We report on Hubble Space Telescope ( HST ) observations of the late-time afterglow and host galaxy of GRB 021004 ( z = 2.33 ) . Although this gamma-ray burst ( GRB ) is one of the best observed so far in terms of sampling in the time domain , multi-wavelength coverage and polarimetric observations , there is substantial disagreement between different interpretations of data sets on this burst in the literature . We have observed the field of GRB 021004 with the HST at multiple epochs from 3 days until almost 10 months after the burst . With STIS prism and G430L spectroscopy we cover the spectral region from about 2000 Å to 5700 Å corresponding to 600–1700 Å in the restframe . From the limit on the flux recovery bluewards of the Lyman-limit we constrain the H i column density to be above 1 \times 10 ^ { 18 } cm ^ { -2 } ( 5 \sigma ) . Based on ACS and NICMOS imaging we find that the afterglow evolved achromatically within the errors ( any variation must be less then 5 % ) during the period of HST observations . The color changes observed by other authors during the first four days must be related to a stochastic phenomenon superimposed on an afterglow component with a constant spectral shape . This achromaticity implies that the cooling break has remained on the blue side of the optical part of the spectrum for at least two weeks after the explosion . The optical–to–X-ray slope \beta _ { \mathrm { OX } } is consistent with being the same at 1.4 and 52.4 days after the burst . This indicates that the cooling frequency is constant and hence , according to fireball models , that the circumburst medium has a constant density profile . The late-time slope of the lightcurve ( \alpha _ { 2 } , F _ { \nu } \propto t ^ { - \alpha _ { 2 } } ) is in the range \alpha _ { 2 } = 1.8 – 1.9 , and is inconsistent with a single power-law . This could be due to a late-time flattening caused by the transition to non-relativistic expansion or due to excess emission ( a ‘ bump ’ in the lightcurve ) about 7 days after burst . The host galaxy is like most previously studied GRB hosts a ( very ) blue starburst galaxy with no evidence for dust and with strong Ly \alpha emission . The star-formation rate of the host is about 10 M _ { \sun } yr ^ { -1 } based on both the strength of the UV continuum and on the Ly \alpha luminosity . The spectral energy distribution of the host implies an age in the range 30–100 Myr for the dominant stellar population . The afterglow was located very close ( \sim 100 pc ) to the center of the host implying that the progenitor was possibly associated with a circumnuclear starburst .