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I am trying to get my hands on the basics of algebraic geometry, and I am confused about the following.

Let $(\mathbb{C}^{\times})^n \subset \mathbb{C}^n$ denote the complex torus. What I know is that to any given algebraic variety $X \subset \mathbb{C}^n$, one can associate its vanishing ideal $$ I(X) := \lbrace P \in \mathbb{C}[x_1,...,x_n] \ | \ P(x_1,...,x_n) = 0, \ \forall (x_1,...,x_n) \in X \rbrace. $$ Then the ring of regular functions on $X$ is defined as the quotient $$ \mathbb{C}[x_1,...,x_n] / I(X). $$ For instance, the affine space $\mathbb{C}^n$ has vanishing ideal $\{0\}$, and therefore its ring of regular functions is the whole ring of polynomials.

For the complex torus, I was told that the ring of regular functions is given by the ring of Laurent polynomials $$ \mathbb{C}[x_1,...,x_n,x_1^{-1},...,x_n^{-1}]. $$

My questions are the following:

  1. How can I get the ring of Laurent polynomials from the above reasoning ?
  2. Let I(X) be the vanishing ideal of a subvariety $X \subset \mathbb{C}^n$. I have read somewhere that intersecting subvarieties corresponds to adding vanishing ideals. I there a way to interpret the quotient $$ \mathbb{C}[x_1,...,x_n,x_1^{-1},...,x_n^{-1}] / I(X)\mathbb{C}[x_1,...,x_n,x_1^{-1},...,x_n^{-1}] $$ in terms of the intersection / union (or something else) of $X$ and $(\mathbb{C}^{\times})^n$ ? If it helps, one can suppose that $X$ is a linear subspace of $\mathbb{C}^n$.

Thanks a lot !

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The algebraic torus $(\mathbb{C}^\times)^n$ is not a closed subvariety of affine $n$-space. However, we can see $(\mathbb{C}^\times)^n$ as the affine subvariety of $\mathbb{C}^{2n}$ with variables $x_1, x_2, \ldots, x_n, y_1,y_2, \ldots, y_n$ cut out by the $n$ polynomials $x_iy_i - 1$. Its affine coordinate ring is then isomorphic to the ring you wrote down.

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