  A deformation $U$, of a graded $K$-algebra $A$ is said to be of PBW type if
$grU$ is $A$. It has been shown for Koszul and $N$-Koszul algebras that the
deformation is PBW if and only if the relations of $U$ satisfy a Jacobi type
condition. In particular, for these algebras the determination of the PBW
property is a {\it finite} and explicitly determined linear algebra problem. We
extend these results to an arbitrary graded $K$-algebra, using the notion of
central extensions of algebras and a homological constant attached to $A$ which
we call the {\it complexity} of $A$.
