# Notation of Affine space.

I am following Fulton's book on Algebraic curves.It starts with the notion of affine spaces denoted by $$\mathbb A^n(k)$$ where $$k$$ is a field.The author defines the affine space $$\mathbb A^n(k)=k^n$$ the $$n$$-fold cartesian product of the field $$k$$ and there is no additional structure on it.So,my question is about the notation.Why cannot we simply denote the affine space by $$k^n$$.Why we denote it with a more complicated notation $$\mathbb A^n(k)$$?Algebraic curves is almost new topic to me and I have no idea why this is defined by such a notation.Can someone clarify?

## 2 Answers

If you write $$k^n$$ it will most likely be interpreted as the $$n$$-dimensional vector space over $$k$$. The $$n$$-dimensional affine space over $$k$$ is a different object, it’s not a vector space, and although both objects can be identified as sets they belong to different categories and have different morphisms (affine space is commonly thought as a scheme). So the notation $$\mathbb{A^n}(k)$$ is preferred because it is less ambiguos, and it is consistent with the notation $$\mathbb{P}^n(k)$$ for projective space.

Yes, $$\mathbb A^n(k)$$ is just $$k^n$$ but Fulton's motivation for the notation is probably to distinguish the set of $$n-$$tuples of members of $$k$$ from the projective space $$\mathbb P^n(k)$$, which is $$k^{n+1}\(0, ... ,0)$$ modulo the equivalence relation of scalar multipication by a non-zero member of $$k$$.