  We study inequalities between graded Betti numbers of ideals in a standard
graded algebra over a field and their images under embedding maps, defined
earlier by us in [Math. Z. 274, (2013), no. 3-4, pp. 809-819; arXiv:1009.4488].
We show that if graded Betti numbers do not decrease when we replace ideals in
an algebra by their embedded versions, then the same behaviour is carried over
to ring extensions. As a corollary we give alternative inductive proofs of
earlier results of Bigatti, Hulett, Pardue, Mermin-Peeva-Stillman and Murai. We
extend a hypersurface restriction theorem of Herzog-Popescu to the situation of
embeddings. We show that we can obtain the Betti table of an ideal in the
extension ring from the Betti table of its embedded version by a sequence of
consecutive cancellations. We further show that the lex-plus-powers conjecture
of Evans reduces to the Artinian situation.
