Direct and inverse limits; definition & uniqueness

Posted: January 12, 2011 in Direct and Inverse Limit, Noncommutative Ring Theory Notes
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Definition 1. Let \{M_i, f_{ij} \} be a direct system in a category \mathcal{C}. Suppose there exists an object M and morphisms f_i : M_i \longrightarrow M, for every i, such that the following conditions are satisfied:

1) f_j f_{ij}=f_i, whenever i \leq j,

2) (The universal property) Suppose that there exist an object X and morphisms g_i : M_i \longrightarrow X, \ i \in I, such that g_j f_{ij}=g_i whenever i \leq j. Then there exists a unique morphism f: M \longrightarrow X such that ff_i = g_i, \ i \in I.

Then M is called the direct limit of our direct system and we will just write \varinjlim M_i = M.

Definition 2. Let \{M_i, f_{ij} \} be an inverse system in a category \mathcal{C}. Suppose that there exists an object M and morphisms f_i : M \longrightarrow M_i, for every i, such that the following conditions are satisfied:

1) f_{ij}f_j=f_i, whenever i \leq j,

2) If there exists some object X and morphisms g_i : X \longrightarrow M_i, for every i, such that f_{ij}g_j=g_i whenever i \leq j, then there must exist a unique morphism f : X \longrightarrow M such that f_if = g_i for all i.

Then M is called the inverse limit of our inverse system and we will just write \varprojlim M_i = M.

In our definitions I wrote “the” direct limit and “the” inverse limit. That is because “the” direct (resp. inverse) limit of a direct (resp. inverse) system, if it exists, is unique, up to isomorphism of course, as we are going to see:

Fact. The direct (resp. inverse) limit of a direct (resp. inverse) system is unique up to isomorphism if it exists.

Proof. I will prove the fact for direct limit only. The proof for inverse limit is similar. Suppose that both \{M, f_i \} and \{M', f'_i \} satisfy the conditions in Dedinition 1. Then, by condition 2), there exist (unique) morphisms f: M \longrightarrow M' and f': M' \longrightarrow M such that ff_i=f'_i and f'f'_i=f_i. Thus the morphism f'f : M \longrightarrow M satisfies (f'f)f_i=f_i. But we also have that the identity morphism \text{id}_M : M \longrightarrow M satisfies \text{id}_M f_i = f_i. Thus, by the uniqueness condition in 2), we must have f'f = \text{id}_M. Similarly ff'= \text{id}_{M'} and hence f is an isomorphism and f' is its inverse.

Remark. You might have found the relationships between the morphisms in the definition of direct (resp. inverse) systems and limits complicated but they are not! If you draw diagrams, then you will see that they just mean that those diagrams are commutative.

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