Definition 6.25.1. A ringed space is a pair $(X, \mathcal{O}_ X)$ consisting of a topological space $X$ and a sheaf of rings $\mathcal{O}_ X$ on $X$. A morphism of ringed spaces $(X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ is a pair consisting of a continuous map $f : X \to Y$ and an $f$-map of sheaves of rings $f^\sharp : \mathcal{O}_ Y \to \mathcal{O}_ X$.
6.25 Ringed spaces
Let $X$ be a topological space and let $\mathcal{O}_ X$ be a sheaf of rings on $X$. We are supposed to think of the sheaf of rings $\mathcal{O}_ X$ as a sheaf of functions on $X$. And if $f : X \to Y$ is a “suitable” map, then by composition a function on $Y$ turns into a function on $X$. Thus there should be a natural $f$-map from $\mathcal{O}_ Y$ to $\mathcal{O}_ X$, see Definition 6.21.7 and Lemma 6.21.8. For a precise example, see Example 6.25.2 below. Here is the relevant abstract definition.
Example 6.25.2. Let $f : X \to Y$ be a continuous map of topological spaces. Consider the sheaves of continuous real valued functions $\mathcal{C}^0_ X$ on $X$ and $\mathcal{C}^0_ Y$ on $Y$, see Example 6.9.3. We claim that there is a natural $f$-map $f^\sharp : \mathcal{C}^0_ Y \to \mathcal{C}^0_ X$ associated to $f$. Namely, we simply define it by the rule Strictly speaking we should write $f^\sharp (h) = h \circ f|_{f^{-1}(V)}$. It is clear that this is a family of maps as in Definition 6.21.7 and compatible with the $\mathbf{R}$-algebra structures. Hence it is an $f$-map of sheaves of $\mathbf{R}$-algebras, see Lemma 6.23.1. Of course there are lots of other situations where there is a canonical morphism of ringed spaces associated to a geometrical type of morphism. For example, if $M$, $N$ are $\mathcal{C}^\infty $-manifolds and $f : M \to N$ is a infinitely differentiable map, then $f$ induces a canonical morphism of ringed spaces $(M, \mathcal{C}_ M^\infty ) \to (N, \mathcal{C}^\infty _ N)$. The construction (which is identical to the above) is left to the reader.
\begin{eqnarray*} \mathcal{C}^0_ Y(V) & \longrightarrow & \mathcal{C}^0_ X(f^{-1}V) \\ h & \longmapsto & h \circ f \end{eqnarray*}
It may not be completely obvious how to compose morphisms of ringed spaces hence we spell it out here.
Definition 6.25.3. Let $(f, f^\sharp ) : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ and $(g, g^\sharp ) : (Y, \mathcal{O}_ Y) \to (Z, \mathcal{O}_ Z)$ be morphisms of ringed spaces. Then we define the composition of morphisms of ringed spaces by the rule Here we use composition of $f$-maps defined in Definition 6.21.9.
\[ (g, g^\sharp ) \circ (f, f^\sharp ) = (g \circ f, f^\sharp \circ g^\sharp ). \]
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