  Multigraded Betti numbers are one of the simplest invariants of
multiparameter persistence modules. This invariant is useful in theory -- it
completely determines the Hilbert function of the module and the isomorphism
type of the free modules in its minimal free resolution -- as well as in
practice -- it is easy to visualize and it is one of the main outputs of
current multiparameter persistent homology software, such as RIVET. However, to
the best of our knowledge, no bottleneck stability result with respect to the
interleaving distance has been established for this invariant so far, and this
potential lack of stability limits its practical applications. We prove a
stability result for multigraded Betti numbers, using an efficiently computable
bottleneck-type dissimilarity function we introduce. Our notion of matching is
inspired by recent work on signed barcodes, and allows matching bars of the
same module in homological degrees of different parity, in addition to
matchings bars of different modules in homological degrees of the same parity.
Our stability result is a combination of Hilbert's syzygy theorem, Bjerkevik's
bottleneck stability for free modules, and a novel stability result for
projective resolutions. We also prove, in the $2$-parameter case, a
$1$-Wasserstein stability result for Hilbert functions with respect to the
$1$-presentation distance of Bjerkevik and Lesnick.
