  A significant part of modern topological data analysis is concerned with the
design and study of algebraic invariants of poset representations -- often
referred to as multi-parameter persistence modules. One such invariant is the
minimal rank decomposition, which encodes the ranks of all the structure
morphisms of the persistence module by a single ordered pair of
rectangle-decomposable modules, interpreted as a signed barcode. This signed
barcode generalizes the concept of persistence barcode from one-parameter
persistence to any number of parameters, raising the question of its bottleneck
stability. We show in this paper that the minimal rank decomposition is not
stable under the natural notion of signed bottleneck matching between signed
barcodes. We remedy this by turning our focus to the rank exact decomposition,
a related signed barcode induced by the minimal projective resolution of the
module relative to the so-called rank exact structure, which we prove to be
bottleneck stable under signed matchings. As part of our proof, we obtain two
intermediate results of independent interest: we compute the global dimension
of the rank exact structure on the category of finitely presentable
multi-parameter persistence modules, and we prove a bottleneck stability result
for hook-decomposable modules. We also give a bound for the size of the rank
exact decomposition that is polynomial in the size of the usual minimal
projective resolution, we prove a universality result for the dissimilarity
function induced by the notion of signed matching, and we compute, in the
two-parameter case, the global dimension of a different exact structure related
to the upsets of the indexing poset. This set of results combines concepts from
topological data analysis and from the representation theory of posets, and we
believe is relevant to both areas.
