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Recently, it has been suggested that the CO line width (FWHM(CO)) is asurrogate for the bulge velocity dispersion ($\sigma$) of the host galaxies ofhigh-redshift quasars, and the black hole -- bulge ($M_{BH}-\sigma$) relationobtained with this assumption departs significantly from the $M_{BH}-\sigma$relation in the local universe. In this study, we first present aninvestigation of the correlation between the CO line width and the bulgevelocity dispersion using a sample of 33 nearby Seyfert galaxies. We find thatthe formula adopted in previous studies, $\sigma=\rm{FWHM(CO)}/2.35$, isgenerally not a good approximation. Using it, one may underestimate the valueof bulge velocity dispersion significantly when the CO line is narrower than400 $km s^{-1}$. By involving the galactic inclination angle $i$ as anadditional parameter, we obtain a tight correlation between theinclination-corrected CO line width and the bulge velocity dispersion, namely,$\rm {FWHM(CO)}/\sin i=-67.16\pm80.18+(3.62\pm0.68)\sigma$. Using this newrelation, we can better estimate the bulge velocity dispersion from the CO linewidth if the galactic inclination is known. We apply this new relation to ninehigh-redshift quasars with CO line detections and find that they are consistentwith the local $M_{BH}-\sigma$ relation if their inclination angles are around$15^o$. The possible smaller inclinations of the high-redshift quasars arepreferred because of their relatively greater likelihood of detection, and arealso consistent with their relatively smaller CO line widths compared tosubmillimeter galaxies (SMGs) at high redshift having a similar total amount ofmolecular gas. Future observations are needed to confirm these results.Comment: 18 pages, 4 figures, accepted by the Astrophysical Journa

CO Line Width and the Black Hole–Bulge Relationship at High Redshift