### Cosmology

We adopted a Chabrier preliminary mass perform (IMF)^{31} to estimate star formation fee and assumed cosmological parameters of *H*_{0} = 70 km s^{−1} Mpc^{−1}, *Ω*_{M} = 0.3, and *Ω*_{Λ} = 0.7.

### Pattern choice

#### The BH pattern

The pattern for galaxies with immediately measured BH lots is primarily from ref. ^{11}, which incorporates 91 central galaxies collected from refs. ^{19,20,21}. We excluded 18 sources with BH lots measured with reverberation mapping and saved solely these measured with dynamical strategies. We then added one other 63 galaxies with measured BH lots from latest literature, which have been matched with the group catalogue^{32} of close by galaxies to pick solely central galaxies. We obtained the HI flux densities and lots more and plenty of this pattern by crossmatching with the close by galaxy database, HyperLeda^{33}. Our ultimate pattern contains 69 central galaxies with 41 from ref. ^{11} and the remaining from the compilation of latest literature. In Prolonged Knowledge Desk 3, we listing the fundamental properties of our BH pattern.

#### The galaxy pattern

The pattern for galaxies with HI measurements and oblique BH mass measurements are from the prolonged GALEX Arecibo SDSS Survey (xGASS; ref. ^{34}) and HI-MaNGA programme^{35,36}, which embrace HI observations in direction of a consultant pattern of about 1,200 and 6,000 galaxies with 10^{9}*M*_{⊙} < *M*_{⋆} < 10^{11.5}*M*_{⊙}, respectively. The depth of the survey additionally permits for stringent constraints on the higher limits for the HI non-detections, enabling a complete evaluation of *f*_{HI} for your complete pattern. We restricted the redshift z < 0.035 to make sure excessive HI-detection charges even on the highest stellar lots and BH lots. We chosen solely group central galaxies, which embrace not less than one satellite tv for pc galaxy of their teams, based mostly on the crossmatch with the group catalogue^{37,38,39}. Remoted central galaxies missing any satellites of their teams are discarded as a result of they could have in all probability suffered from extra environmental results^{40}. We derived the BH lots for the xGASS and HI-MaNGA samples with their velocity dispersion^{21} from SDSS DR17^{37} (*σ*_{SDSS}, and we require *σ*_{SDSS} ≥ 70 km s^{−1}):

$$log left(frac{{M}_{{rm{BH}}}}{{M}_{odot }}proper)=(8.32pm 0.04)+(5.35pm 0.23)log left(frac{{sigma }_{{rm{SDSS}}}}{200,{rm{km}},{{rm{s}}}^{-1}}proper).$$

(1)

### Bodily parameters of the BH and galaxy pattern

#### Stellar lots

The stellar lots for the galaxy pattern are taken from the MPA-JHU catalogue^{41,42}, that are derived from SED becoming based mostly on SDSS information. For the BH pattern, as a result of most of them lack the identical photometric protection because the galaxy pattern, we derive their stellar lots from their Ok-band luminosity and velocity dispersion-dependent Ok-band mass-to-light ratio following ref. ^{21}:

$${M}_{star }/{L}_{{rm{Ok}}}=0.1{sigma }_{{rm{e}}}^{0.45}.$$

(2)

As an correct dedication of *σ*_{e} shouldn’t be out there for all galaxies, we derived *σ*_{e} for the complete BH pattern from the tight correlation in ref. ^{21}:

$$start{array}{l}log left(frac{{sigma }_{{rm{e}}}}{{rm{km}}},{{rm{s}}}^{-1}proper)=(2.11pm 0.01)+(0.71pm 0.03)log left(frac{{L}_{{rm{Ok}}}}{1{0}^{11}{L}_{odot }}proper) ,,,,,,,+(-0.72pm 0.05)log left(frac{{R}_{{rm{e}}}}{5,{rm{kpc}}}proper).finish{array}$$

(3)

To discover whether or not there are systematic variations between the 2 strategies, we evaluate the stellar lots of the galaxy pattern taken from the MPA-JHU catalogue and people derived from equation (2). A median mass distinction 0.32 dex is discovered between the 2 strategies (Prolonged Knowledge Fig. 6), which can be attributed to the lean from the basic airplane past the mass-to-light ratio, for instance, the darkish matter element within the efficient radius. We corrected these systematic mass variations for the BH pattern to match that of the galaxy pattern.

