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$ 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

$\displaystyle \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

$\displaystyle \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

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

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

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

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

$\displaystyle \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.


next up previous
Next: locally free sheaves of Up: Topics in Non commutative Previous: quasi coherent sheaves
2007-12-11