Questions tagged [formal-completions]

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complete ring is always Hausdorff? [duplicate]

I think complete ring $R$ with respect to some ideal $I⊂R$ is always Hausdorff, because the topology is inherited from direct some of discrete space, which is Hausdorff. For example, well known ...
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23 views

Does taking integral closure and completion commute?

For a Commutative ring $R$ with total ring of fractions ( https://en.m.wikipedia.org/wiki/Total_ring_of_fractions) $K$, let $\overline R$ denote the integral closure of $R$ in $K$. Let $(R,\mathfrak m)...
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29 views

Few questions regarding formal vector bundles on formal completions.

This question is regarding formal vector bundles on formal schemes. The formal schemes that I'm interested are formal completion of a scheme along a closed sub-scheme. The category of formal coherent ...
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1answer
27 views

Completion of nonstandard numbers allowed and isomorphic to non-negative, non-infinitesimal, integer hyperreals?

I recently learned about nonstandard number systems satisfying Peano's axioms (when they use only first order logic). And this gives us what is often described by "$\mathbb N$ with densely many ...
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36 views

Completion of $R[x,x^{-1}]$ at the ideal $(1+x)$ [duplicate]

Consider the ring $R[x,x^{-1}]$ where $R$ is a commutative ring. I want a description of the ring after completing at the ideal $(1+x)$. So my guess is that it is isomorphic to $R[[y]]$, where the ...
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1answer
53 views

Error in definition of convergence of sequence in completion? [closed]

From Eisenbud's Commutative Algebra page 194 We have the $\frak m$-adic filtration for some ideal ${\frak m} \subset R$ and we have $\hat{\frak m}_n=\ker(R\to \hat{R})$. We shall say that a sequence ...
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30 views

General element of completion of a ring [duplicate]

I want to prove the statement that completion of a ring w.r.t. maximal ideal is local. Let $\mathfrak{m}$ be a maximal ideal of a ring $A$, and $\hat{A}$ be an $\mathfrak{m}$-adic completion of $A$. I ...
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1answer
41 views

If $\widehat{M}$ is a free $\widehat{R}$-module of rank $n$ then $M$ has a generating set of $n$ elements as an $R$-module.

With reference to my last question If $\widehat{M}$ is a free $\widehat{R}$-module, then $M$ is a free $R$-module, $R$ is a Zariski ring. I want to ask the following question. Let $R$ be a Zariski ...
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2answers
72 views

Complete DVR containing a field isomorphic to residue field

Let $(R, \mathfrak m, k)$ be a ($\mathfrak m$-adically) complete DVR containing $k\cong R/\mathfrak m$. Also assume $k$ is algebraically closed. Then, is it true that we always have a $k$-algebra ...
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81 views

$\mathbb{Z}_p=\varprojlim \mathbb{Z}/p^{n}\mathbb{Z}$ is uncountable.

The ring of $p$-adic integers is given by $\mathbb{Z}_p=\varprojlim \mathbb{Z}/p^{n}\mathbb{Z}$. From this description how can we conclude that $\mathbb{Z}_p$ is uncountable ? It follows from the ...
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1answer
58 views

Relationship Between the Dimension of an $R / \mathfrak m$-Vector Space and Its Completion in the $\mathfrak m$-adic Topology

Consider a Noetherian local ring $(R, \mathfrak m, k).$ We will denote by $\widehat -$ the completion of $-$ with respect to the $\mathfrak m$-adic topology, i.e., the topology on $R$ in which a ...
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1answer
71 views

If $\widehat{M}$ is a free $\widehat{R}$-module, then $M$ is a free $R$-module, $R$ is a Zariski ring.

Let $R$ be a Zariski ring with $I$-adic topology, $I \subset J(R)$. Let $M$ be a finitely generated $R$-module. Now I have to show that if the $I$-adic completion $\widehat{M}$ is a free $\widehat{R}$-...
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1answer
95 views

Defining p-adic numbers via formal competion: Question about continuity of multiplication

Consider the following: Why does the multiplication function $$\cdot: \Bbb{Q} \times \Bbb{Q} \to \Bbb{Q}$$ extend to the completion? I thought the universal property of completion says that uniformly ...
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1answer
71 views

Defining p-adic numbers via a formal completion.

