3
votes
2answers
62 views

Canonical isomorphism between Cauchy sequence completion and inverse limit

I'm studying chapter 10 of Atiyah Macdonald. The book introduces two ways to construct the completion of an abelian topological group: Equivalence classes of Cauchy sequences and inverse limit. I can ...
0
votes
0answers
73 views

Explanation of a passage in Atiyah / Macdonald

On page 105 the authors show that $\hat{\hat{G}} \cong \hat{G}$ (Proposition 10.5) and conclude that the canonical homomorphism $\phi : \hat{G} \to \hat{\hat{G}}$ is an isomorphism. How does the fact ...
2
votes
0answers
33 views

Is an ideal generated by a compact subset finitely generated?

Let $R$ be a commutative topological ring and let $K$ be a compact subset of $R$. Denote by $I$ the ideal generated by $R$. Then is it true (or under what assumptions on $R$ (besides Noethernity)) is ...
2
votes
2answers
98 views

continuous images of Cauchy sequences in topological groups

on page 102 of Atiyah and MacDonald's "Introduction to Commutative Algebra", they state that if $G$ and $H$ are abelian topological groups and $f$ is a continuous homomorphism from $G$ to $H$, then ...
1
vote
1answer
137 views

Questions about the intersection of all neighborhoods of $0$ in a topological abelian group.

Let $H$ be the intersection of all neighborhoods of $0$ in a topological abelian group. On page 102 of the book introduction to commutative algebra by Atiyah and Macdonald, the fourth line of the ...
7
votes
1answer
98 views

Closed subgroups of n copies of the p-adic integers

What do closed subgroups of $\mathbb{Z}_p \oplus \cdots \oplus \mathbb{Z}_p$ look like (where there are $n$ summands in the direct sum)?
6
votes
1answer
175 views

Addition and multiplication are continuous in the $I$-adic topology

Can you tell me if this is correct? Let $R$ be a ring and let it have the $I$-adic topology for some ideal $I$ in $R$. I want to show that $+: R \times R \to R$ is continuous at $(x_0, y_0)$. Proof: ...
3
votes
2answers
137 views

Adic Completion of a direct sum

This is a question related to this. Let $G= \mathbb Z / p \mathbb Z$ for some prime $p$. Let $A = \bigoplus_{n\in \mathbb N} G$, that is, all sequences in $G$ with all but finitely many terms zero. ...
3
votes
1answer
39 views

Quick question about Hausdorffness of completion

Let $G$ be the usual: a topological Abelian group with a topology induced by a countable neighbourhood basis $G_n$ of zero such that $G = G_1 \supset G_2 \supset \dots$. Let $\widehat{G}$ denote the ...
3
votes
1answer
99 views

Quick question about completions and inverse limits

Let $G$ be the usual: a topological Abelian group with a topology induced by a countable neighbourhood basis $G_n$ of zero such that $G = G_1 \supset G_2 \supset \dots$. Let $\widehat{G}$ denote the ...
2
votes
3answers
215 views

Isomorphism from $\widehat{G}$ to $\displaystyle \lim_{\longleftarrow} G/G_n$

Let $G$ be a topological abelian group and let $\widehat{G}$ denote its completion (i.e. equivalence classes of Cauchy sequences). Let $G_n$ be a descending sequence of subgroups, i.e. $G = G_0 ...
7
votes
1answer
193 views

Intersection of neighborhoods of 0. Subgroup?

Repeating for my exam in commutative algebra. Let G be a topological abelian group, i.e. such that the mappings $+:G\times G \to G$ and $-:G\to G$ are continuous. Then we have the following Lemma: ...