direct limit
Idea
A direct limit is a special kind of colimit. It’s naming is a consequence of them being defined before categorical formalism, and also for the fact that one can guess what diagram the colimit is over, etc.
Definition
- A directed set is a partially ordered set where each finite subset has an upper bound.
- A directed system of (say) abelian groups $(A_i, f_{ij})_{i,j\in I}$ is
- an abelian group $A_i$ for eachh $i\in I$
- a homomorphihsm $f_{ij}:A_i\to A_j$ whenever $i\leq j$
- $f_{ii}=\text{id}_{A_i}$
- $f_{ij}=f_{jk}\circ f_{ij}$, i.e. the following diagram commutes:
- A direct limit the colimit. Explicitly, it is the universal object $A$ equipped with morphisms $\phi_i:A_i\to A$ such that the following diagram commutes:
Morphisms
Consider two directed systems $(A_i,f_{ij})$, $(B_i, g_{ij})$ indexed by the same set $I$. A morphism $$\phi:(A_i,f_{ij})\to (B_i, g_{ij})$$ is a natural transformation between the directed systems, regarded as categories. Explicitly,
- for eachh $i\in I$, a morphism $\phi_i:A_i\to B_i$ such that the following diagram commutes:
This indduces a homomorphism between the direct limits $$\lim_\to\phi:\lim_\to A_i\to \lim_\to B_i.$$
Examples
2 (subextensions) Given an algebraic extension $K/F$, the set of all subextensions $K/L/F$ (we can exclude $K$) is a partially ordered set under inclusions. Furthermore, it is
- directed: $L,L'\subseteq LL'$
- a directed system: wrt inclusion maps
The direct limit of this system is $K$ itself. It expresses the fact that $K$ is the union of all intermediate fields $K/L/F$.
2(b) (Galois extensions) Let $K/k$ be Galois. Then for every $\alpha\in K$, the minimal polynomial $P_\alpha$ is separable and all of its roots are contained in $K$. Hence $k_{P_\alpha}\subset L$, whence $k_{P_\alpha}/k$ is finite Galois. Thus $$K/k\cong\lim_\to L/k$$ indexed over all $L/k$ finite Galois.
3 (uniqueness of algebraic closure) By Example 2,
4 (uniqueness of separable closure)