  We perform the analytical next-to-leading order calculation of the process
$p+A\to \gamma^{*}+\mathrm{jet}+X$, at forward rapidities and low $x$. These
kinematics justify a hybrid approach, where a quark from the \textquoteleft
projectile' proton scatters off the gluon distribution of the \textquoteleft
target', which can be a nucleus or a highly boosted proton. By using the Color
Glass Condensate effective theory approach, this gluon distribution is allowed
to be so dense that the quark undergoes multiple scattering. Moreover, large
high-energy logarithms in the ratio of the hard scale and the center-of-mass
energy are resummed by the Balitsky, Kovchegov, Jalilian-Marian, Iancu,
McLerran, Weigert, Leonidov, Kovner or BK-JIMWLK evolution equations. We
demonstrate that all ultraviolet divergences encountered in the calculation
cancel, while the high-energy divergences are absorbed into BK-JIMWLK. The
remaining singularities are collinear in nature and can be either absorbed into
the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi evolution of the incoming quark,
when they stem from initial-state radiation, or else can be treated by
introduction of a jet function in case they are caused by final-state
emissions. The resulting cross section is completely finite and expressed in
function of only a small set of color operators.
