Are Saturated Ideals Radical?

Question

I am reading the section 5.5 of Gathmann's note. In the note, saturated ideal is defined as follows:

For a homogeneous ideal $I \subseteq S = k[x_0, x_1, \dots, x_n]$, the saturation ideal of $I$ is defined by

$\overline{I} = \{\, s \in S : \forall i \, \exists m > 0, \ x_i^ms \in I \,\}.$

In the Gathmann's note Lemma 5.5.9, it is stated that for homogeneous ideals $I, J \subseteq S$

1. $\overline{I}$ is homogeneous

2. $V_+(I) = V_+(\overline{I})$

3. $V_+(\overline{I}) = V_+(\overline{J}) \Leftrightarrow \overline{I} = \overline{J}$

From these I concluded that for any homogeneous ideal $I \subseteq S_+$

$V_+(\overline{ \sqrt{I} }) = V_+(\sqrt{I}) = V_+(I) = V_+(\overline{I})$, hence $\overline{\sqrt{I}} = \overline{I}.$

And there for letting $J = \overline{I}$ we have

$J = \overline{J} = \overline{\sqrt{J}} \supseteq \sqrt{J} \supseteq J$ and $J = \sqrt{J}.$

However, I have a counterexample where $S = k[x_0, x_1]$ and

$I = (x_0^2), \ \sqrt{I} = (x_0), \ J = \overline{I} = (x_0^2), \ \sqrt{J} = (x_0)$

so $J \neq \sqrt{J}.$

Where did I go wrong in these arguments?

Answer 1

As I understand it, different homogeneous ideals can give rise to the same closed subscheme, and for a homogeneous ideal $I$, the largest ideal yielding the same closed subscheme as $I$ is its saturation $\bar I$. So Lemma 5.5.9 (3) is really about subschemes.

Thus there is a bijection between saturated homogeneous ideals in $S$ and closed subschemes of $\mathbb P^n$.

On the other hand, different closed subschemes can have the same closed subset, and the radical of $I$ is the largest homogeneous ideal yielding the same closed subset as $I$.

Thus there is a bijection between the homogeneous radical ideals of $S$ and the closed subsets of $\mathbb P^n$.

This agrees with the easy implication that radical implies saturated for homogeneous ideals.

Note this plays a role in the Hilbert scheme, which parameterises closed subschemes of projective space (having the same Hilbert polynomial). One way of constructing this is by parameterising the corresponding homogeneous saturated ideals, which can be done using Grassmannians on the subspace of monomials of some suitable degree.