Is the annihilator of a minimal prime ideal principal?

Question

My setup is as follows: $X$ is a projective, reduced curve (which is not integral) with a finite morphism onto $\mathbb{P}_k^1$. $\DeclareMathOperator{\Ann}{Ann}$ Let $R$ be a coordinate ring of $X$ which is finite free over $k[x]$ (since $X$ has more than one irreducible component, $R$ has at least two minimal prime ideals). Let $P$ be a minimal prime of $R$ corresponding to an irreducible component of $X$.

Do we always have that $\Ann(P)$ is pincipal?

What I tried:

• Every example I was able to come up satisfied the above property. Hence I did not find any counter-example.

• There is some $b \in R$ such that $P = \Ann(b)$ and hence $b \in \Ann(\Ann(b))$. Thus a necessary condition is that such a generator $a \in R$ of $\Ann(P)$ must divide every $b \in R$ that satisfies $\Ann(b) = P$. Since $\Ann(b) = \Ann(bf)$ for all $f \in k[x]$ (since $R$ is torsion-free over $k[x]$) we may assume that no element $f \in k[x]$ divides $b$ in $R$. That's where I am stuck at.

I am grateful for any kind of help, counter-example or hints.

Answer 1

This is false. To see why, consider the following lemma.

Lemma. Let $R$ be a Noetherian ring with exactly two minimal primes $\mathfrak p$ and $\mathfrak q$ such that $\mathfrak p \mathfrak q = 0$. Then $\operatorname{Ann}(\mathfrak p) = \mathfrak q$.

The assumption is in particular satisfied if $\mathfrak p \cap \mathfrak q = 0$, which is equivalent to $R$ being reduced.

Proof. Clearly $\mathfrak q \subseteq \operatorname{Ann}(\mathfrak p)$, since $\mathfrak p \mathfrak q = 0$. Since $\mathfrak p \not\subseteq \mathfrak q$, we have $\mathfrak p_{\mathfrak q} = R_{\mathfrak q}$, so any element killing $\mathfrak p$ better be in $\mathfrak q$. (See also Tag 00L2.) $\square$

Thus, it suffices to construct such a ring $R$ where $\mathfrak q$ is not principal. Basically anything you write down will work.

Example. Let $E$ be an elliptic curve over an algebraically closed field $k$, and let $p \in E$ be a closed point. Glue two copies of $E$ at $p$, i.e. consider the union $X = (E \times p) \cup (p \times E) \subseteq E \times E$. This admits a finite flat map to $\mathbb P^1$ given by $E \times E \to E \to \mathbb P^1$ where the first map is $(x,y) \mapsto x+y$ and the second is any nonconstant map.

Removing any other point $q$ gives an affine open $((E \setminus q) \times p) \cup (p \times (E \setminus q)) \subseteq X$, and on its coordinate ring $R$ we have two minimal prime ideals $\mathfrak p$ and $\mathfrak q$ corresponding to the components $(E \setminus q) \times p$ and $p \times (E \setminus q)$ respectively.

If $\mathfrak q$ were principal, then the same is true for its restriction to $R/\mathfrak p = \Gamma(E\setminus q,\mathcal O)$. But the map \begin{align} \operatorname{Pic}(E \setminus q) &\to E(k) = \operatorname{Pic}^0(E)\\ r &\mapsto r-q \end{align} is an isomorphism, and if $p \in E \setminus q$ were principal this implies that $p - q = 0$, which is absurd. $\square$