  The hot electrons in the intra-cluster medium produce a spectral distorsion
of the cosmic microwave background (CMB) black body emission, the thermal
Sunyaev-Zel'dovich effect (tSZ). This characteristic spectral distorsion is now
commonly used to detect and characterize the properties of galaxy clusters. The
tSZ effect spectral distorsion does not depends on the redshift, and is only
slightly affected by the galaxy cluster properties via the relativistic
corrections, when the electrons reach relativistic velocities. The present work
proposes a linear component separation approach to extract the tSZ effect
Compton parameter and relativistic corrections for next-generation CMB
experiments. We demonstrated that relativistic corrections, if neglected, would
induce a significant bias on galaxy cluster Compton parameter, tSZ scaling
relation slope, and tSZ angular power spectrum shape measurements. We showed
that tSZ relativistic corrections mapping can be achieved at high
signal-to-noise ratio with a low level of contamination up to $\ell=3000$ for
next generation CMB experiments. At smaller angular scales the contamination
produced by infra-red emission will be a significant source of bias. Such tSZ
relativistic corrections mapping enables the study of galaxy cluster
temperature profile via the tSZ effect only.
