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On the Universal Coefficient Formula and Derived $\varprojlim ^{(i)} $ Functor

Anzor Beridze, Leonard Mdzinarishvili

Abstract

It is known that homology and inverse limit functors do not commute. In the paper we consider this very problem and find its application for various homology theories. In particular, on the category of general topological spaces, there are defined exact homology functors induced by different non-free cochain complexes. Relation between them and other classical homology theories are given. In addition, for the defined homology functors the tautness and the continuous properties are obtained.

On the Universal Coefficient Formula and Derived $\varprojlim ^{(i)} $ Functor

Abstract

It is known that homology and inverse limit functors do not commute. In the paper we consider this very problem and find its application for various homology theories. In particular, on the category of general topological spaces, there are defined exact homology functors induced by different non-free cochain complexes. Relation between them and other classical homology theories are given. In addition, for the defined homology functors the tautness and the continuous properties are obtained.

Paper Structure

This paper contains 3 sections, 19 theorems, 71 equations.

Table of Contents

  1. Universal Coefficient Formula
  2. Some properties of inverse limit and its derived functors
  3. Applications in homology theory

Key Result

Theorem 1

For each cochain complex $C^*$ and $R$-module $G$ over a fixed principal ideal domain $R$, there exists a short exact sequence

Theorems & Definitions (31)

  • Theorem 1: Universal Coefficient Formula
  • proof
  • Corollary 1
  • Theorem 2
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 21 more