space k^n. But on wikipedia there is a whole abstract definition of it, and I don't know whether it corresponds to what algebraic geometry books refer to (there is no mention of fields, and in the books there is no mention of vector spaces, so I wouldn't know how to apply it).

Is it an actual algebraic structure or just a set or?

 

[Rasalhague]

An affine space, A, is a tuple, (A,V,f), where A is a nonempty set, the underlying set or point set of this affine space, whose elements we call points.

V is a vector space, (V,K,+,s), where V is a nonempty set whose elements we call vectors; K is its underlying field, + is vector addition, obeying the axioms of a commutative group, and s is the scalar multiplication function, s:K x V --> V, by which this vector space is defined. K, the field, is defined similarly, as a tuple, (K,+,*), where K is a nonempty set whose elements we call scalars, + is the field's addition function, and * its multiplication.

f is a function f:V x A --> A which obeys the affine space axioms, as listed at Wikipedia: Affine space in two equivalent versions, and at Wolfram Mathworld and many other sites.

The set A on its own doesn't constitute an affine space. It needs this extra structure to meet the requirements of the definition. But when a particular choice of V and f are "understood" (understood meaning: expected to be obvious from the context), people will often refer to the set A, colloquially/sloppily, as an affine space, just as they may refer to the set V as a vector space, when it's taken for granted that listeners/readers will know which particular field and which particular vector addition function are being used.

A well known type of vector space, sometimes called a coordinate space, is a vector space of the form (Kⁿ,K,+,s) where + is itemwise/entrywise/termwise addition, the ith term of a + b being the sum of the ith terms of a and b, and s multiplies each term of its vector input by its scalar input. Not every vector space is a coordinate space, but every n-dimensional vector space over the field K is isomorphic to (Kⁿ,K,+).

The point set of an affine space doesn't have to be a set of the form Kⁿ. But given any coordinate space V = (Kⁿ,K,+,s), we have an affine space (Kⁿ,V,+).

In fact, more generally, given any vector space, (V,K,+,s), where + is any vector addition function obeying the axioms of a commutative group, and s any scalar multiplication function obeying the vectors space axioms, we have an affine space (V,V,+). This is what Wikipedia: Affine space means by "any vector space is an affine space over itself."

 

[micromass]

The problem with affine spaces is that there are several definitions of this. And none of these definitons give the same answer. So when reading a book, you always need to see which definition they are using...

Now, I know three definitions of an affine space:
1) an affine space used in ordinary geometry (not algebraic geometry). This consists of a set X, a vector space V, and an action of V on X. This gives us an affine space over a certain vector space.
If I'm not mistaken, this is what Rasalhague tries to say in his post. And this is also what appears in the wikipedia article. But this is not the affine space of algebraic geometry!

2) In algebraic geometry, when you work with varieties. Then we are given a field k. The affine n-space over k, is simply defined as kⁿ. We often denote this as $$\mathbb{A}^n(k)$$.
What they do next, is look at $$k[X_1,...,X_n]$$ (the polynomials over k with n indeterminates). Every set of polynomials determine a set in $$\mathbb{A}^n(k)$$, namely it's set of zeroes. Formally, given a set $$S\subseteq k[X_1,...,X_n]$$, then we define

$$V(S)=\{(a_1,...,a_n)\in \mathbb{A}^n(k)~\vert~\forall f\in S:~f(a_1,...,a_n)=0\}$$

These sets actually determine a topology on $$\mathbb{A}^n(k)$$ (= the Zariski topology).

So, if you do algebraic geometry, always apply (2). If you do ordinary geometry, then you must use (1).

 

[Bleys]

Thank you both for the reply!

Ok so the definition I'm interested in is (2). But why the need to call it affine n-space? Is there background or reason to this? I know "affine" means 'analogous' or 'similar' and I could understand from definition (1) why it would be called that (the analogy of Eucledian n-space) but why use it in (2)? Just because it's n-tuples of elements in a field?

Anyway, thanks for the help, both of you!

 

[micromass]

Good question! To be honest, I don't really know the answer. But I have a guess:

In algebraic geometry, there are two important kinds of geometry: $$\mathbb{A}^n(k)$$ and $$\mathbb{P}^n(k)$$, which both have different properties (but which are sometimes quite alike!).

In the space $$\mathbb{A}^n(k)$$, we can talk about lines. And these lines satisfy the following: two lines which are parallel don't intersect.
However, in the space $$\mathbb{P}^n(k)$$, all lines intersect! Even if the lines are parallel! (we say that two parallel lines intersect "at infinity")

So these two spaces are very different from the view of parallel lines. This is a situation which was already familiar from ordinary geometry! Indeed, in ordinary geometry, one can talk about affine spaces (this are the spaces from (1)) and these have the property that two parallel lines don't intersect.
But in ordinary geometry, we can also regard projective spaces, and these are spaces where every line intersect.

And I think that the analogy with ordinary geometry has motivated algebraic geometers to call $$\mathbb{A}^n(k)$$ the affine space of dimension n and to call $$\mathbb{P}^n(k)$$ the projective space of dimension n.

 

[Rasalhague]

Sorry, Bleys, I didn't realize there were multiple conflicting definitions.

micromass, just out of curiosity, what's the third definition besides (1) the definition I'd heard of, and (2) "Cartesian power of the underlying set of a field"?

 

[micromass]

Ah yes, I forgot the third definition. The third definition is actually a generalization of (2). The second definition works well with algebraic varieties. But if you want to work with schemes, then a new definition is in order.

Let R be an arbitrary ring (not a field now), then $$R[X_1,...,X_n]$$ denotes the polynomial ring. We can build a topological space out of $$R[X_1,...,X_n]$$ as follows:
the underlying set is $$Spec(R[X_1,...,X_n])=\{\mathfrak{p}~\vert~\mathfrak{p}~\text{is a prime ideal of}~R[X_1,...,X_n]\}$$. Let I be an ideal of $$R[X_1,...,X_n]$$, then let $$V(I)=\{\mathfrak{p}\in Spec(R[X_1,...,X_n])~\vert~I\subseteq \mathfrak{p}\}$$. The collection of V(I) form the closed sets of a topology on $$R[X_1,...,X_n]$$.

Thus we have a topological space. Now we can define a certain sheaf on this topological space. The topological space together with that sheaf is called $$\mathbb{A}^n_R$$ and is called the affine space of dimension n over R.

This is a very complicated definition, but it is the central point of study in algebraic geometry.