# Questions tagged [completion]

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### Malcev completion of free groups

Let $K$ be a field with $\operatorname{char} K=0$, $\hat{L}_n$ the complete free Lie algebra of $n$ variables $x_1,\dotsc,x_n$ and $\exp(\hat{L}_n)$ its associated group with the product given by BCH ...
1 vote
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### Completions of infinite rank Kac-Moody algebras

I am reading Kac's Infinite dimensional Lie algebras, third edition. In section 7.12, Kac discusses completions of Kac-Moody algebras of infinite rank, and define $\bar{\mathfrak{g}}(A)$, for any ...
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### Why is $\operatorname{Sym}(V)^*$ the completion of $\operatorname{Sym}(V^*)$?

Let $k$ be a characteristic zero field and $V$ be a finite-dimensional $k$-vector space. What's an "algebraic" proof that $\operatorname{Sym}(V)^*$ is naturally isomorphic to the completion ...
1 vote
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### On Noetherianity and local ness of a completed tensor product

Let $R$ be a regular local complete (with respect to the maximal ideal) ring with field of fraction $K$. Let $S\cong R[[x_1,\cdots, x_n]]/J$ (this is a Noetherian local ring which is an $R$-algebra) ...
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### $M^\wedge_I \to N^\wedge_I$ an isomorphism if $S_P^{-1}M^\wedge_P \to S_P^{-1}N^\wedge_P$ is an isomorphism for all primes $P$ containing $I$

Let $R$ be a Noetherian ring, $I \subseteq R$ an ideal, and $S \subseteq R$ a multiplicative set. Lemma 2.3 of Adam, Haeberly, Jackowski, and May's paper A generalisation of the Segal conjecture ...
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### Does CZF prove there is a minimal cauchy completion of the rationals?

In IZF, we can easily prove there is a minimal cauchy complete field extending the rationals: the dedekind reals are cauchy complete, so just intersect all of its cauchy complete subfields. CZF can ...
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### Regularity before and after completion - reference request

Put $R=\mathbb{Z}[x_1,\dotsc,x_n]$ and $I=(x_1,\dotsc,x_n)$. Let $M$ be an $R$-module that is probably not finitely generated. Suppose that the sequence $x_1,\dotsc,x_n$ is regular on $M$; I believe ...
142 views

### Exactness of $I$-adic completion in a certain non-finitely generated case

I would like the functor $$(-\otimes_{\mathbb Z} F)\hat{}: \mathbb{Z}[x_1,\dots,x_r]\text{-Mod}_{\mathrm{f.g.}}\longrightarrow \mathbb{Z}[x_1,\dots,x_r]\text{-Mod}$$ to be exact, where completion is w....
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### Does Grothendieck's algebraization imply existence of colimits of schemes?

I am a little bit confused about two lemmas regarding Grothendieck's algebraization. Assume all algebras are defined over some field. Here is the short version of my question: Does Tag 09ZT ("...
142 views

### Colimits of the infinitesimal neighborhoods of symmetric product in the category of schemes

This problem is highly related to this one and in fact it is the same question applied to a very specific situation. Given a smooth projective curve $C$, let $\text{Sym}^i(C)$ be the $i$-th symmetric ...
1 vote
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### Pro-p completion of a quotient of $U/w(U)$ is virtually nilpotent for a finitely generated free group $U$

Let $w$ be a word of a free group. Assume that $H/\overline{w(H)}$ is virtually nilpotent for every finitely generated pro-$p$ group $H$. Let $U$ be a finitely generated free group and $T$ the maximal ...
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### Rank of a finite group and its representations

$\DeclareMathOperator\Rep{Rep}\DeclareMathOperator\rank{rank}$Let $G$ be a finite group, and $C=\Rep(G)$ be the monoidal category of complex finite-dimensional representations of $G$. As $C$ is finite ...
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### Is Cauchy completion the largest extension with the same free cocompletion?

EDIT Title has been edited. Let $C$ be a category, and $$\hat{C} = [C^{op}, (Set)]$$ be its free cocompletion. Despite its name, the free cocompletion of free cocompletion is not equivalent to the ...
1 vote
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### Characteristic of ring completions

This may be a completely trivial question, but I haven’t seen it stated in any of the references I checked. Is the characteristic of a ring $R$ equal to that of its completions? This is true for the ...
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### Completion/Compactification of a Kähler metric on $\mathbb C^2$

Consider $\mathbb{C}^{2}$ equipped with the Kähler form $$\omega_{\mu}=\frac{i}{2} \partial \bar{\partial} \log \left(\left(1+|z|^{2}\right)^{\mu}+|w|^{2}\right),$$ where $\mu$ is a positive real ...
426 views

### Completed stalks of the pushforward of the structure sheaf

Let $\pi:X\to S$ be a proper morphism of Noetherian schemes. Is it possible that $\pi_*\mathcal{O}_X\neq \mathcal{O}_S$ but the natural map $\mathcal{O}_s^\wedge\to (\pi_*\mathcal{O}_X)_s^\wedge$ is ... 496 views

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### Completion and extension by scalars

