Questions tagged [completion]
The completion tag has no usage guidance.
63
questions
4
votes
1
answer
204
views
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
0
answers
65
views
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 ...
0
votes
0
answers
121
views
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
0
answers
100
views
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) ...
2
votes
0
answers
65
views
$M^\wedge_I \to N^\wedge_I$ 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 ...
3
votes
1
answer
120
views
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 ...
2
votes
0
answers
91
views
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 ...
0
votes
0
answers
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....
7
votes
1
answer
555
views
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 ("...
2
votes
0
answers
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
0
answers
67
views
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 ...
5
votes
1
answer
295
views
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 ...
3
votes
1
answer
265
views
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
0
answers
59
views
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 ...
3
votes
1
answer
109
views
Vanishing tate of a $p$-complete spectra
I was told: if $X$ is bdd below and $p$-complete spectra then $X^{tC_q}$ vanishes for primes $q \not= p$.
I do not see how this holds.
I am aware from I.2.9 that if $X$ is bdd. below, then $X^{tC_q} \...
1
vote
0
answers
78
views
Completion of $K$-algebra of finite type with respect to the residue norm
Let $K$ be a non-archimedean field. For $n \in \mathbb{N}$ let
\begin{equation*}
T_n=\{ \sum_{\nu_1, \ldots, \nu_n \geq 0 } a_{\nu_1, \ldots, \nu_n }X_1^{\nu_1}\cdots X_n^{\nu_n} \in K[\![X_1,\ldots, ...
8
votes
0
answers
139
views
What is the relationship between free bicompletion and the Isbell envelope?
Given a small category $\mathbb C$, we can form the free cocompletion $\mathbf y : \mathbb C \to \mathcal P(\mathbb C)$ and the free completion $\mathbf y^\circ : \mathbb C \to \mathcal P^\circ(\...
6
votes
0
answers
145
views
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 ...
0
votes
1
answer
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 ...
2
votes
1
answer
496
views
Varieties with everywhere good reduction that are isomorphic over every completion have isomorphic generic fibers
Let $R$ be the ring of integers in a number field. Let $X$ and $Y$ be smooth and proper schemes over $R$. For a maximal ideal $\mathfrak{m}\subset R$ denote the completion of the localization at $\...
2
votes
0
answers
314
views
Henselization and completions of local rings & schemes
That's the second part of my coarse becoming acquainted with Henselizations of fields and local rings. (in this question we focus on local rings as it is more algebro geometric motivated). So let $(R,...
2
votes
1
answer
223
views
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 $M$ ...
0
votes
0
answers
157
views
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
0
answers
87
views
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
0
answers
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 ...
3
votes
0
answers
517
views
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 ...
4
votes
0
answers
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 ...
5
votes
1
answer
376
views
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 ...
4
votes
1
answer
350
views
Mapping cone and derived tensor product
This question is in some sense a continuation to this question: Derived Nakayama for complete modules
For the setting: Let $A$ be a ring and let $I$ be some finitely generated ideal in $A$. Let $f\...
6
votes
2
answers
1k
views
Derived Nakayama for complete modules
I have encountered the following "Nakayama Lemma" recently:
Let $A$ be a ring and $I$ some finitely generated ideal. Let $\mathcal
C_\bullet$ be a chain complex of $I$-(derived) complete $A$-...
2
votes
0
answers
88
views
Reference request : $I$-adic smoothness
The following result has been know for a while now:
Let $f:\mathfrak X \rightarrow \mathfrak Y$ be an adic morphism of (locally) Noetherian formal schemes and $K\subset \mathcal O_{\mathfrak Y}$ and ...
1
vote
0
answers
114
views
Completion in the non-noetherian case
Let $A$ be a non-noetherian, commutative $\mathbb{C}$-algebra and $X, Y$ be noetherian affine $\mathbb{C}$-schemes. Denote by $X_A:=X \times_{\mathbb{C}} \mbox{Spec}(A)$ and $Y_A:=Y \times_{\mathbb{C}}...
1
vote
0
answers
556
views
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 $pA_p$-...
3
votes
2
answers
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 ...
4
votes
0
answers
372
views
completion of finitely generated module over non-Noetherian ring
Let $A$ be a commutative ring with unity and fix $f \in A$. Any $A$-module $M$ has its $f$-adic completion, the $\hat{A}$-module $\hat{M} = \underset{n}{\lim} M/f^nM$. There is a canonical map $\hat{A}...
6
votes
2
answers
730
views
Why is $K_{\upsilon}|K$ separable for a global field $K$?
I asked this question on math.stackexchange but maybe it fits here better. If not, I apologize in advance and will remove the question.
Let $K$ be a global field and $\upsilon$ a prime of $K$. Then ...
5
votes
0
answers
229
views
Can we recover the completion of a local ring $R$ if its associated graded is the coordinate ring of a Veronese variety?
Suppose $R$ is a localization of a normal closed point of a variety of dimension $n$ over an algebraically closed field $k$ with maximal ideal $\mathfrak{m}$. Suppose also that the associated graded $\...
6
votes
1
answer
211
views
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 ...
8
votes
1
answer
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 ...
13
votes
1
answer
814
views
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
1
answer
365
views
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 ...
3
votes
0
answers
239
views
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 $\hat{R}$...
3
votes
0
answers
91
views
Example of R-bad space
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 ...
4
votes
1
answer
267
views
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 $n\ge0$, $r_n\in R$ ...
8
votes
0
answers
787
views
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 ...
5
votes
1
answer
289
views
Enriched Cauchy completions and underlying categories
The ordinary Cauchy completion $\overline{C}$ of a small category $C$ satisfies a number of conditions: Every idempotent in $\overline{C}$ splits, there's an equivalence of categories $[C^{op}, Set] \...
0
votes
1
answer
317
views
Is there a complete local analogue of the Artin-Tate lemma?
The Artin-Tate lemma states that if $A \subseteq B \subseteq C$ are commutative rings where $A$ and $C$ are Noetherian, $C$ is finitely generated as an $A$-algebra, and $C$ is finitely generated as a $...
5
votes
2
answers
211
views
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 ...
4
votes
3
answers
891
views
How is a MacNeille completion "universal" like a beta-compactification is "universal"?
The beta-compactification of a topological space is characterized as the largest space such that every mapping from the original space to another (range) space can be extended through to a mapping ...
17
votes
4
answers
4k
views
Completion of a local ring of a curve
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 $\mathcal{O}_p(X)$ ...