Displaying similar documents to “The Z/2 Z cohomology of the universal ordinary distribution”

Free ℤ-module

Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama (2012)

Formalized Mathematics

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In this article we formalize a free ℤ-module and its rank. We formally prove that for a free finite rank ℤ-module V , the number of elements in its basis, that is a rank of the ℤ-module, is constant regardless of the selection of its basis. ℤ-module is necessary for lattice problems, LLL(Lenstra, Lenstra and Lovász) base reduction algorithm and cryptographic systems with lattice [15]. Some theorems in this article are described by translating theorems in [21] and [8] into theorems of...

Torsion Z-module and Torsion-free Z-module

Yuichi Futa, Hiroyuki Okazaki, Kazuhisa Nakasho, Yasunari Shidama (2014)

Formalized Mathematics

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In this article, we formalize a torsion Z-module and a torsionfree Z-module. Especially, we prove formally that finitely generated torsion-free Z-modules are finite rank free. We also formalize properties related to rank of finite rank free Z-modules. The notion of Z-module is necessary for solving lattice problems, LLL (Lenstra, Lenstra, and Lov´asz) base reduction algorithm [20], cryptographic systems with lattice [21], and coding theory [11].

Bounding the orders of finite subgroups.

Ian J. Leary, Brita E. A. Nucinkis (2001)

Publicacions Matemàtiques

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We give homological conditions on groups such that whenever the conditions hold for a group G, there is a bound on the orders of finite subgroups of G. This extends a result of P. H. Kropholler. We also suggest a weaker condition under which the same conclusion might hold.