All Questions
Tagged with local-rings localization
53 questions
2
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0
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42
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Localization of profinite integers
Let $p$ be a prime number, $\mathbb{Z}_{(p)}$ the localization of $\mathbb{Z}$ at $p$ and $\widehat{\mathbb{Z}} = \prod_{q\text{ prime}} \mathbb{Z}_q$ the profinite completion of the integers. Is the ...
- 3,030
0
votes
1
answer
96
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Is a localization of a local ring local?
I'm trying to answer the following question: " If $R$ is a local ring and $S$ a multiplicative set, is $S^{-1}R$ a local ring?"
I found out in this other question that a localization is ...
1
vote
1
answer
56
views
Does the uniformizer $\pi$ generate the extension $K/T$ where $T$ is the inertia field?
I'm trying to prove the equivalence of the statements
(1) $\sigma(x) \equiv x$ mod $\mathfrak{p}^{i+1}$ (that is, $\sigma \in V_i$, the $i^{th}$ ramification group) for all $x\in \mathcal{O}_{K,\...
- 539
0
votes
0
answers
62
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Using Nakayama's lemma in non-local ring
Let $R$ be a Noetherian integral domain of dimension one and $\mathfrak{m}$ an ideal such that $\text{dim }\mathfrak{m}/\mathfrak{m}^2=1$ as an $R/\mathfrak{m}$-vector space. The localization of $R$ ...
- 323
1
vote
1
answer
40
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How to compute a minimal spanning set and the minimal spanning number of $\Bbb C[x,y]_{(x-1,y-1)}/(x^3-y^2)$?
Let $A=\frac{\mathbb C[X,Y]}{(X^3-Y^2)}$. I am asked to show that $\mathbf m=(\overline X-1,\overline Y-1)$ is a maximal ideal of $A$ which I have shown successfully. Now I am asked to compute the $\...
- 3,801
0
votes
0
answers
81
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Free module over local ring $R$. [duplicate]
People often say that a module $M$ over a not necessarily neotherian local ring $R$ being projective is flat, and also free. However, some refer to the finitely-generatedness of $M$, i.e. $M$ being ...
- 893
3
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0
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114
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Is the ring of formal power series a localization of some non local ring at prime ideals?
Consider $F = K[[x]],$ the formal series over field $K.$ We know that $K$ is a local ring with maximal ideal $(x).$ Does there exist a non local ring $R$ and prime ideal $p$ such that $R_p = K[[x]]?$ ...
- 793
4
votes
2
answers
163
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Multiplicity of intersection of $y^2=x^3$ and $x^2=y^3$ at the origin
Those curves intersect at the origin with multiplicity 4, if I did everything correctly. In fact, parametrizing by $t \mapsto (t^2,t^3)$ the first curve and plugging into the second, yields $t^4=t^9$, ...
- 1,587
2
votes
1
answer
210
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stalk isomorphism + homeomorphism = isomorphism of varieties
I'm going over some old exercises in Chapter I of Hartshorne and am stuck on the reverse direction of Exercise I.3.3(b), which should follow straight from definitions. If $\varphi:X\to Y$ is a ...
- 265
1
vote
0
answers
79
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Definition of $k$-rational points on an algebraic set
I am learning introductory algebraic geometry by myself. Probably I am misunderstanding something. Could you point out where I mistake?
Let $k$ be a field that is not necessarily an algebraically ...
- 425
1
vote
1
answer
38
views
Dimension of $(A[[X]])_\mathfrak{m} \geq k+1$, regular ring, chain of prime ideals
Assume $A$ is a regular ring and $m$ a maximal ideal of $A$. We define $R= A[[X]]$.Then $\mathfrak{M}=mR + XR$ is a maximal ideal of $R$. Lets assume $h(mA_m) = k$.
I want to show, that
\begin{align*}
...
- 25
1
vote
1
answer
58
views
Localization in power series
Let $A$ be a comm. ring with unity.
Say $\mathfrak{M}$ is a maximal ideal of $A[[X]]$.
Is following statement generally true?
\begin{align*}
(A[[X]])_\mathfrak{M} \cong (A_{\mathfrak{M} \cap A}[[X]])_\...
- 25
1
vote
0
answers
93
views
Why is a semilocal domain $A$ a UFD, if its localisations at maximal ideals $A_{\mathfrak m}$ are UFDs?
This is Exercise 20.2 in Matsumura's Commutative Ring Theory:
Let $A$ be an integral domain. We say that $A$ is locally UFD if $A_{\mathfrak m}$ is a UFD for every maximal ideal $\mathfrak m$. If $A$ ...
