where I am not fully satisfied that $K^n$ is a field, rather $n$ pieces of fields under cartesian product such that $K\times K \times K \times \dots \times K$ where $n$ pieces of $K$. Also $K^n$ contains $n$-length tuples where each parameter in $K$. How is $K^n$ usually defined?
1 Answer
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It is not a field. As a ring, it is not even an integral domain, since $(0,1,\ldots)\cdot (1,0,\ldots)=0$. As a set, it consists of $n$-length tuples of elements of $K$.
Depending on your setting, you may want to view $K^n$ as an $n$-dimensional $K$-vector space or an $n$-dimensional $K$-algebra.
In algebraic geometry, thinking of it as $n$-length tuples is sufficient, since:
- You don't really use the ring/group structure of $K^n$, and
- As a ring $K^n$ has special points ($0$, $1$ etc). In algebraic geometry, you're not interested in these. As such, it is common notation to denote this set by $\mathbb A_K^n$ to differentiate it from the other interpretations of $K^n$.
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1$\begingroup$ I'm no algebraic geometer, but I've seen people write ${\bf A}^n$ to mean the scheme (as opposed to the set $n$-tuples). $\endgroup$– tomaszMar 12, 2016 at 21:13
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$\begingroup$ @tomasz I'm also no algebraic geometer. On page 1 of Hartshorne it's defined in this way, and called "affine $n$ space over $K$". In chapter $2$, it is indeed used to refer to the scheme. $\endgroup$ Mar 12, 2016 at 21:32