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 $X$?