Let $X$ be an affine smooth variety over the complex numbers, $X^{an}$ its associated smooth complex analytic space, and $\mathcal{O}$, resp. $\mathcal{O}^{an}$ the respective structure sheaves.

> Is the natural map

> $$\mathcal{O}(X)\to \mathcal{O}^{an}(X^{an})$$

> injective?

Denoting by $f : X^{an}\to X_{Zar}$ the morphism of sites, this map is given as the composition

$$\Gamma(X_{Zar}, \mathcal{O})\to \Gamma(X^{an},f^{-1}\mathcal{O})\to \Gamma(X^{an},\mathcal{O}^{an})$$
and in this composition we know the last map is injective (see Serre’s GAGA paper, Prop. 10(b)).

What about the first map?

Morally, this map should send a regular function on $X$ to the holomorphic function it induces on $X^{an}$ in an evident way, so I would expect the answer to this question to be trivially yes, but I want to make sure I’m not missing anything.