direct image of a sheaf

DEFINITION 9.8   Let $X,Y$ be topological spaces. Let $f:X\to Y$ be a continuous map. Let $\mathcal F$ be a sheaf on $X$. Then we define its direct image with respect to $f$ by

$$f_*(\mathcal F)(U)=\mathcal F(f^{-1}(U))$$

with obvious restriction maps.

PROPOSITION 9.9   Let $X,Y$ be topological spaces. Let $f:X\to Y$ be a continuous map. Let $\mathcal F$ be a sheaf on $X$. Let $\mathcal G$ be a sheaf on $Y$. Then we have a natural isomorphism.

$$\operatorname{Hom}(\mathcal G,f_*\mathcal F)\cong \operatorname{Hom}(f^{-1}\mathcal G,\mathcal F)$$

PROOF.. We first define an adjoint map

$$\iota: f^{-1} f_*\mathcal F \to \mathcal F$$

and construct the isomorphism using it.

$\qedsymbol$

PROPOSITION 9.10   Let $X,Y$ be (locally) ringed spaces. Let $f:X\to Y$ be a morphism of (locally) ringed spaces. Let $\mathcal F$ be a sheaf of $\mathcal{O}_X$-modules. Let $\mathcal G$ be a sheaf on $\mathcal{O}_Y$-modules. Then we have a natural isomorphism of modules.

$$\operatorname{Hom}_{\mathcal{O}_Y}(\mathcal G,f_*\mathcal F) \cong \operatorname{Hom}_{\mathcal{O}_X}(f^*\mathcal G,\mathcal F)$$

PROOF..

$$\operatorname{Hom}_{\mathcal{O}_Y} (\mathcal G,f_*\mathcal F)\cong \operatorname{Hom}_{f^{-1}\mathcal{O}_Y}(f^{-1}\mathcal G,\mathcal F)$$

$$\cong \operatorname{Hom}_{\mathcal{O}_X} (\mathcal{O}_X \otimes_{... ...,\mathcal F) \cong \operatorname{Hom}_{\mathcal{O}_X}(f^*\mathcal G,\mathcal F)$$

$\qedsymbol$

EXAMPLE 9.11   Let $A,B$ be rings. Let $\varphi:A \to B$ be a ring homomorphism. We put $f=\operatorname{Spec}(\varphi)$ be the continuous map $Y=\operatorname{Spec}(B)\to \operatorname{Spec}(A)=X$ corresponding to $\varphi$. We note that $B$ carries an $A$-module structure via $\varphi$. Accordingly, we have the corresponding sheaf $\mathcal{O}_X \otimes_A B$ on $X$. We may easily see that this sheaf coincides with $f_*\mathcal{O}_Y$. The map $\varphi:A \to B$ then may also be regarded as a homomorphism of $A$-modules. We have thus an $\mathcal{O}_X$ module homomorphism

$$\varphi_\char93 : \mathcal{O}_X \to f_*\mathcal{O}_Y$$

of sheaves on $X$. By the adjoint relation (Proposition 9.9), we obtain a sheaf homomorphism

$$\operatorname{Spec}(\varphi)^\char93 : f^\char93 \mathcal{O}_X \to \mathcal{O}_Y.$$

of sheaves of rings.