$X_f$ of locally ringed space $(X, O_X)$.

Question

Let $(X, O_X)$ be a locally ringed space. $f \in \Gamma(X,O_X)$ be a global section. $$X_f:= \{ x \in X \, ; \, f_x \text{ is invertible in } O_{X,x} \} $$

It is claimed that

1. $X_f$ is an open subset
2. The image of $f$ in $\Gamma(X_f, O_X)$ is invertible.

How does one see this?

Answer 1

If $f_x$ is invertible in $\mathcal{O}_{X, x}$, then by definition of the stalk, there is an open neighborhood of $U$ of $x$ and a section $g \in \Gamma(U, \mathcal{O}_X)$ with $fg = 1$ on $U$.

It follows that if $y \in U$ is any other point in $U$, then $f_y$ is invertible in $\mathcal{O}_{X, y}$, which shows that the set of all such points is open.

The fact that the image of $f$ is invertible follows from the sheaf axioms: you can cover $X_f$ with opens $U$ on which you have local inverses $g$ for $f$, and these local inverses agree on the overlaps because inverses are unique (when they exist) in any ring.