Zariski topology
Lagt inn: 28/11-2016 15:17
In this exercise we let $A = \mathbb{Z}$ and consider $Spec{\mathbb{Z}}$.
Let $I = (12) ⊆ \mathbb{Z}$. What is $X = \nu (I) ⊆ Spec \mathbb{Z}$?
What is $I(X) ⊆\mathbb{Z}$ ? What is $rad(I)$?
Is $X$ an irreducible closed subset of $Spec \mathbb{Z}$?
$X = \nu (I) = \{P \in Spec\mathbb{Z} | P \supset I\} = \{ (2), (3) \}$
[tex]I(X) = \bigcap _{P \in X} P \implies I(X) = (2) \cap (3) = (6)[/tex]
$12 = 2^2 * 3$
$rad I = (2)*(3) = (6) \implies rad I = I(X)$
$\implies X \ irreducible$
Litt usikker på det siste punktet, som sier X irreducible. Trengs det flere betingelser eller er dette nokk?
Let $I = (12) ⊆ \mathbb{Z}$. What is $X = \nu (I) ⊆ Spec \mathbb{Z}$?
What is $I(X) ⊆\mathbb{Z}$ ? What is $rad(I)$?
Is $X$ an irreducible closed subset of $Spec \mathbb{Z}$?
$X = \nu (I) = \{P \in Spec\mathbb{Z} | P \supset I\} = \{ (2), (3) \}$
[tex]I(X) = \bigcap _{P \in X} P \implies I(X) = (2) \cap (3) = (6)[/tex]
$12 = 2^2 * 3$
$rad I = (2)*(3) = (6) \implies rad I = I(X)$
$\implies X \ irreducible$
Litt usikker på det siste punktet, som sier X irreducible. Trengs det flere betingelser eller er dette nokk?