Definition of relative Cartier divisors

Question

I'm a bit confused about the definition of relative Cartier divisors which I found in Hida's book "Geometric Modular Forms". In Chapter $2$, section $2.1$, he says that, given a locally Noetherian Scheme $S$ and a proper flat reduced curve $C$ over $S$, a relative Cartier divisor is a closed subscheme $D\subseteq X$ such that $D$ is flat over $S$ and the ideal sheaf of $D$, say $I(D)$ is invertible over $C$. Then, at the end of the section he gives the following, apparently trivial, example. He considers $D_N=\text{Spec}(\mathbb{Z}/N\mathbb{Z})\subseteq\text{Spec}(\mathbb{Z})$ where $N>1$ is an integer, and he says it is a Cartier divisor. I assume he uses $\text{Spec}(\mathbb{Z})$ both as the basis $S$ and as the curve $C$. By the way, since $\mathbb{Z}$ is a PID, flatness coincides with torsion freenes, and clearly $\mathbb{Z}/N\mathbb{Z}$ is not torsion free as a $\mathbb{Z}$-module, since $N$ is nonzero in $\mathbb{Z}$ and it annihilates every element in $\mathbb{Z}/N\mathbb{Z}$. Am I misunderstanding something? What is the mistake in all this picture? Thank you very much for any suggestion!

Answer 1

Yes, $\DeclareMathOperator{\Spec}{Spec}\newcommand{\Z}{\mathbb{Z}}\Spec(\Z/N\Z)\hookrightarrow\Spec(\Z)$ is not a relative effective Cartier divisor over $\Spec(\Z)$ because it is not flat over $\Spec(\Z)$. Here's another way to see this. Relative effective divisors are stable under base change by [Tag 056P]. Hence, given a relative effective Cartier divisor $D\to S$, each fiber $D_s$ has to be a relative effective Cartier divisor over $\Spec(\kappa(s))$. This is not the case here.

However, $\Spec(\Z/N\Z)$ is still an effective Cartier divisor on $\Spec(\Z)$ in the absolute sense. This just means that it is a closed subscheme of $\Spec(\Z)$ whose ideal sheaf is invertible. See [Tag 01WQ] for more information. This is most likely what the author meant.