Basic question related to sheaf of a scheme

Question

Suppose I have a scheme $X$. And some non-empty open set $U \subseteq X$. Does it then follow that $O_X(U)$ is not the trivial $0$-ring by any chance?

Answer 1

Yes.

If $V$ is an affine scheme, it is easy to see that $\mathcal O_V(V) = \mathbf{0}$ if and only if $V=\emptyset$, if and only if $1=0$ in $\mathcal O_V(V)$.

Choose an open affine cover $\{V_i\}$ of $U$, and suppose that at least one of the $V_i$'s, say $V_0$, is nonempty. For each $i$, consider the section $1 \in \mathcal O_{V_i}(V_i)$. On $V_0$, we have $1 \neq 0$ because $V_0$ is nonempty. These sections glue on intersections, to give a section $1 \in \mathcal O_U(U)$, which restricts on $V_0$ to the nonzero section $1 \in \mathcal O_{V_0}(V_0)$. It follows that $1 \neq 0 $ in $\mathcal O_U(U)$, hence $\mathcal O_U(U) \neq \mathbf{0}$.