Divisible ℤ-modules

Divisible Z-modules

In this article, we formalize the definition of divisible Z-module and its properties in the Mizar system [3]. We formally prove that any non-trivial divisible Z-modules are not finitely-generated. We introduce a divisible Z-module, equivalent to a vector space of a torsion-free Z-module with a coefficient ring Q. Z-modules are important for lattice problems, LLL (Lenstra, Lenstra and Lovász) b...

Dieudonné Modules and p-Divisible Groups

The notion of `-adic Tate modules, for primes ` away from the characteristic of the ground field, is incredibly useful. The analogous notion at the prime p is that of Dieudonné modules. At finite level, Dieudonné modules classify commutative finite group schemes of p-power order over a field of characteristic p. Dieudonné modules can be used to determine the local Brauer invariant of the endomo...

Divisible Uniserial Modules over Valuation Domains

MODULES FOR Z/p× Z/p

We investigate various aspects of the modular representation theory of Z/p × Z/p with particular focus on modules of constant Jordan type. The special modules we consider and the constructions we introduce not only reveal some of the structure of (Z/p× Z/p)-modules but also provide a guide to further study of the representation theory of finite group schemes.

Recognizing Badly Presented Z-modules

Finitely generated Z-modules have canonical decompositions. When such modules are given in a finitely presented form there is a classical algorithm for computing a canonical decomposition. This is the algorithm for computing the Smith normal form of an integer matrix. We discuss algorithms for Smith normal form computation, and present practical algorithms which give excellent performance for m...