I am currently working through a paper (related to tropical geometry, but this is not important in the following context) which utilizes the concept of very affine varieties in the following way (I'll give a quick summary of the concept):

Let $X$ be a variety over a field $K$, i.e. an irreducible, reduced, separated $K$-scheme of finite type. Then an open subset $U$ of $X$ is called very affine if $U$ has a closed immersion to some multiplicative split torus $\mathbb G^r_m := Spec(K[T_1^{\pm 1}, \dots, T_r^{\pm 1}])$.

The author states: "[...] The very affine open subsets of $X$ form a basis for the Zariski topology."

I am wondering why this is the case. Here is what I know so far, if any of it helps: First, let $U$ be any open affine subset of $X$. Then \begin{equation} \text{Hom}_{K-\text{Sch.}}(U, \mathbb G^r_m) \cong \text{Hom}_{K-\text{alg.}}(K[T_1^{\pm 1}, \dots, T_r^{\pm 1}], \mathcal O_X(U)). \end{equation}

Any $K$-algebra morphism $f \colon K[T_1^{\pm 1}, \dots, T_r^{\pm 1}] \to \mathcal O_X(U)$ is uniquely given by the units $f(T_i) \in \mathcal O_X(U)^*$.
Furthermore it is known that the abelian group $\mathcal O_X(U)^* / K^*$ is free of finite rank, say $r$. Choose representatives $\varphi_1, \dots, \varphi_r$ in $\mathcal O_X(U)^*$ of a basis, we get a (more or less canonical) morphism $$\varphi \colon K[T_1^{\pm 1}, \dots, T_r^{\pm 1}] \to \mathcal O_X(U), T_i \mapsto \varphi_i.$$

So far so good. Using these identities it is easy to show that the following statements are equivalent:

a.) $U$ is very affine;
b.) $\mathcal O_X(U)$ is generated as a $K$-algebra by $\mathcal O_X(U)^*$;
c.) the canonical map $\varphi$ above is surjective.

Also I have shown that intersections of very affine opens are very affine again.

My approach now would be to take any point $x \in X$, choose any affine open $U$ around it and then try to 'make it very affine' by successively localizing until I get an open subscheme $V$ of $U$ that satisfies statement b.) above. However I can't really get it to work.

Long explanation, but if anyone could help or has any comments, that would be super awesome!

Thanks a lot in advance!


I thought about it once more and I think I have a solution which might work:

Claim: Let $X$ be variety over field $K$. Then the very affine open subsets of $X$ form a basis for the Zariski topology.

Proof: Let $x \in X$ and $U \subseteq X$ be an open neighborhood of $x$. It suffices to show that there is very affine $V$ around $x$ with $V \subseteq U$.
As open subschemes of varieties are varieties again, and by possibly passing to a smaller open neighborhood, we can assume that $U$ is affine with $U = \text{Spec}(A)$, where $A$ is a $K$-algebra of finite type, i.e. it is of the form $A = K[T_1, \dots, K_n]/I$ for some ideal $I$. Let $\mathfrak p$ denote the prime ideal of $A$ corresponding to $x$. Let $\overline{T_i}$ denote the class in $A$ of $T_i \in K[T_1, \dots, K_n]$.

We first suppose that there is no $i \in \{1, \dots, n\}$ with $\overline{T_i} \in \mathfrak p$. Then $V := D(\overline{T_1}) \cap \dots \cap D(\overline{T_n}) = D(\overline{T_1 \cdot \cdots \cdot T_n}) \subseteq U = \text{Spec}(A)$ is open around $x$ and corresponds to localization $A[\frac{1}{\overline{T_1 \cdot \cdots \cdot T_n}}] = A[\frac{1}{\overline{T_1}}]\cdots[\frac{1}{\overline{T_n}}] =: B$.
As $\overline{T_i} \in B^* \ \forall i \in \{1, \dots, n\}$ we obtain a surjective $K$-algebra morphism $$ K[T^{\pm 1}_1, \dots, T^{\pm 1}_n] \to B, T_i \mapsto \overline{T_i}, $$ which shows that $V$ is very affine.

On the other hand, suppose $\exists i$ with $\overline{T_i} \in \mathfrak p$. Denote $J \subset \{1, \dots, n\}$ the set of such indices. For any $j \in J$ we then must have $\overline{T_j} + 1 \notin \mathfrak p$ and we can proceed as above by successively localizing to obtain $V$ (however instead of localizing at $\overline{T_j}$ we use $\overline{T_j} + 1$ for all $j \in J$). Then again obtain surjective $K$-algebra morphism $$ K[T^{\pm 1}_1, \dots, T^{\pm 1}_n] \to B, T_i \mapsto \begin{cases}\overline{T_i}\ \ \text{for } i \notin J \\ \overline{T_i} + 1 \ \ \text{for } i \in J \\ \end{cases}, $$ which shows the claim.


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