The gamma-construction and permanence properties of the (relative) $F$-rational signature

We study some permanence properties of the relative $F$-rational signature defined and studied by Smirnov–Tucker. We show that this invariant is compatible with the gamma-construction, and then derive other main results from the $F$-finite case established by Smirnov–Tucker. We also obtain limited results about the $F$-rational signature defined and studied by Hochster–Yao. We explore some features of the gamma-construction along the way, which may be of independent interest.