$K$ -valued points and fibers

DEFINITION 2.1   Let $K$ be a field. a $K$ -valued point $P$ of a scheme $X$ is a morphism

$$\iota_P:\operatorname{Spec}(K)\to X$$

of schemes. Let $\mathcal{F}$ be a quasi coherent $\mathcal{O}_X$ -module. Then a fiber of $\mathcal{F}$ on a $K$ -valued point $P$ is the pullback $\iota_P^*(\mathcal{F})$ . We often identify it with a $K$ -vector space

$$\iota_P^*(\mathcal{F})(K).$$

EXAMPLE 2.2   Let $X=\operatorname{Spec}(A)$ be an affine scheme. Then a $K$ -valued point $P$ of $X$ is given by a ring homomorphism

$$\operatorname{eval}_P:A\to K.$$

a quasi coherent $\mathcal{O}_X$ -module $\mathcal{F}$ is given by an $A$ -module $M$ .

$$\mathcal{F}\cong \mathcal{O}_X \otimes_A M$$

The fiber of $\mathcal{F}$ is then given by

$$\iota_P^*(\mathcal{O}_X \otimes_ A M)=K\otimes_A M.$$

In general, we study quasi coherent sheaves on a scheme from three different point of view. Namely, we may study them

1. section wise,
2. stalk wise, or
3. fiber wise.

Each view point is useful.