Definition of a vertical ideal sheaf and a vertical fractional ideal sheaf

Question

I'm working through the paper Singular semipositive metrics in non-Archimedean geometry by Sebastien Boucksom, Charles Favre and Mattias Jonsson (J. Algebraic Geom. 25 (2016), 77-139, doi:10.1090/jag/656, arXiv:1201.0187).

In Section 1, subsection 1.1, they define vertical ideal sheaves and vertical fractional ideal sheaves. Here are my questions: Let $\mathfrak{a}$ be an ideal sheaf on $X$, by saying $\mathfrak{a}$ is co-supported on the special fiber $X_0$ of $X$, do they mean $\text{supp} \, (\mathcal{O}_X / \mathfrak{a}) \subseteq X_0 $?

What is the definition of a fractional ideal sheaf? I only found one definition stating it is a coherent subsheaf of the sheaf of total quotient rings $\mathscr{K}_X$. If i define a fractional ideal sheaf this way and i know my Scheme $X$ is integral, then $\mathscr{K}_X(U) = \mathcal{O}_{X,\eta}$ is the function field for all $U$ open in $X$, so $\mathfrak{a}(U)$ is just a subring of the function field for all $U$ open in $X$?

Answer 1

In general, a fractional ideal sheaf on an integral scheme $X$ is a coherent $\mathcal{O}_X$-submodule $\mathfrak{a}$ of the function field of $X$, thought of as a constant sheaf. In particular, for every open $U \subseteq X$, $\mathfrak{a}(U)$ is an $\mathcal{O}_X(U)$-submodule of the function field, not necessarily a subring.

Now in the setting of the paper of Boucksom-Favre-Jonsson, let $R$ be a dvr with uniformizer $\varpi$ and we'll work with a flat, integral $R$-scheme $\mathscr{X}$ of finite type. If $\mathfrak{a} \subseteq \mathcal{O}_{\mathscr{X}}$ is a coherent sheaf of ideals on $\mathscr{X}$, then the condition that "$\mathfrak{a}$ be cosupported on the special fibre" is precisely saying that the support of the quotient sheaf $\mathcal{O}_{\mathscr{X}}/\mathfrak{a}$ lies in the special fibre $\mathscr{X}_0 = V(\varpi)$.

In this setting, the vertical fractional ideals are the $\mathcal{O}_{\mathscr{X}}$-submodules of the function field of $\mathscr{X}$ that become actual vertical ideals after multiplying by a sufficiently-high power of $\varpi$.

A good example to have in mind is the following: take the $R$-scheme $$ \mathscr{X} = \mathrm{Spec}\left( \mathcal{A} = \frac{R[T_1,T_2]}{(T_1T_2-\varpi)}\right). $$ The generic fibre of $\mathscr{X}$ is a torus over the fraction field of $R$, while the special fibre consists of two lines meeting in a node. For example, the ideal $\mathfrak{a} = (T_1,T_2)$ is vertical: it cuts out the node of the special fibre. However, if you take the $\mathcal{A}$-submodule of $\mathrm{Frac}(\mathcal{A})$ generated by $T_1^{-1}$, then this is a vertical fractional ideal sheaf: after multiplying by $\varpi$, you get the ideal $(T_2) \subseteq \mathcal{A}$, which cuts out one of the components of the special fibre.