Consider following fragment of the definition of p-adic numbers in "A course in abstract harmonic analysis" by Folland: So we have that $+: \Bbb{Q} \times \Bbb{Q} \to \Bbb{Q}$ is continuous ...
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45 views

Does the exist any ring embedding from $\mathbb{Z}_p$, the ring of $p$-adic integers to $\mathbb{R}$?

Does there exist any ring embedding from $\mathbb{Z}_p$, the ring of $p$-adic integers to $\mathbb{R}$ ? One can realise $\mathbb{Z}_p=\varprojlim \mathbb{Z}/p^{n}\mathbb{Z}$. I can't produce any ...
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1answer
69 views

Completion of a polynomial ring over a complete ring

I'm learning about ring completions, and this question came to mind: If $R$ is a complete local ring with maximal ideal $\mathfrak{m}$ (e.g. $R = \mathbb{Z}_p$ or $R = k[[x]]$), is the completion of $...
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80 views

$R$ be a Noetherian semi local ring such that $R/N(R)$ is complete $\mathrm{Jac}(R)$-adically, then $R$ is complete $\mathrm{Jac}(R)$-adically.

Let $R$ be a Noetherian semi local ring, and $I=\text{N}(R)$, $J=\text{Jac}(R)$. If $R/I$ is complete w.r.t. $J$-adic filtration then we have to show that $R$ is complete in $J$-adic filtration. Thus ...
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42 views

If $I$ is contained in the Jacobson radical of a Noetherian ring $A$ then $A\to \widehat{A}$ is faithfully flat.

Let $A$ is a noetherian ring, $I\subseteq A$ is an ideal, and $\widehat{A}$ is the $I$-adic completion of $A$. If $I$ is contained in the Jacobson radical of $A$ then I have to show that $A\to \...
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34 views

How to show that $\hat{R} \cong \prod_{i=1}^{r}\hat{R_{m_i}}$.

Let $\mathfrak m_1,\ldots,\mathfrak m_r$ be distinct maximal ideals of a Noetherian ring $R$ and $I=\bigcap_{i=1}^{r}\mathfrak m_{i}$. Let $\widehat{R}$ be the $I$-adic completion of $R$ and $\widehat{...
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Kernel of $\mathbb{Z}_p \to \mathbb{Z}/p^{n}\mathbb{Z}$ equals to $p^n \mathbb{Z}_p$.

Let $\mathbb{Z}_p$ denotes the ring of $p$-adic integers, i.e., $\mathbb{Z}_p:= \varprojlim \mathbb{Z}/p^{n}\mathbb{Z}$. Then consider the projection map $\pi_{n}: \mathbb{Z}_p \to \mathbb{Z}/p^{n}\...
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1answer
22 views

In a filtered module $\sum x_n$ converges if $x_n$ tends to 0 in $M$

Let $M$ be a filtered module which is Hausdorff and complete with respect to the topology defined by the filtration. I want to show that if the sequence $\{x_n\}$ tends to $0$ the series $\sum x_n$ ...
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1answer
30 views

Formal power series: are subgroups of the multiplicative group closed?

I have a subgroup $G$ of the multiplicative subgroup of the ring of formal power series on n coefficients, $(\mathbb{Z}[[X_1,...,X_n]])^*$, and an element $a = \sum_{i \in I} a_i X_1^{i_1} ... X_n^{...
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40 views

Sufficient condition for completeness of a Noetherian Local ring.

I am trying to show if $(R,\mathfrak m)$ is a Noetherian local ring and $I\subset R$, a nilpotent ideal is such that $R/I$ is $\mathfrak m$-adically complete, then $R$ is $\mathfrak m$-adically ...
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Grillet, Proposition VI.9.2 (minor detail)

Having a quick problem with a line in Grillet's Abstract Algebra, Prop. VI.9.2 on p. 267-8 on the topic of Filtrations and Completions. He says that the ideal $$\widehat{\mathfrak{a}}_j = \{ (x_1 + \...
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36 views

Completion of finite local homomorphism.

