  The nature of the ordering of the one-dimensional Heisenberg spin-glass model
with a long-range power-law interaction is studied by extensive Monte Carlo
simulations, with particular attention to the issue of the spin-chirality
decoupling/coupling. Large system sizes up to $L=4096$ are studied. With
varying the exponent $\sigma$ describing the power-law interaction, we observe
three distinct types of ordering regimes. For smaller $\sigma$, the spin and
the chirality order at a common finite temperature with a common
correlation-length exponent, exhibiting the standard spin-chirality coupling
behavior. For intermediate $\sigma$, the chirality orders at a temperature
higher than the spin, exhibiting the spin-chirality decoupling behavior. For
larger $\sigma$, both the spin and the chirality order at zero temperature. We
construct a phase diagram in the $\sigma$ versus the temperature plane, and
discuss implications of the results. Critical properties associated with both
the chiral-glass and the spin-glass transitions are also determined.
