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The following is definition of them, from Eisenbud, Commutative algebra with a view toward algebraic geometry.

  1. $M$ be any module over ring $R$ and $P$ be a minimal prime ideal over the $\mathrm{ann}(M)$. Then the submodule $M'$ of $M$ defined by $$M' := \ker(M \longrightarrow M_P)$$ is called the $P$-primary component of $0$ in $M$.

  2. Let $P$ be a prime ideal in $R$. Then $P$-primary component of the $n$th power of $P$ is called the $n$th symbolic power of $P$, written by $P^{(n)}$

But I can't understand how two definitions are compatible, since applying 1, $P^{(n)}=\ker(P^n\longrightarrow P^nR_P)$, so it must be a submodule of $P^n$, but author says it contains $P^n$. I think there is suitable definition over 1, saying primary component of $0\neq M'$ in $M$, but I can't find the definition in google. Can anyone help me?

user26857
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1 Answers1

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The first definition is of the $P$-primary component of $0$.

The second definition uses of the $P$-primary component of the non-zero ideal $P^n$.

So how do we square these? (At least if $R$ is a Noetherian ring). If we have a module $M$ and a submodule $N$, then the $P$-primary component of $0$ in the module $M/N$ is some module of the form $N'/N$. (At least if $P$ is a minimal prime of the annihilator of $M/N$.) Then $N'$ is the $P$-primary component of $N$ in $M$.

So, for $R$ Noetherian, $P$ is a minimal prime of the annihilator of $R/P^n$, and the kernel of $(R/P^n)\to(R/P^n)_P$ is $Q/P^n$ where $Q$ is an ideal of $R$. Then $P^{(n)}=Q$.

Angina Seng
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