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What follows comes from Hulek, Elementary Algebraic Geometry, first chapter.

Definition Let $V$ be an affine variety in $\mathbb{A}^n_{k}$. A polynomial function on $V$ is a map $f:V\longrightarrow k$ such that there is a polynomial $F\in k[x_1,\ldots,x_n]$ with $f(P)=F(P)$ for every $P\in V$.

Definition The coordinate ring of $V$ is defined by $k[V]:=k[x_1,\ldots,x_n]/I(V)$ and can be identificated with the set of polynomial functions on $V$.

Definition Let $W\subseteq\mathbb{A}^m_k$ be another variety. A map $f:V\longrightarrow W$ is called a polynomial map if there are polynomials $F_1,\ldots,F_m\in k[x_1,\ldots,x_n]$ such that $f(P)=(F_1(P),\ldots,F_m(P))\in W\subseteq \mathbb{A}^m_k$, for all points $P\in V$.

I cannot understand the utility of the following Lemma

Lemma Let $y_1,\ldots,y_m$ be the coordinate functions in $\mathbb{A}^m_k$. A map $f:V\longrightarrow W$ is a polynomial map if and only if $f_j:=y_j\circ f\in K[V]$ for all $j$.

What is the meaning of this Lemma? Does it simply say that the $i$-th component of a tuple of polynomials is a polynomial? Why a separate Lemma for this triviality? Am I missing something deeper?

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  • $\begingroup$ It means that the coordinate functions of a morphism (regular map) between two varieties are regular functions. In some books morphism is defined like that. $\endgroup$ – mesel Feb 19 '14 at 9:13

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