Associated primes of the symbolic power and ordinary power

Question

I am struggling to understand the following quote from a paper of Arsie and Vatne "A note on symbolic powers and ordinary powers of homogeneous ideals":

Our interest in the symbolic power stems from the fact that taking the symbolic power of a homogeneous ideal gives the saturated ideal for a scheme which is the scheme defined by the ordinary power, having subtracted the embedded components. Thus, in some special cases, via the symbolic power we can obtain the saturation of the ordinary power.

If $R$ is a commutative Noetherian ring, then the $m$th symbolic power is defined to be $I^{(m)} = R \cap \left(\bigcap_{P \in \text{Min}_R (R/I)} I^mR_P \right) \subseteq R_{(0)}$.

I interpret "having subtracted the embedded components" to mean that the embedded primes of $I^m$ do not contain $I^{(m)}$.

1. Is this interpretation correct and why is it true that the embedded primes of $I^m$ do not contain $I^{(m)}$? I am sure it is obvious from the definition so forgive me.
2. Are the only associated primes of $I^{(m)}$ the minimal primes of $I^m$? Thanks.

Answer 1

1. It suffices to consider when $I$ is a prime ideal. More concretely, let $R = k[x,y,z]/(xy-z^2)$ and $I = (x,z)$. Then $I^{(2)} = R \cap I^2R_I$. The primary decomposition of $I^2 = I^{(2)} \cap Q$ where $\sqrt{Q} = m = (x,y,z)R$. Hence $Q$ is the embedded component. When they say an embedded components (not embedded prime), I believe that they refer to $Q$ not $m$. Of course, $Q$ cannot contain $I^{(2)}$ when you have a "irredundant" primary decomposition. However, since an embedded component always contains a minimal component. Hence its radical contains the symbolic power.

2. This is true: if $I \subseteq p$ is minimal, then $\sqrt{IR_p} = pR_p$. Hence $IR_p$ is $pR_p$-primary. A contraction of a primary ideal is primary. Therefore, the intersection on the RHS of $I^{(m)}$ is a primary decomposition of $I^{(m)}$. This gives the set of associate primes of $I^{(m)}$.

As far as I understand the definition of symbolic powers of non-primary ideal is not universal. By the way, I do not understand what you mean by $R_{(0)}$.