#### HI fraction and higher limits

The HI-detection restrict relies upon not solely on the sensitivity but in addition on the width of the HI line. To acquire extra real looking higher limits, we first derived the anticipated HI line width for every HI non-detection. The width of the HI line signifies the round velocity of the host galaxy, which ought to be proportional to the stellar lots. We explored this utilizing the HI detections from the xGASS pattern. Prolonged Knowledge Fig. 1 reveals the relation between *M*_{⋆} and the noticed line width, in addition to *M*_{⋆} and inclination-corrected line width. It signifies that the inclination-corrected line width is tightly correlated with *M*_{⋆}, which is additional used to derive the anticipated line width for the HI non-detections. Combining the sensitivity of the HI observations and the anticipated line width, we derived the higher limits for all of the HI non-detections in our BH and galaxy samples.

#### Morphology

For BH pattern, the morphology indicator *T* is obtained from the HyperLEDA database^{33}. It may be a non-integer as a result of for many objects the ultimate *T* is averaged over varied estimates out there within the literature. For the galaxy pattern, we categorised them in to the early varieties and late varieties based mostly on the Sérsic index (from NASA-Sloan Atlas catalogue; NSA: Blanton M.; http://www.nsatlas.org) bigger or smaller than 2.

#### Star formation charges

The particular star formation charges (SSFR) of the galaxy pattern are from the MPA-JHU catalogue based mostly on ref. ^{42}. The SSFR for the BH pattern is taken from the unique reference.

#### Bulge lots

The bulge data is from refs. ^{43,44} for the BH pattern and galaxy pattern, respectively. Extra particularly, we calculate the bulge mass for the galaxy pattern utilizing r-band *B*/*T*.

#### Stellar mass floor density

We calculated the Ok-band efficient radius for each the BH and the galaxy pattern in response to ref. ^{21}: log *R*_{e} = 1.16 log *R*_{K_R_EFF} + 0.23 log *q*_{K_BA}, the place *R*_{e} is the corrected obvious efficient measurement, *R*_{K_R_EFF} and *q*_{K_BA} are Ok-band obvious efficient radius and Ok-band axis ratio from 2MASS. After changing the obvious sizes to the bodily sizes, the stellar mass floor density was derived as ({varSigma }_{{rm{star}}}={M}_{star }/(2{rm{pi }}{R}_{{rm{e}}}^{2})).

#### H_{2} lots

We collected H_{2} lots from xCOLD GASS survey^{18} and ref. ^{45} for xGASS and MaNGA galaxies, respectively. We acknowledge that not less than within the close by Universe, the molecular-to-atomic gasoline mass ratio will increase solely weakly with stellar lots and stays comparatively low over a large stellar mass vary, with (Requiv {M}_{{{rm{H}}}_{2}}/{M}_{{rm{HI}}} sim 10-20 % ) at 10^{9}*M*_{⊙} < *M*_{⋆} < 10^{11.5}*M*_{⊙}. We calculate the overall gasoline fractions as ({mu }_{{rm{HI}}+{{rm{H}}}_{2}},=)(({M}_{{rm{HI}}}+{M}_{{{rm{H}}}_{2}})/{M}_{star }). For central galaxies (remoted centrals plus group centrals), we evaluate the *M*_{BH}–*μ*_{HI} and *M*_{BH}–({mu }_{{rm{HI}}+{{rm{H}}}_{2}}) relation in Prolonged Knowledge Fig. 4. The *M*_{BH}–({mu }_{{rm{HI}}+{{rm{H}}}_{2}}) relation displays a stronger correlation with the smaller scatter than the *M*_{BH}*–μ*_{HI} relation. We acknowledge that, based mostly on molecular hydrogen gasoline content material traced by way of mud extinction, earlier research present an *M*_{BH}–({f}_{{{rm{H}}}_{2}}) correlation^{12}. Future research with extra direct measurements of molecular hydrogen gasoline for giant samples might be wanted to look at intimately whether or not *M*_{BH} additionally performs a basic half in regulating the molecular gasoline content material in galaxies.

