  Motivated by recent experimental LHC measurements on the forward inclusive
jet productions and based on our previous calculations on forward hadron
productions, we calculate single inclusive jet cross-section in $pA$ collisions
at forward rapidity within the color glass condensate framework up to the
next-to-leading-order. Moreover, with the application of jet algorithm and
proper subtraction of the rapidity and collinear divergences, we further
demonstrate that the resulting next-to-leading-order hard coefficients are
finite. In addition, in order to deal with the large logarithms that can
potentially spoil the convergence of the perturbative expansion and improve the
reliability of the numerical predictions, we introduce the collinear jet
function and the threshold jet function and resum these large logarithms hidden
in the hard coefficients.
