free module if it can be decomposed as a direct sum of free interval modules. Properties Finite Type Conditions A persistence module indexed over is said to be of finite type if the following conditions hold for all : Each vector space is finite-dimensional. There exists an integer such that the map is an isomorphism for all . If satisfies the first condition, then is commonly said to be pointwise finite-dimensional (p.f.d.). The notion of pointwise finite-dimensionality immediately extends to arbitrary indexing sets. The definition of finite type can also be adapted to continuous indexing sets. Namely, a module indexed over is of finite type if is p.f.d., and contains a finite number of unique vector spaces. Formally speaking, this requires that for all but a finite number of points there is a neighborhood of such that for all , and also that there is some such that for all . A module satisfying only the former property is sometimes labeled essentially discrete, whereas a module satisfying both properties is known as essentially finite. An -persistence module is said to be semicontinuous if for any and any sufficiently close to , the map is an isomorphism. Note that this condition is redundant if the other finite type conditions above are satisfied, so it is not typically included in the definition, but is relevant in certain circumstances. Structure Theorem One of the primary goals in the study of persistence modules is to classify modules according to their decomposability into interval modules. A persistence module that admits a decomposition as a direct sum of interval modules is often simply called "interval decomposable." One of the primary results in this direction is that any p.f.d. persistence module indexed over a totally ordered set is interval decomposable. This is sometimes referred to as the "structure theorem for persistence modules." The case when is finite is a straightforward application of the structure theorem for finitely generated modules over a principal ideal domain. For modules indexed over , the first known proof of the structure theorem is due to Webb. The theorem was extended to the case of (or any totally ordered set containing a countable subset that is dense in with the order topology) by Crawley-Boevey in 2015. The generalized version of the structure theorem, i.e., for p.f.d. modules indexed over arbitrary totally ordered sets, was established by Botnan and Crawley-Boevey in 2019. References Mathematics