  The commutative and homological algebra of modules over posets is developed,
as closely parallel as possible to the algebra of finitely generated modules
over noetherian commutative rings, in the direction of finite presentations,
primary decompositions, and resolutions. Interpreting this finiteness in the
language of derived categories of subanalytically constructible sheaves proves
two conjectures due to Kashiwara and Schapira concerning sheaves with
microsupport in a given cone. The motivating case is persistent homology of
arbitrary filtered topological spaces, especially the case of multiple real
parameters. The algebraic theory yields computationally feasible, topologically
interpretable data structures, in terms of birth and death of homology classes,
for persistent homology indexed by arbitrary posets. The exposition focuses on
the nature and ramifications of a suitable finiteness condition to replace the
noetherian hypothesis. The tameness condition introduced for this purpose
captures finiteness for variation in families of vector spaces indexed by
posets in a way that is characterized equivalently by distinct topological,
algebraic, combinatorial, and homological manifestations. Tameness serves both
the theoretical and computational purposes: it guarantees finite primary
decompositions, as well as various finite presentations and resolutions all
related by a syzygy theorem, and the data structures thus produced are
computable in addition to being interpretable. The tameness condition and its
resulting theory are new even in the finitely generated discrete setting, where
being tame is materially weaker than being noetherian.
