PlanetMath: inverse limit

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inverse limit

(Definition)

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

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ {G_0} & {G_1} \ar@{->}[l]_{f_1} ... ...->}[l]_{f_2} & {G_3} \ar@{->}[l]_{f_3} & {\ldots} \ar@{->}[l]_{f_4}} } \end{xy}$

Definition 1 The inverse limit of , denoted by

$\displaystyle \varprojlim(G_i,f_i),$ or $\displaystyle \quad \varprojlim G_i$

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

$\displaystyle (\ g_0,\ g_1,\ g_2,\ g_3,\ \ldots),\$ with $\displaystyle \quad g_i\in G_i, \quad f_i(g_i)=g_{i-1}$

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

Examples:

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 , for $i\geq 2$ , to be “reduction modulo $p^{i-1}$ ” i.e.
$\displaystyle f_i\colon \mathbb{Z}/p^i\mathbb{Z}\to \mathbb{Z}/p^{i-1}\mathbb{Z}$

$\displaystyle f_i(x \operatorname{mod}p^i)= x \operatorname{mod} p^{i-1}$
which are obviously surjective homomorphisms. The inverse limit of $(\mathbb{Z}/p^i\mathbb{Z}, f_i)$ is called the -adic integers and denoted by
$\displaystyle \mathbb{Z}_p=\varprojlim\mathbb{Z}/p^i\mathbb{Z}$
Let be an elliptic curve defined over $\mathbb{C}$ . Let be a prime and for any natural number write for the -torsion group, i.e.
$\displaystyle E[n]=\{ Q\in E \mid n\cdot Q=O\}$
In this case we define , and
$\displaystyle f_i\colon E[p^i] \to E[p^{i-1}], \quad f_i(Q)=p\cdot Q$
The inverse limit of is called the Tate module of and denoted
$\displaystyle T_p(E)=\varprojlim E[p^i]$

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

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

be a collection of morphisms satisfying:

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

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

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

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

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

"inverse limit" is owned by alozano.

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See Also:

-adic integers, Galois representation, infinite Galois theory, profinite group, category associated to a partial order, inverse limit, cohomology of small categories

Other names:

inverse system, projective limit

Also defines:

inverse limit

Keywords:

inverse limit

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Cross-references: infinite Galois theory, composition, identity, morphisms, objects, collection, category, directed set, Tate module, natural number, elliptic curve, integers, prime, product, subgroup, subset, homomorphisms, surjective, chain, groups, sequence
There are 13 references to this entry.

This is version 7 of inverse limit, born on 2003-08-26, modified 2005-10-08.
Object id is 4655, canonical name is InverseLimit.
Accessed 8349 times total.

Classification:

AMS MSC:

20F22 (Group theory and generalizations :: Special aspects of infinite or finite groups :: Other classes of groups defined by subgroup chains)

Pending Errata and Addenda

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