Locally ringed space

Definition

Let $(X, \mathcal{O}_X)$ be a ringed space, that is, a topological space $X$ with a sheaf $\mathcal{O}_X$ of commutative rings on $X$. It is a locally ringed space if for every point $x \in X$ the stalk $\mathcal{O}_{X,x}$ is a local ring. The maximal ideal of $\mathcal{O}_{X,x}$ is written $\mathfrak{m}_x$.

Let $(X, \mathcal{O}_X)$ and $(Y, \mathcal{O}_Y)$ be locally ringed spaces. A morphism of locally ringed spaces $(X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ is a pair $(f, f^\#)$ such that:

1. $f \colon X \to Y$ is a continuous map.
2. $f^\# \colon \mathcal{O}_Y \to f_* \mathcal{O}_X$ is a morphism of sheaves of rings on $Y$.
3. For every $x \in X$, the induced ring map $f^\#_x \colon \mathcal{O}_{Y, f(x)} \to \mathcal{O}_{X,x}$ is a local homomorphism, that is, $(f^\#_x)^{-1}(\mathfrak{m}_x) = \mathfrak{m}_{f(x)}$.