- 10.4k
1
vote
1
answer
73
views
When is the maximal ideal of the localization principal?
My friend asked me the following exercise in commutative algebra:
Let $F\in \mathbb{C}[X,Y]$, and $R:=\mathbb{C}[X,Y]/(F)$. Denote by $x,y$ the residue classes of $X,Y$ in $R$, and let $\mathfrak{m}:=...
- 1,511
0
votes
1
answer
85
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Why $\mathfrak{p}A_\mathfrak{p} = 0$, where $A_{\mathfrak p}$ is the localization at the kernel $\mathfrak p$ of a surjective ring homomorphism.
Let $A$ be a commutative, Noetherian, local ring, $\mathfrak{O}$ a discrete valuation ring and $\lambda : A \rightarrow \mathfrak{O}$ be an epimorphism. Let $\mathfrak{p}=\ker(\lambda)$, and consider ...
- 763
2
votes
1
answer
251
views
Local rings of $V(y^2-x^3)$
I want to find the local rings of $V(y^2-x^3)$, and establish if it's isomorphic to $K[x]_{(x)}$, or maybe some other ring which I don't know.
We take $p=(t^2,t^3) \in V$ and we want to find $O_{(t^2,...
- 533
0
votes
0
answers
68
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Condition for conormal module of commutative, Noetherian, local ring to have finite length
Let $A$ be a commutative, Noetherian, local ring, $O$ a discrete valuation ring and $\lambda : A \rightarrow O$ be an epimorphism. Let $p=\ker(\lambda)$, and consider the conormal $A$-module $p/p^2$. ...
- 763
1
vote
1
answer
110
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Computing residue fields of affine schemes
I have taken a course on schemes, so I am familiar with the basic definitions, but I'm very rusty and I've forgotten how to do this (if I ever knew).
Basically, I want to compute the residue fields of ...
- 3,931
1
vote
1
answer
79
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If $\operatorname{Idem}B\rightarrow \operatorname{Idem}(B/mB)$ is surjective then $B$ is a product of local rings
Let $A$ be a local ring with maximal ideal $m$ and $B$ a finite $A$-algebra (by finite I mean that $B$ is a finitely generated $A$-module). If we denote by $\operatorname{idem}B$ (respectively $\...
- 1,046
1
vote
1
answer
89
views
When an element of the total ring of fractions of a noetherian local ring $R$ is integral over $R$?
Let be $(R,\mathfrak{m},k)$ a Noetherian local ring. Denote by $Q$ the total ring of fractions of $R$. Let $I$ be an ideal of $R$ and $r \in I$ a $R$-regular element.
I have the following question: ...
- 185
1
vote
1
answer
253
views
Maximal Cohen-Macaulay modules of full support, over non-artinian local Cohen-Macaulay rings, are faithful?
Let $(R,\mathfrak m)$ be a local Cohen-Macaulay ring of positive dimension. Let $M$ be a finitely generated maximal Cohen-Macaulay module i.e. $\operatorname{depth} M=\dim R$.
If $\operatorname{Supp}(...
- 3,715
6
votes
1
answer
129
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Let $A$ be an integrally closed noetherian domain, and $(R, \mathfrak{m})$ local with $A \subseteq R \subseteq K(A)$. Is $R$ a localization of $A$?
Let $K(A)$ denote the fraction field of $A$. For context, I'm trying to prove $A = \bigcap_{\text{ht}(\mathfrak{p}) = 1} A_{\mathfrak{p}}$ for an integrally closed domain $A$, from Atiyah-Macdonald's ...
- 5,809
3
votes
2
answers
163
views
Question in a proof from Gathmann's notes on Algebraic Geometry: The tangent space.
I am studying chapter 10 from Gathmann's notes about algebraic geometry and there is something I don't understand in the following proof.
I don't really understand the equality $\frac{g}{f} = c g$. ...
- 4,368
1
vote
0
answers
65
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Properties of $k[x(x-1)]_{\langle x(x-1) \rangle} \subseteq k[x]_{\langle x \rangle}$
Let $k$ be aa arbitrary field.
Let $R=k[x(x-1)]_{\langle x(x-1) \rangle}$ and let $S=k[x]_{\langle x \rangle}$,
$m=x(x-1)R$, $n=xS$, $k(m)=R/m$, $k(n)=S/n$.
We have, $mS = n$ (since $x-1$ is ...
- 6,937
1
vote
2
answers
250
views
Is a localization at a maximal ideal of a polynomial ring a perfect ring?