Suppose that $(A,\mathfrak{m}_A)$ and $(B,\mathfrak{m}_B)$ are local rings and that $\varphi:A\rightarrow B$ is a local homomorphism (I am happy to assume that this morphism is quasifinite). Let $M$ ...
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1answer
119 views

On descending chain of ideals with zero intersection in a complete semi-local ring

Let $R$ be a Noetherian semi-local ring, let $\mathfrak m_1,...,\mathfrak m_n$ be the finitely many maximal ideals. Let $J=\mathfrak m_1\cap ...\cap \mathfrak m_n$ denote the Jacobson radical of $R$ . ...
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36 views

Polynomial ring over an analytically unramified ring is locally analytically unramified

Let us call a Noetherian local ring $(R, \mathfrak m)$ to be analytically unramified if the $\mathfrak m$-adic completion of $R$ is reduced i.e. has no non-zero nilpotent. https://en.m.wikipedia.org/...
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1answer
66 views

image in formal completion ring

Let $X$ be a scheme and $C\subset X$ be an integral subscheme and $I$ be a sheaf of ideals over $X$. Denote $mult_C I:=Sup\{n\in \Bbb N: I\mathcal O_{C,\xi_C} \subset \mathfrak m_{\xi_C}^n\}$ the ...
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1answer
48 views

On the associated graded ring , corresponding to an $\mathfrak m$-primary ideal , of the $\mathfrak m$-adic completion

Let $(R, \mathfrak m)$ be a Noetherian local ring. Let $\hat R$ be the $\mathfrak m$-adic completion of $R$. $J$ be an ideal of $R$ with $\sqrt J=\mathfrak m$. Let $\hat J =J\hat R$ be the $\mathfrak ...
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1answer
59 views

Construction of enveloping group of a monoid

Let $M$ be a monoid and let $G$ be the group which it generates. $G$ can be described as the group obtained from $M$ by adjoining formal inverses. Despite this simple description, I am trying to ...
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93 views

On the integral closure of Noetherian local domain in a finite extension field of the fraction field

Let $(R, \mathfrak m)$ be a Noetherian local domain such that the $\mathfrak m$-adic completion of $R$ is also an integral domain. Let $K$ be the fraction field of $R$ . Let $L$ be a finite extension ...
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73 views

convergence of the p-adic log in $pZ_p$

I've been trying to solve the following problem: Show that the series $\log(1 + x) = \sum_{n = 1}^{\infty} (-1)^{n+1} x^n/n$ converges on $p\mathbb{Z}_p$ with respect to $|.|_p$ (p-adic absolute ...
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1answer
107 views

Does the completion of a local Gorenstein ring has finite injective dimension over the original ring?

Let $(R, \mathfrak m)$ be a local Gorenstein ring and $\hat R$ be its $\mathfrak m$-adic completion. So we have a canonical map $R \to \hat R$ which makes $\hat R$ into an $R$-module. My question is: ...
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1answer
130 views

On homological dimensions of finitely generated modules over a local ring and its completion

Let $(R,\mathfrak m)$ be a Noetherian local ring. Let $\hat R$ be the $\mathfrak m$-adic completion of $R$ so we have a canonical map $R \to \hat R$, which makes $\hat R$ into a $R$-algebra, so every $...
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1answer
70 views

finitely tailed Laurent series as the completion of $k(T)$

I've been trying to solve the following problem: Show that $k((1/T))$, the ring of finitely tailed Laurent series in $1/T$ with coefficients in $k$, is the completion of $k(T)$ with respect to a ...
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1answer
35 views

Completion $K \otimes_R \text{lim} R /I^n$ is a field

Let $K$ be a field with valuation ring $R \subset K$. Let $\mathfrak{m}$ be the maximal ideal of this valuation ring. Form the inverse limit $\hat{R} = \text{lim}_{I \text{ a nonzero ideal of } R} R/...
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1answer
97 views

Completion of $\mathbb{Q}$. Is this the Adele Ring?

