  Suppose that $X$ is an integral scheme (quasi-)projective over a complete
local ring of mixed characteristic. Using ideas of Takamatsu-Yoshikawa and
Bhatt-Ma-et. al, we define a notion of a $+$-test ideal on $X$, including for
divisors and linear series. We obtain global generation results in this setting
that generalize the well known global generation results obtained via
multiplier ideal sheaf techniques in characteristic $0$ and via test ideals in
characteristic $p>0$. We also obtain applications to the order of vanishing of
linear series and to the diminished base locus in mixed characteristic similar
to results of Ein-Lazarsfeld-Mustata-Nakamaye-Popa, Nakayama, and Mustata in
the equal characteristic case.