There are several equivalent definitions for a perfect ring $R$ (not necessarily a commutative ring), for example: Every flat left $R$-module is projective;
see wikipedia.
Also, there is the notion of ...
- 6,937
0
votes
0
answers
122
views
Flat and algebraic (non-integral) local rings extension $R \subseteq S$ with $m_RS=m_S$
Let $R \subseteq S$ be two Noetherian local rings (not necessarily regular) which are integral domains,
with $m_RS=m_S$, namely, the ideal in $S$ generated by $m_R$ (= the maximal ideal of $R$) is $...
- 6,937
5
votes
1
answer
178
views
A sandwich theorem for local rings
The following question seems natural to ask in view of this question and its comments/answers:
Let $R \subseteq S$ be commutative Noetherian rings, let $q$ be a maximal ideal of $S$,
$p$ a maximal of $...
- 6,937
4
votes
1
answer
3k
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Definition of homomorphism of local rings
Let $(R,m_R)$, $(S,m_S)$ be two local rings.
By definition, a local homomorphism of local rings is a ring homomorphism $f: R \to S$
such that the ideal generated by $f(m_R)$ in $S$ is contained in $...
- 6,937
0
votes
0
answers
105
views
Flatness of localizations
Let $R \subseteq A$ be two $\mathbb{C}$-algebras, $P$ a prime ideal of $R$, $Q$ a prime ideal of $A$, and $Q \cap R = P$. Assume that $(A_Q,QA_Q)$ is flat over $(R_P,PR_P)$.
'When' $A$ is flat over $...
- 6,937
0
votes
0
answers
63
views
A result concerning local rings
The following result appears in the book 'Commutative Algebra with a View Toward Algebraic Geometry' by David Eisenbud:
Theorem 18.16* Let $(R,P)$ be a regular local ring, and let $(A,Q)$ be a local ...
- 6,937
2
votes
1
answer
257
views
Show that $R^* = R\setminus \mathfrak{m}.$
Here is the question I want to answer:
A commutative ring $R$ is local if it has a unique maximal ideal $\mathfrak{m}.$In this case, we say $(R, \mathfrak{m})$ is a local ring. For example, if $R$ is ...
user778657
0
votes
2
answers
47
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Examples of local ring with principal group units $U_{1}=U_{1}^{2}$
Let $A$ be a local ring with maximal ideal $M$. if we consider $U_{1}=1+m$.
I am trying to find some examples of local rings where the condition $U_{1}=U_{1}^{2}$ holds.
I was thinking like local ...
- 683
2
votes
1
answer
94
views
Local behaviour of a module localized at a prime ideal
Let $R$ be a commutative ring and $p,q$ be two prime ideals of $R$ with $q\subset p$. We know $(R_p)_{qR_p}\cong R_q $ as rings. Let $M$ be an $R$-module. Is it true that $(M_p)_{qR_p}\cong M_q$ as $...
- 1,574
0
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1
answer
81
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Does a closed point of a scheme have an affine open environment with the same dimension?
Consider a scheme $X$, and an a closed point $x\in X$. I am wondering whether there is an affine open neighborhood $x\in U\subseteq X$ such that
$$\dim \mathcal O_{X,x}=\dim U.$$
I tried the ...
- 340
2
votes
0
answers
170
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Nagata rings: localization and finite algebra
A Nagata (pseudo-geometric) ring is a Noetherian ring $R$, such that $R/\mathfrak{p}$ is Japanese (N-2) for every prime $\mathfrak{p} \subseteq R$.
Matsumura claims in his book "Commutative Algebra" ...
- 126
1
vote
1
answer
265
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Localization in a ring which is not an integral domain
In an integral domain $A$, localization by a prime ideal $\mathfrak p$ (obtaining the local ring $A_\mathfrak p$) can intuitively be thought of as simply formally inverting all the elements of $A\...
- 3,587
0
votes
1
answer
202
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Example for being an integral domain is not a local property.
Let $K$ be a field and $R=K\times K$ the product ring. We know this ring is not integral domain. But for all $P \in Spec(R)$, $R_P$ (the localization of $R$ at $P$) is integral domain. I know this ...
0
votes
0
answers
30
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Difference between two localizations
What is the difference between $\mathbb{C}[x]_{(x)}[y]$ and $\mathbb{C}[x,y]_{(x)}$? To me they are both equal to:
$\{ \frac{f(x, y)}{g(x,y)} | g(0, y) \neq 0 \}$
Is this true and if it isn't can ...
