  Strong scaling relations between host galaxy properties (such as stellar
mass, bulge mass, luminosity, effective radius etc) and their nuclear
supermassive black hole's mass point towards a close co-evolution. In this
work, we first review previous efforts supporting the fundamental importance of
the relation between supermassive black hole mass and stellar velocity
dispersion ($M_{\rm BH}$-$\sigma_{\rm e}$). We then present further original
work supporting this claim via analysis of residuals and principal component
analysis applied to some among the latest compilations of local galaxy samples
with dynamically measured supermassive black hole masses. We conclude with a
review of the main physical scenarios in favour of the existence of a $M_{\rm
BH}$-$\sigma_{\rm e}$ relation, with a focus on momentum-driven outflows.
