localization

DEFINITION 2.5   Let $f$ be an element of a commutative ring $A$. Then we define the localization $A_f$ of $A$ with respect to $f$ as a ring defined by

$$A_{f}=A[Y]/(Y f -1)$$

where $Y$ is a indeterminate.

LEMMA 2.6   When $K$ is a field, then we have a canonical identification

$$\operatorname{Spec}(A_f)(K)=\{P\in \operatorname{Spec}(A)(K); \operatorname{eval}_P(f)\neq 0\}.$$