### Quiescent fraction

To estimate the quiescent fraction at completely different *M*_{BH}, we chosen galaxies from the MPA-JHU catalogue of SDSS galaxies with the identical standards because the galaxy pattern, besides that we restricted the rate dispersion to better than 30 km s^{−1} to cowl broader *M*_{BH} and we made no constraints on the HI detection. We categorised the pattern galaxies into star-forming and quiescent ones, separated at SSFR = −11. In every *M*_{BH} bin, the quiescent fraction was calculated because the ratio between the variety of quiescent galaxies and that of all galaxies. The result’s proven in Prolonged Knowledge Fig. 5, which is in step with that of earlier work^{29,46}.

### Linear least squares approximation

We applied linear regression for the BH pattern and the galaxy pattern utilizing Python package deal LTS_LINEFIT launched in ref. ^{47}, which is insensitive to outliers and can provide the intrinsic scatter across the linear relation with corresponding errors of the fitted parameters.

### Linear becoming together with higher limits

To include each detections and higher limits within the galaxy pattern, we utilized the Kaplan–Meier non-parametric estimator to derive the cumulative distribution perform at completely different *M*_{BH} bins (with Python package deal Reliability^{48}), and carried out 10,000 random attracts from the cumulative distribution perform at every bin to suit the relation between *f*_{gasoline} and *M*_{BH}. The linear relation and its corresponding errors are taken as the most effective becoming and commonplace deviations of those fittings (Prolonged Knowledge Desk 2). The non-detection fee of HI is comparatively low throughout a lot of the *M*_{BH} vary and turns into important just for galaxies with probably the most huge BHs (reaching about 50% at *M*_{BH} > 10^{8}*M*_{⊙}).

### Partial least sq. regression

To derive probably the most important bodily parameters in figuring out *μ*_{HI} statistically, we used the Python package deal Scikit-learn^{49} with partial least squares (PLS) Regression perform, which makes use of a non-linear iterative partial least squares (NIPALS)^{50} algorithm. The PLS algorithm generalizes a couple of latent variables (or principal parts) that summarize the variance of impartial variables, which is used to search out the basic relation between a set of impartial and dependent variables. It has benefits in regression amongst extremely correlated predictor variables. It calculates the linear combos of the unique predictor datasets (latent variables) and the response datasets with maximal covariance, then matches the regression between the projected datasets and returns the mannequin:

the place *X* and *Y* are predictor and response datasets, *B* is the matrix of regression coefficients and *F* is the intercept matrix.

We constructed the *X* and *Y* matrices because the set of *M*_{BH}, *M*_{⋆}, *Σ*_{star}, *M*_{bulge} and the set of *μ*_{HI}. For the BH and galaxy samples, this returns the pattern measurement of 45 and 189, respectively. The optimum variety of latent variables (linear combos of predictor variables) in PLS Regression is set by the minimal of imply squared error from cross-validation (utilizing perform cross_val_predict in Scikit-learn) at every variety of parts. We discover that the optimum variety of latent variables for each the BH and the galaxy pattern converges to 1. Additional growing the variety of latent variables yields only some share adjustments within the imply squared errors, and *M*_{BH} stays probably the most important predictor parameter. Following appendix B in ref. ^{51}, the variance contribution from completely different parameters to *μ*_{HI} is decomposed as

$${rm{Var}}(Y)=mathop{sum }limits_{i=1}^{4}{rm{Var}}({X}_{i}{B}_{i})+{rm{Var}}(F),$$

(5)

the place Var is a measure of the unfold of a distribution. The portion of every parameter variance is proven within the final column of the Prolonged Knowledge Desk 3, which reveals that *M*_{BH} dominates the variance. Additional growing the variety of latent variables outcomes solely in a couple of share adjustments within the imply squared errors, and *M*_{BH} stays probably the most important predictor parameter.