# inverse limit

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

 $\xymatrix{{G_{0}}&{G_{1}}\ar@{->}[l]_{f_{1}}&{G_{2}}\ar@{->}[l]_{f_{2}}&{G_{3}% }\ar@{->}[l]_{f_{3}}&{\ldots}\ar@{->}[l]_{f_{4}}}$
###### Definition 1.

The inverse limit of $(G_{i},f_{i})$, denoted by

 $\varprojlim(G_{i},f_{i}),\quad\text{ or }\quad\varprojlim G_{i}$

is the subset of $\prod_{i=0}^{\infty}G_{i}$ formed by elements satisfying

 $(\ g_{0},\ g_{1},\ g_{2},\ g_{3},\ \ldots),\ \text{with}\quad g_{i}\in G_{i},% \quad f_{i}(g_{i})=g_{i-1}$

Note: The inverse limit of $G_{i}$ can be checked to be a subgroup of the product $\prod_{i=0}^{\infty}G_{i}$. See below for a more general definition.

Examples:

1. 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 $i\geq 2$, to be “reduction modulo $p^{i-1}$” i.e.

 $f_{i}\colon\mathbb{Z}/p^{i}\mathbb{Z}\to\mathbb{Z}/p^{i-1}\mathbb{Z}$
 $f_{i}(x\operatorname{mod}p^{i})=x\operatorname{mod}p^{i-1}$

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

 $\mathbb{Z}_{p}=\varprojlim\mathbb{Z}/p^{i}\mathbb{Z}$
2. 2.

Let $E$ be an elliptic curve defined over $\mathbb{C}$. 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

 $f_{i}\colon E[p^{i}]\to E[p^{i-1}],\quad f_{i}(Q)=p\cdot Q$

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

 $T_{p}(E)=\varprojlim E[p^{i}]$

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

 $\{f_{\alpha,\beta}\colon G_{\beta}\to G_{\alpha}\mid\alpha,\beta\in S,\quad% \alpha\leq\beta\}$

be a collection of morphisms satisfying:

1. 1.

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

2. 2.

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

###### Definition 2.

The inverse limit of $(\{G_{\alpha}\}_{\alpha\in S},\{f_{\alpha,\beta}\})$, denoted by

 $\varprojlim(G_{\alpha},f_{\alpha,\beta}),\quad\text{ or }\quad\varprojlim G_{\alpha}$

is defined to be the set of all $(g_{\alpha})\in\prod_{\alpha\in S}G_{\alpha}$ such that for all $\alpha,\beta\in S$

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

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