inverse image of a sheaf

DEFINITION 8.1   Let $f:X\to Y$ be a continuous map between topological spaces. Let $\mathcal F$ be a sheaf on $Y$. Then the inverse image $f ^{-1} \mathcal F$ of $\mathcal F$ by $f$ is the sheafication of a presheaf $\mathcal G$ defined by

$$\mathcal G(U)=\varinjlim_{V\supset f(U)} \mathcal F(V).$$

LEMMA 8.2   Let $f:X\to Y$ be a continuous map between topological spaces. Let $\mathcal F$ be a sheaf on $Y$. Then we have a natural isomorphism

$$f^{-1}(\mathcal F)_x \cong \mathcal F_{f(x)}$$

for each point $x \in X$.

PROOF.. Let $\mathcal G$ be the presheaf defined as in the previous Definition. Since sheafication does not affect stalks, we have a natural isomorphism

$$f^{-1}(\mathcal F)_x \cong \mathcal G_{x}$$

On the other hand, we have

$$\mathcal G_x = \varinjlim_{U \ni x} \mathcal G(W) = \varinjlim_{U \ni x} \left( \varinjlim_{V\supset f(U)}\mathcal F(V) \right)$$

Then since the map $f$ is continuous, the injective limit at the right hand side may be replaced by

$$\varinjlim_{V \ni f(x)}\mathcal F(V)=\mathcal F_{f(x)}$$

$\qedsymbol$

DEFINITION 8.3   A ringed space $(X,\mathcal{O}_X)$ is a topological space $X$ with a sheaf of rings $\mathcal{O}_X$ on it. A locally ringed space is a ringed space whose stalks are local rings.

DEFINITION 8.4   Let $(X,\mathcal{O}_X)$ $(Y,\mathcal{O}_Y)$ be ringed spaces.

1. A morphism $(f,f^\char93 ):X \to Y$ as ringed spaces is a continuous map $f:X\to Y$ together with a sheaf homomorphism
   $$f^\char93 : f^{-1}(\mathcal{O}_Y) \to \mathcal{O}_X.$$
   (Note that $f^\char93$ gives a ring homomorphism
   $$f^\char93 _x: O_{Y,f(x)}\to O_{X,x}$$
   for each point $x \in X$. We call it an “associated homomorphism”.)
2. Let us further assume that $X,Y$ are locally ringed space. Then a morphism $(f,f^\char93 )$ of ringed spaces is said to be a morphism of locally ringed spaces if the associated homomorphism $f^\char93 _x$ is a local homomorphism for each point $x \in X$.

It goes without saying that when $X$ is a (locally) ringed space, then its open set $U$ also carries a structure of (locally) ringed space in a natural way, and that the inclusion map $U\to X$ is a morphism of (locally) ringed space.