  The following article is an application of commutative algebra to the study
of multiparameter persistent homology in topological data analysis. In
particular, the theory of finite free resolutions of modules over polynomial
rings is applied to multiparameter persistent modules. The generic structure of
such resolutions and the classifying spaces involved are studied using results
spanning several decades of research in commutative algebra, beginning with the
study of generic structural properties of free resolutions popularized by
Buchsbaum and Eisenbud. Many explicit computations are presented using the
computer algebra package Macaulay2, along with the code used for computations.
This paper serves as a collection of theoretical results from commutative
algebra which will be necessary as a foundation in the future use of
computational methods using Gr\"obner bases, standard monomial theories, Young
tableaux, Schur functors and Schur polynomials, and the classical
representation theory and invariant theory involved in linear algebraic group
actions. The methods used are in general characteristic free and are designed
to work over the ring of integers in order to be useful for applications and
computations in data science. As an applications we explain how one could apply
2-parameter persistent homology to study time-varying interactions graphs
associated to quadratic Hamiltonians such as those in the Ising model or
Kitaev's torus code and other surface codes.
