  Homological algebra of modules over posets is developed, as closely parallel
as possible to that of finitely generated modules over noetherian commutative
rings, in the direction of finite presentations and resolutions. Centrally at
issue is how to define finiteness to replace the noetherian hypothesis which
fails. 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 theoretical and
computational purposes: it guarantees finite presentations and resolutions of
various sorts, all related by a syzygy theorem, amenable to algorithmic
manipulation. Tameness and its homological theory are new even in the finitely
generated discrete setting of $\mathbb{N}^n$-gradings, where tame is materially
weaker than noetherian. In the context of persistent homology of filtered
topological spaces, especially with multiple real parameters, the algebraic
theory of tameness yields topologically interpretable data structures in terms
of birth and death of homology classes.
