Give an example of a direct system of modules, {Ai, αj^i}, over some directed index set I, for w

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Give an example of a direct system of modules, {Ai, αj^i},...

Give an example of a direct system of modules, $\left\{A_{i}, \alpha_{j}^{i}\right\}$, over some directed index set $I$, for which $A_{i} \neq\{0\}$ for all $i$ and $\lim _{\rightarrow} A_{i}=\{0\}$.

Key Concepts

Direct System

A direct system is a collection of modules indexed by a directed set together with transition maps between these modules that satisfy certain coherence conditions (such as identity and transitivity). This setup allows one to systematically 'glue together' the modules along the directed system, forming a framework from which a global object, the direct limit, can be constructed.

Direct Limit (Colimit)

The direct limit (or colimit) is the module that results from ‘merging’ the modules in a direct system according to the transition maps. It is constructed by taking the disjoint union of all the modules and then factoring by an equivalence relation determined by the maps. The direct limit captures the asymptotic behavior of the system and can sometimes be drastically smaller (even trivial) than the individual modules, depending on the nature of the maps.

Directed Index Set

A directed index set is a partially ordered set with the property that every pair (or finite set) of indices has an upper bound. This feature is essential in ensuring that for any two modules in a direct system, there exists a later module to which both can map, which is crucial for defining the direct limit in a coherent manner.

Module Homomorphisms (Structure Maps)

The structure maps in a direct system are module homomorphisms that connect the individual modules according to the directed index set. They must satisfy the condition that for any indices i ? j ? k, the composition of the maps from i to j and j to k equals the direct map from i to k. These maps control how elements are related and potentially identified in the formation of the direct limit.

Nontrivial Modules with Trivial Limit

This concept illustrates that even if all individual modules in a direct system are nonzero, the amalgamation process governed by the transition maps can lead to a direct limit that is the zero module. This occurs when the structure maps are such that every element eventually maps to zero, emphasizing the decisive role of the homomorphisms in determining the nature of the direct limit.

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