- 157
2
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0
answers
49
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Is the localization of a complete $A$-module $B$ complete? (Where $(A,\mathfrak{m})$ is $\mathfrak{m}$-adically complete)
Let $(A, \mathfrak{m})$ be a local noetherian ring and let $A \subset B$ be a finite $A$-algebra. Assume that $A$ is $\mathfrak{m}$-adically complete. It follows that $B$ is $\mathfrak{m}$-adically ...
- 127
3
votes
2
answers
818
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A criterion for a module over a local ring to be of finite length
Let $R$ be a local ring with maximal ideal $\mathfrak{m}$ and let $M$ be an $R$-module. For a prime ideal $\mathfrak{p}\in R$, $M_p$ shall denote the localization of $M$ at $p$.
I've read the ...
user501184
0
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1
answer
312
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Localization of $A$ at the multiplicative set of of all functions that do not vanish outside of $V(f)$ --- Vakil's 4.1.1
I am confused about the Definition 4.1.1 and Exercise 4.1.A in Vakil's notes on Algebraic Geometry.
4.1.1. Definition. Define $\mathcal{O}_{\mathrm{Spec}A}(D(f))$ to be the localization of $A$ at the ...
- 17.1k
0
votes
0
answers
97
views
How can I show flatness of this morphism using local algebra?
Consider the obviously non-flat morphism of schemes
$$
\text{Spec}(\mathbb{C}[x,y]/(xy)) \to \text{Spec}(\mathbb{C}[x])
$$
How can I show that it is not flat using local algebra? I've looked at a few ...
- 3,303
1
vote
0
answers
112
views
On simplifying $mR_m/m^2R_m$ where $m$ is maximal ideal of $R$
Let $R$ be a commutative ring and $m$ be a maximal ideal ; consider the local ring $R_m$ got from localizing $R$ at $m$ ; then $mR_m$ is its unique maximal ideal . Then is it true that $m/m^2 \cong ...
user456828
-4
votes
1
answer
53
views
Minimal no. of generators of $mA_m$ , where $m$ is the maximal ideal $(\bar x -1 , \bar y -1)$ of $A=\mathbb C[x,y]/(x^3-y^2)$ [closed]
Let $A=\mathbb C[x,y]/(x^3-y^2)$ and consider the maximal ideal $m=(\bar x -1 , \bar y -1)$ of $A$ . Then how to compute the minimal no. of generators , $\mu(mA_m)$ , of $mA_m$ ?
- 4,476
2
votes
3
answers
470
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A question of the uniqueness of the prime ideal lying over in an integral extension (Corollary 5.9 in Atiyah's Commutative Algebra)
I am reading Atiyah's Commutative Algebra.
I can understand all the proof of Corollary 5.9 except the sentence $\mathfrak{n}^c=\mathfrak{n}'^c=\mathfrak{m}$.
Question 1.
Why $\mathfrak{n}^c=\mathfrak{...
- 3,841
3
votes
0
answers
1k
views
Localization and faithfully flat algebras
Let $A,B$ be commutative rings with unit.
I'm trying to prove that if $f:A\to B$ is flat and $q\subset B$ is a prime ideal with $p=f^{-1}(q)$, then $\hat{f}:A_p\to B_q$ is faithfully flat, where $\hat{...
- 10.5k
0
votes
1
answer
52
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Equality between rank and dimension [closed]
Let $M$ a finitely generated module over a noetherian commutative ring $A$. Assume that $M_{\mathfrak p}$ is a free $A_{\mathfrak p}$-module. For any prime $\mathfrak p$ and let's put $k(\mathfrak p):...
- 1,937
0
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1
answer
1k
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What is the localization of $\mathbb{Z}_{(p)}$ at $p\mathbb{Z}_{(p)}$?
Let $p$ be prime so that $\mathbb{Z}_{(p)}$ is the local ring with unique maximal ideal $\mathfrak{m}=p\mathbb{Z}_{(p)}$. What is the localization $(\mathbb{Z}_{(p)})_\mathfrak{m}$? Intuitively I ...
- 257
4
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0
answers
165
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When is $RS^{-1}$ a local ring?
Suppose we have a noncommutative ring $R$ and multiplicatively closed set that is both right Ore, and right reversible, i.e. it is a right denominator set. Now, we can localize $R$ at $S$ to form $RS^{...
- 1,431
1
vote
0
answers
1k
views
Power of maximal ideal in the localization of $k[x,y]$ at $(x,y)$
Let $k$ be an algebraically closed field, let $R_p$ be the localization of $R=k[x,y]$ at the maximal ideal $p:=(x,y)$, i.e. $R_p:=\{F/G \in k(x,y) : g(0,0)\ne 0\}$. Then $R_p$ is a local ring with $...
user228168