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

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 $$X$$?

• I gave a more complete reference and better links :-) – David Roberts Dec 18 '18 at 6:17

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.