Proposition:

Let $A$ be a ring. We have the following properties:

1. For any pair of ideals $I,J$ of $A$, we have $V(I)\cup V(J)=V(J\cap J)$,
2. Let $(I_{\alpha})_{\alpha}$ be any family of ideals of $A$. Then $\cap_{\alpha}V(I_{\alpha})=V(\sum_{\alpha}I_{\alpha})$, and
3. $V(A)=\emptyset$ and $V(0)=Spec(A)$.

In particular, there exists a unique topology on $Spec(A)$ whose closed subsets are the sets of the form $V(I)$ for an ideal of $A$. moreover, the sets of the form $D(f)$ , $f\in A$, constitute a base of open subsets on $Spec(A)$.

Lemma:

Let $A$ be a ring. Let $I,J$ be two ideals of $A$. The following propositions are true.

1. $\sqrt I=\cap_{P\in V(I)}P$, (i.e the radical of $I$ equals the intersection of the ideals $P\in V(I)$ )
2. We have $V(I)\subseteq V(J)$ if and only if $J\subseteq \sqrt I$.

Lemma:

Let $\varphi$ be a rings homomorphism. Let $f=Spec\varphi$ be the map associated to $\varphi$ as above. Then we have:

1. The map $f$ is continuous,
2. If $\varphi$ is surjective, then $f$ induces a homeomorphism from $Spec(B)$ onto the closed subset $V(\ker\varphi)$ of $Spec(A)$,
3. If $\varphi$ is a localization morphism $A\to S^{-1}S$, then $f$ is a homeomorphism from $Spec(S^{-1}A)$ onto the subspace $\{P\in Spec(A) \ , \ P\cap S=\emptyset\}$ of $Spec(A)$.

Definition:

Let $X$ be a topological space. A presheave $\mathcal F$ (of abelian groups) on $X$ consists of the following data:

• an Abelian group $\mathcal F(U)$ for every open subset $U$of $X$, and
• a groups homomorphism (restriction map) $\rho_{UV}:\mathcal F(U)\to \mathcal F(V)$ for every pair of open subsets $V\subseteq U$.
  Which verify the following conditions:
  1. $\mathcal F(\emptyset)=0$;
  2. $\rho_{UU}=Id$;
  3. If we have three open subsets $W\subseteq V\subseteq U$, then $\rho_{UW}=\rho_{VW}\circ\rho_{UV}$.