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Score of 6
1 answer
208 views

Let $G$ be a group and $k$ a field. The Malcev completion $G \to \hat G$ over $k$ is (at least in this post) defined to be the group-like part $\mathbb{G}\widehat{k[G]}$ of the completed group ...
Score of 2
0 answers
262 views

$\def\C{\mathrm{C}} \def\P{\mathrm{P}} \def\AR{\mathrm{AR}}$(All rings are commutative and unital.) I am trying to prove or disprove the following: ($\C$). Iterated adic completion amounts to joint ...
Score of 3
1 answer
345 views

Let $X$ be a scheme, $Z$ be a closed subscheme. If $Z$ is affine, is the formal completion $\hat X$ of $X$ along $Z$ necessarily an affine formal scheme? I have heard this claim (at least when $Z$ is ...
Score of 3
0 answers
291 views

At a high level, algebra is the study of morphisms. In many nice categories, there are two types of morphisms — monomorphisms and epimorphisms — because every morphism decomposes as the application of ...
Score of 2
0 answers
140 views

$\DeclareMathOperator\CS{CS}\DeclareMathOperator\NS{NS}$Let $G$ be an abelian first-countable topological group. Define $$\CS(G) := \{(x_n) \in G^{\mathbb{N}} \ | \ (x_n) \subset G \text{ is Cauchy-...
Score of 4
0 answers
173 views

Let $N$ be a norm on the Schwartz space $\mathcal{S}(\mathbb{R})$, and let $B(\mathbb{R})$ be the Banach space completion of $\mathcal{S}(\mathbb{R})$ under $N$, with the assumption that $B(\mathbb{R})...
Score of 2
1 answer
350 views

$\def\C{\mathcal{C}} \def\D{\mathcal{D}} \def\Z{\mathbb{Z}}$Let $A$ be a commutative unital ring. Let $I\subset A$ be a finitely generated ideal. Then the derived $I$-completion of an object $K\in D(A)...
Score of 1
0 answers
72 views

I have the following question. Suppose $(R, \mathfrak{m})$ is a local excellent ring, where $(0)$ is a primary ideal with $\mathfrak{n}=\sqrt{(0)}$, such that $R/\mathfrak{n}$ is normal. Is it true ...
Score of 1
0 answers
229 views

$\def\sA{\mathcal{A}} \def\Hom{\operatorname{Hom}}$ Lemma-Definition. Let $\sA$ be an abelian category. A complex $I\in K(\sA)$ is said to be K-injective if any of the following equivalent conditions ...
Score of 2
1 answer
725 views

I was reading the proof of Lemma 10.12 in this paper. In the second sentence, the following fact is used implicitly: Let $(R,\mathfrak{m})$ be a commutative local ring. Let $\widehat{R}$ be its $\...
Score of 7
2 answers
740 views

$\def\D{\mathcal{D}} \def\ind{\operatorname{Ind}} \def\K{\mathcal{K}} \def\A{\mathcal{A}}$Inside [GW, Remark F.168, p. 794], we find: [Let $\K$ be a category and let $\K_S$ be its localization with ...
Score of 3
1 answer
431 views

$\def\A{\mathcal{A}} \def\C{\mathcal{C}} \def\D{\mathcal{D}} \def\ind{\operatorname{Ind}} \def\op{\mathrm{op}} \def\Hom{\operatorname{Hom}}$I am trying to understand the following result from ...
Score of 2
1 answer
234 views

Let $k$ be a field, $A$ a finitely generated $k$-algebra and $I \subset A$ an ideal with $I$-adic completion $\hat{A} = \varprojlim A/I^n$. Is every finitely generated $\hat{A}$-module the completion ...
Score of 3
0 answers
130 views

I know that a function between uniform spaces $\varphi:X\to X'$ is an Hausdorff completion if and only if: $X'$ is complete Hausdorff uniform space; $\varphi$ is a dense and initial uniformly ...
Score of 2
0 answers
352 views

I would like to prove/disprove the following claim: Let $A$ be an equicharacteristic local domain, and denote by $\widehat{A}$ its completion with respect to its maximal ideal. If $\widehat{A}$ is ...

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