"Coordinate ring'' of the algebraic $n$-torus

Question

I'm trying to calculate the "coordinate ring'' of the algebraic $n$-torus $(\mathbb C^n)^\ast$:

If $X:=\mathbb C^n$ and $U:=(\mathbb C^n)^\ast$, we have that $X$ is an irreducible affine algebraic set ad $U$ is an open subset (with the Zariski topology). By definition we have that the ring of regular functions on $U$ is:

$$\mathcal O_X(U)=\bigcap_{x\in U}\Gamma(X)_{\mathfrak m_x}$$

where $\Gamma(X)=\mathbb C[T_1,\ldots,T_n]$ is the coordinate ring of $X$ and $\mathfrak m_x=\{f\in\Gamma(X)\,:\, f(x)=0\}$. But clearly $$U=X\setminus V(T_1,\ldots,T_n)= D(T_1)\cup\ldots\cup D(T_n) $$

so $$\mathcal O_X(U)=\mathcal O_X(D(T_1))\cap\ldots\cap\mathcal O(D(T_n))$$

and by the fact that $\mathcal O_X(D(f))=\Gamma(X)_f$ we can conclude finally that:

$$\mathcal O_X(U)=\mathbb C[T_1,\ldots, T_n]_{T_1}\cap\ldots\cap\mathbb C[T_1,\ldots, T_n]_{T_n}$$

Now I have two questions.

1. (technical question). How can I describe formally the ring $\mathbb C[T_1,\ldots, T_n]_{T_1}\cap\ldots\cap\mathbb C[T_1,\ldots, T_n]_{T_n}$? Why it should be $\mathbb C[T_1^{\pm 1},\ldots, T_n^{\pm 1}]$?
2. With the coordinate ring of an algebraic affine set $X$ in $\mathbb C^n$, I intend the ring $\mathbb C[T_1,\ldots,T_n]/I(X)$; in this case $U$ is not an affine variety but is a prevariety (finite union of affine variety), so what is the meaning of saying "the coordinate ring of $U$"? If $X$ is an affine variety then $\mathcal O_X(X)=\Gamma(X)$, so the global sections of the structural sheaf are the elements of the coordinate ring, why in this case $\mathcal O_X(U)$ should be the coordinate ring of $U$?

Answer 1

1. You seem to have misunderstood the definition of the torus. It is given by $\mathbb{C}^* \times \dotsc \times \mathbb{C}^*$, it is not the whole affine space minus a point. So it equals $D(T_1) \cap \dotsc \cap D(T_n) = D(T_1 \cdot \dotsc \cdot T_n)$ with coordinate ring $k[T_1,\dotsc,T_n]_{T_1 \cdot \dotsc \cdot T_n} = k[T_1^{\pm 1},\dotsc,T_n^{\pm 1}]$. Alternatively, you can use the fact that the coordinate ring of a product of affine varieties is the tensor product of the coordinate rings, so that in this case we get $\bigotimes_{i=1}^{n} k[T_i^{\pm 1}] = k[T_1^{\pm 1},\dotsc,T_n^{\pm 1}]$. What you have tried to compute is the coordinate ring of the punctured affine space $\mathbb{A}^n \setminus \{0\}$, which is in fact $\cap_{i=1}^{n} k[T_1,\dotsc,T_n]_{T_i}=k[T_1,\dotsc,T_n]$ for $n \geq 2$ (using that $T_1,T_2$ are coprime in the UFD $k[T_1,\dotsc,T_n]$; this is a special case of the algebraic Hartog's Lemma) and $k[T_1,T_1^{-1}]$ for $n=1$.

2. This is just a matter of terminology. If $X$ is any ringed space, then $\mathcal{O}_X(X)$ is sometimes called the ring of (global) regular functions on $X$, but also sometimes the coordinate ring of $X$. Of course one cannot expect to reconstruct $X$ from this ring, unless $X$ is an affine scheme. For example $\mathbb{A}^2$ has the same coordinate ring as $\mathbb{A}^2$ minus a closed point, as remarked above.