  Recent ALMA observations suggest that the highest velocity part of molecular
protostellar jets are launched from the dust-sublimation regions of the
accretion disks (<0.3 au). However, formation and survival of molecules in
inner protostellar disk winds, in the presence of a harsh FUV radiation field
and the absence of dust, remain unexplored. We aim at determining if simple
molecules can be synthesized and spared in fast and collimated dust-free disk
winds or if a fraction of dust is necessary to explain the observed molecular
abundances. This work is based on the Paris-Durham shock code designed to model
irradiated environments. Fundamental properties of the dust-free chemistry are
investigated from single point models. A laminar 1D disk wind model is then
built using a parametric flow geometry. This model includes time-dependent
chemistry and the attenuation of the radiation field by gas-phase
photoprocesses. We show that a small fraction of H2 (< 1e-2), primarily formed
through the H- route, can efficiently initiate molecule synthesis such as CO
and SiO above TK ~ 800 K. The attenuation of the radiation field by atomic
species (eg. C, Si, S) proceeds through continuum self-shielding. This process
ensures efficient formation of CO, OH, SiO, H2O through neutral-neutral
reactions, and the survival of these molecules. Class 0 dust-free winds with
high mass-loss rates ($\dot{M}_w >$ 2e-6 Msun/yr) are predicted to be rich in
molecules if warm (TK > 800 K). The molecular content of disk winds is very
sensitive to the presence of dust and a mass-fraction of surviving dust as
small as 1e-5 significantly increases the H2O and SiO abundances. Chemistry of
high-velocity jets is a powerful tool to probe their content in dust and
uncover their launching point. Models of internal shocks are required to fully
exploit the current (sub-)millimeter observations and prepare future JWST
observations.
