Let $V$ be an affine toric variety (where the base field is $\mathbb C$), i.e., $V$ is an irreducible affine variety containing a torus $T$ as a Zariski open subset and $T$ has an action on $V$.
The action if $T$ on $V$ is given by a morphism $T\times V\rightarrow V$. Let $f\in\mathbb C[V]$, a regular function on $V$. For $t\in T$, we define $t\cdot f$ which is given by $p\mapsto f(t^{-1}\cdot p)$.
My question is why $t\cdot f\in\mathbb C[V]$.
This is from the book 'Toric varieties' by Cox, Little, Schenck; page 19, Theorem 1.1.17
Thank you.
Best Answer
Recall that a function in $\Bbb C[V]$ is the same as a regular map $V\to \Bbb A^1_\Bbb C$, and that the composition of regular maps is a regular map.
The map $i:T\to T$ given by $t\mapsto t^{-1}$ is a regular map, and the map $m:T\times V$ given by $(t,p)\mapsto t\cdot p$ is a regular map. Thus $V\stackrel{t_0\times id}{\to} T\times V \stackrel{i\times id}{\to} T\times V \stackrel{m}{\to} V$ for a fixed $t_0$ is given by $v\mapsto t_0^{-1}\cdot v$. Further, post-composing with $f: V\to \Bbb A^1_\Bbb C$ gives us that the composite map is $p\mapsto f(t_0^{-1}\cdot p)$ and since this is a composition of regular maps, it is a regular map from $V\to \Bbb A^1_\Bbb C$ and thus an element of $\Bbb C[V]$.