sheaves

Affine spectrum $\operatorname{Spec}(A)$ of a ring $A$ carries one more important structure. Namely, its structure sheaf.

We will firstly review some definitions and first properties of sheaves.

To illustrate the idea, we recall an easy lemma in topology.

LEMMA 07.17 (Gluing lemma)   Let $X,Y$ be a topological spaces. Let $\{U_\lambda\}_{\lambda \in \Lambda}$ be an open covering of $X$.

1. If we are given a collection of continuous maps $\{f_\lambda: U_\lambda \to Y\}_{\lambda\in \Lambda}$ such that
   $$f_\lambda\vert _{U_{\lambda}\cap U_{\mu}} = f_\mu\vert _{U_{\lambda}\cap U_{\mu}}$$
   holds for any pair $(\lambda,\mu)\in \Lambda^2$, then we have a unique continuous map $f:X\to Y$ such that
   $$f\vert _{U_\lambda}=f_\lambda$$
   holds for any $\lambda \in \Lambda$.

2. Conversely, if we are given a continuous map $f:X\to Y$, then we obtain a collection of maps $\{f_\lambda: U_\lambda \to Y\}_{\lambda\in \Lambda}$ by restriction.

PROOF.. (1) It is easy to verify that we have a well-defined map

$$f: X\to Y$$

with

$$f\vert _{U_\lambda}=f_\lambda.$$

The continuity of $f$ is proved by verifying that the inverse image of any open set $V\subset Y$ by $f$ is open in $X$. $\qedsymbol$