Let $I =\{ n \mathbb{Z} : n \in \mathbb{Z} \}$ be the inverse system of nonzero ideals of $\mathbb{Z}$. Define an $I$-cauchy sequence in $\mathbb{Q}$ to be a sequence $\{ a_n \}_{n \in \mathbb{N}_{\...
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1answer
26 views

If $M$ is a $R$-module in a $K$-vector space $V$, is $\widehat M \cap V = M_{\mathfrak p}$?

Let $R$ be a Noetherian integral domain with field of fractions $K$. Let $\mathfrak p$ be a prime ideal of $R$ and assume that the $\mathfrak p$-adic completion $\widehat R$ is a domain. Write $\...
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1answer
53 views

Completion given a partial order

Let $A$ be a ring. Suppose we have a partially ordered set $X$ of nonzero ideals of $A$, which is closed under intersection, summation and product of ideals. We can consider $\text{lim}_{I \in X} A /I$...
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1answer
59 views

$p$-adic completion of $\mathbb Z((t))$

Consider the ring of formal Laurent power series $\mathbb Z((t))$ and let $p$ be a prime. Let $(p)$ be the ideal in $\mathbb Z((t))$ generated by $p$: What is an explicit expression of the $(p)$-...
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51 views

Completion of a stalk not integral domain

We consider the closed subscheme $X:= V(Y^2-X^2(x+1))$ of affine plane $\mathbb{A}^2_k= Spec \text{ } k[X,Y]$. field $k$ is arbitrary. let $x:=(0,0)$ the point representing the prime ideal $(X,Y) \...
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255 views

Etale iff Completions are Isomorphic

I have a question aboutthe proof of proposition 4.3.26 from Liu's "Algebraic Geometry and Arithmetic Curves": Let $Y$ be a locally Noetherian scheme and $f:X \to Y$ a morphism of finite type. Fix a ...
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65 views

Excellent Local Ring: Properties

Let $(R,m)$ be an excellent local ring. Denote by $\phi: R \to \hat{R}$ the canonical completion map and $f: \operatorname{Spec}(\hat{R}) \to \operatorname{Spec}(R)$ the induced map of Specs. Let $Q \...
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1answer
145 views

The free completion of a category under sifted colimits

The free completion of a category under sifted colimit is denoted by $$Sind \ \cal K.$$ If $\cal K$ has sifted colimits we have the functor as in the snippet below:$$\text{colim}:Sind\ \cal K\to K.$$ ...
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1answer
199 views

Questions about Formal Schemes

Set $X$ be a scheme and $Y \subset X$ a closed subscheme given locally by ideal sheaf $I \subset \mathcal{O}_X$. Then there exist formalism constructing from pair $(Y,I)$ the induced formal scheme $\...
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1answer
126 views

Functor of points of the completion of a ring

Let $R$ be a ring, with ideal $I$, and let $\widehat{R}_I$ be the completion $\varprojlim R / I^n$ of $R$. Can I somehow describe the functor $\mathrm{Hom}(\widehat{R}_I,?) : Rings \to Sets$ in an ...
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1answer
167 views

Computation of completion of a local ring

Let $X=\mathrm{Spec}(\mathbb{R}[a,b]/(a^2+b^2+1))$ and consider the closed point $p=(a)$. I would like to compute the completion of $\mathcal{O}_{X,p}$ w.r.t. to its maximal ideal $\mathfrak m$. ...
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1answer
105 views

Is $\hat{G}$ is complete with respect to the induced topology of $G$?

For a topological group $G$ and a given fundamental system of neighbourhoods of $G$ we can define the completion of G and we call it $\hat{G}$. The induced fundamental system of neighbourhoods of $\...
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1answer
176 views

injective homomorphism of local rings and completions

Assume that $f:A\to B$ is an injective and local homomorphism between two local rings. Let $\hat A$ and $\hat B$ be respectively the completions of $A$ and $B$ with respect the maximal ideals. Then we ...
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36 views

A local homomorphism that induces a continuous embedding of complete discrete valuation fields

Let $f:A\to B$ be a local homomorphism between DVR and denote with $K(A)$ and $K(B)$ the function fields of $A$ and $B$ respectively. Moreover assume that we have an embedding $i: K(A)\to K(B)$. My ...