  Let $A$ be a noetherian connected graded algebra. We introduce and study
homological invariants that are weighted sums of the homological and internal
degrees of cochain complexes of graded $A$-modules, providing weighted versions
of Castelnuovo--Mumford regularity, Tor-regularity, Artin--Schelter regularity,
and concavity. In some cases an invariant (such as Tor-regularity) that is
infinite can be replaced with a weighted invariant that is finite, and several
homological invariants of complexes can be expressed as weighted homological
regularities. We prove a few weighted homological identities some of which
unify different classical homological identities and produce interesting new
ones.
