Locally ringed space is reduced if and only if all stalks are reduced

Question

I'll use the following definitions, for a locally ringed space $(X, \mathcal{O}_X)$:

1. $X$ is reduced if each $\mathcal{O}_X(U)$ is reduced.
2. $X$ is locally reduced if each stalk $\mathcal{O}_{X,x}$ is reduced.

For schemes, these two are equivalent, but I'm having trouble seeing whether $2 \implies 1$ without reducing to the affine case. Is this direction still true for locally ringed spaces in general?

Answer 1

Yes; consider the canonical ring homomorphism $\mathcal{O}_X(U) \to \prod_{u\in U} \mathcal{O}_{X,u}$ given by sending a section to its image in the stalk at every point. This is injective by the sheaf condition, and locally reduced implies that the target is reduced, hence the left hand side must be reduced too.