inverse limit

Let ${\{G_i\}}_{i=0}^{\mathrm{\infty }}$ be a sequence of groups which are related by a chain of surjective homomorphisms $f_i:G_i\to G_{i-1}$ such that

Definition 1.

The inverse limit of $\mathrm{(}{G}_i\mathrm{,}{f}_i\mathrm{)}$, denoted by

$$\underleftarrow{\lim }(G_i,f_i),\mathit{\text{ or }}\mathit{\hspace{1em}}\underleftarrow{\lim }{G}_i$$

is the subset of ${\mathrm{\prod }}_{i\mathrm{=}\mathrm{0}}^{\mathrm{\infty }}{G}_i$ formed by elements satisfying

$$(g_0,g_1,g_2,g_3,\mathrm{\dots }),\text{𝑤𝑖𝑡ℎ}\mathit{\hspace{1em}}{g}_i\in G_i,f_i(g_i)=g_{i-1}$$

Note: The inverse limit of $G_i$ can be checked to be a subgroup of the product ${\prod }_{i=0}^{\mathrm{\infty }}{G}_i$. See below for a more general definition.

Examples:

1. Let $p\in \mathbb{N}$ be a prime. Let $G_0=\{0\}$ and $G_i=\mathbb{Z}/p^i\mathbb{Z}$. Define the connecting homomorphisms $f_i$, for $i\ge 2$, to be “reduction modulo $p^{i-1}$” i.e.

   $$f_i:\mathbb{Z}/p^i\mathbb{Z}\to \mathbb{Z}/p^{i-1}\mathbb{Z}$$

   $$f_i(x\mathrm{mod}{p}^i)=x\mathrm{mod}{p}^{i-1}$$

   which are obviously surjective homomorphisms. The inverse limit of $(\mathbb{Z}/p^i\mathbb{Z},f_i)$ is called the $p$-adic integers and denoted by

   $${\mathbb{Z}}_p=\underleftarrow{\lim }\mathbb{Z}/p^i\mathbb{Z}$$

2. Let $E$ be an elliptic curve defined over $ℂ$. Let $p$ be a prime and for any natural number $n$ write $E[n]$ for the $n$-torsion group, i.e.

   $$E[n]=\{Q\in E\mid n\cdot Q=O\}$$

   In this case we define $G_i=E[p^i]$, and

   $$f_i:E[p^i]\to E[p^{i-1}],f_i(Q)=p\cdot Q$$

   The inverse limit of $(E[p^i],f_i)$ is called the Tate module of $E$ and denoted

   $$T_p(E)=\underleftarrow{\lim }E[p^i]$$

The concept of inverse limit can be defined in far more generality. Let $(S,\le )$ be a directed set and let $𝒞$ be a category. Let ${\{G_{\alpha }\}}_{\alpha \in S}$ be a collection of objects in the category $𝒞$ and let

$$\{f_{\alpha ,\beta }:G_{\beta }\to G_{\alpha }\mid \alpha ,\beta \in S,\alpha \le \beta \}$$

be a collection of morphisms satisfying:

1. For all $\alpha \in S$, $f_{\alpha ,\alpha }={\mathrm{Id}}_{G_{\alpha }}$, the identity morphism.

2. For all $\alpha ,\beta ,\gamma \in S$ such that $\alpha \le \beta \le \gamma$, we have $f_{\alpha ,\gamma }=f_{\alpha ,\beta }\circ f_{\beta ,\gamma }$ (composition of morphisms).

Definition 2.

The inverse limit of $\mathrm{(}{\mathrm{\{}{G}_{\alpha }\mathrm{\}}}_{\alpha \mathrm{\in }S}\mathrm{,}\mathrm{\{}{f}_{\alpha \mathrm{,}\beta }\mathrm{\}}\mathrm{)}$, denoted by

$$\underleftarrow{\lim }(G_{\alpha },f_{\alpha ,\beta }),\mathit{\text{ or }}\mathit{\hspace{1em}}\underleftarrow{\lim }{G}_{\alpha }$$

is defined to be the set of all $\mathrm{(}{g}_{\alpha }\mathrm{)}\mathrm{\in }{\mathrm{\prod }}_{\alpha \mathrm{\in }S}{G}_{\alpha }$ such that for all $\alpha \mathrm{,}\beta \mathrm{\in }S$

$$\alpha \le \beta \Rightarrow f_{\alpha ,\beta }(g_{\beta })=g_{\alpha }$$

For a good example of this more general construction, see infinite Galois theory.

Title	inverse limit
Canonical name	InverseLimit
Date of creation	2013-03-22 13:54:20
Last modified on	2013-03-22 13:54:20
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	10
Author	alozano (2414)
Entry type	Definition
Classification	msc 20F22
Synonym	inverse system
Synonym	projective limit
Related topic	PAdicIntegers
Related topic	GaloisRepresentation
Related topic	InfiniteGaloisTheory
Related topic	ProfiniteGroup
Related topic	CategoryAssociatedToAPartialOrder
Related topic	DirectLimit
Related topic	CohomologyOfSmallCategories
Defines	inverse limit