Let $R\subset S$ be commutative rings, $I\trianglelefteq R$ an ideal and $M$ be an $R$-module. Suppose that 1) $R$ is Noetherian and $I$-adically complete. 2) $M$ is a finite $R$-module (hence $M$ ...
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### Is the completion of an infinitely generated module, again infinitely generated

Let $A$ be a local noetherian ring with maximal ideal $m$. Let $M$ be an infinitely generated $A$-module and $\hat{M}$ be the $m$-adic completion of $M$. Denote by $\hat{A}$ the $m$-adic completion of ...
1 vote
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### Completion of infinite projective space

I would like to ask for a reference regarding the completion of infinite-dimensional Projective Space (both Real and Complex). Since in the infinite-dimensional projective space you can take sequences ...
1 vote
145 views

### Riemann–Hurwitz Formula for Normal Projective Curves

My question refers to the proof of Theorem 7.4.16 from Liu's "Algebraic Geometry" at page 290 : QUESTION: Why does $e'_x= length_{\hat{B}}(W_{\hat{B}/\hat{A}}/\hat{B})$ hold? During the proof we ...
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### Induced morphism of completions of local rings

Let $g: A \to B$ be a local ring morphism between local Noetherian (commutative) rings $A,B$ (so $g(m_A) \subset m_B$ for the unique maximal ideals of the corresponding rings). Assume that the induced ...
244 views

### A question related to bousfield localization and nilpotent completion

I am reading Bousfield's paper entitled "The localization of spectra with respect to homology" (MSN). In that paper, Corollary 6.13 states that, if a ring spectrum $E$ has countable homotopy and ...
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### Is completion of isolated singularity isolated?

Let $K$ be an algebraically closed field and let $A=K[x_1,\dots,x_n]/I$ be a $K$-algebra of finite type which has only an isolated singularity at the origin. Let $\mathfrak{m}=(x_1,\dots,x_n)$ and ...
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1 vote
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### Completion of localization of completion

Let $(A,m)$ be a noetherian local ring, and let $p \subseteq A$ be a prime ideal. From this data, we can construct two rings: 1. We may localize $A$ at $p$, and then complete, obtaining the $pA_p$-...
356 views

### Is a filtered colimit of complete module complete?

This is probably a textbook question but i haven't been able to find a reference. Let $R$ be a complete commutative Noetherian local ring and $I$ its unique maximal ideal (I'm mostly interested in the ...
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### Is the completion of a CAT(0) open ball a closed ball?

It is well-known that the completion of a metric space which is homeomorphic to a ball can be very wild; in fact, I think, every compact manifold is the closure of an open ball! But CAT(0) spaces are ...
314 views

### Doesn't completion of a representation ring preserve its indecomposables?

For $G = PSU(3)$, it is known that $\dim I(G;\mathbb Q) / I(G;\mathbb Q)^2 = 3$, while $H^{**}(BG;\mathbb Q)$ is obviously a power series ring in two indeterminates since $G$ has rank 2. This would ...
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### Can an intersection of ideals in a Noetherian ring be replaced by a countable intersection?

Let $(R,\frak m)$ be a Noetherian local ring, and let $X$ be a set of ideals in $R$. Assume $\bigcap_{I \in X} I = 0$. Is there some sequence $\{I_n\}_{n \in \mathbb N}$, with $I_n \in X$ for all $n$...
1 vote
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### link between completion of the universal enveloping algebra and an endomorphism of functor

My question could be resume in the following way : Let $\mathfrak{t} \to \mathrm{End}(V)$ a representation of an abelian Lie algebra into an infinite dimensional vector space. What can we say ...
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### Is the completion of an injective local homomorphism still injective

Let $R$ be a regular local ring (say of dimension 2) with quotient field $F$. Let $A$ be a discrete valuation ring with the same quotient field such that $R\subset A\subset F$, $A\neq F$. Let $\hat{R}$...
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I have been looking around for examples of $R$-bad spaced in the sense of Bousfield and Kan. In their book "Homotopy limits, completions and localizations] they give several examples of such spaces ...
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### Completeness of Localizations of Completions of Commutative Rings

Let $R$ be an integral domain. Let $x,y\in R\setminus\{0\}$ be distinct. Let $\hat R$ be the $x$-adic completion of $R$ (the ring of all sequences $(r_n+Rx^n)_{n\ge0}$ where for $n\ge0$, $r_n\in R$ ...
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### Lemma in Scholze-Weinstein

In the paper "Moduli of $p$-divisible groups" by Scholze and Weinstein (see http://math.bu.edu/people/jsweinst/Moduli/Moduli.pdf), one finds the following claim in Lemma 5.2.7: Lemma: Let $K$ be a ...
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### Lie Algebras over DVRs and basechange to the completion

Let $R$ be a discrete valuation ring containing an algebraically closed field $K$ of characteristic zero and let $L$ be a Lie algebra over $R$ whose underlying $R$-module is finitely generated and ...
Let $X$ be a smooth projective irreducible curve defined over an algebraically closed field $\mathbb{K}$ (of arbitrary characteristic), and let $p\in X$ be a closed point. Denote by $\mathcal{O}_p(X)$ ...