The local-rings tag has no wiki summary.
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Possibilities for dimensions of $\mathfrak{m}^i/\mathfrak{m}^{i+1}$ for a local ring
Let $R$ be a local commutative ring with maximal ideal $\mathfrak{m}$, and denote by $k$ the residue field $R/\mathfrak{m}$. Then we can look at the sequence of $k$-vectorspaces
$$R/\mathfrak{m}, ...
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1answer
121 views
Does the normalization morphism induce isomorphism on residue fields?
The question is basically coming from the following situation:
Let $C$ be an integral curve over a field $k$ (EDIT and assume that $k$ is not algebraically closed) and let $\phi\colon C^N\to C$ be the ...
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2answers
180 views
Condition for a local ring whose maximal ideal is principal to be Noetherian
The statement "a local ring whose maximal ideal is principal is Noetherian" is (I think) false. The ring of germs about $0$ of $C^\infty$ functions on the real line seems to be a counterexample since ...
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101 views
Automorphism on F_2[[X,S]]
Let us define the automorphism $\sigma$ on ${\Bbb F}_2[[X,S]]$ such that
$\sigma \colon S \mapsto S + S^2 + S^3$
$\sigma \colon X \mapsto X + S$.
It is easy to see that the ideal $(S)$ is stable ...
4
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1answer
206 views
Inverse limit of Gorenstein local rings is again Gorenstein?
If we have the system of surjective ring homomorphisms
$f_{i,i+1}: R_{i+1} \twoheadrightarrow R_i$
for an arbitrary $i \geq 0$ such that all $R_i$ are Gorenstein local ring. Let us put
...
3
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1answer
286 views
Automorphisms of complete discrete valuation ring
Let ${\Bbb F}_2[[T]]$ be a c.d.v.r over ${\Bbb F}_2$. We consider the automorphism $\sigma$ of ${\Bbb F}_2[[T]]$ such that $\sigma \colon T \mapsto T + c_2T^2 + c_3T^3 + \cdots$ with $c_i \in {\Bbb ...
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81 views
Super-Gorenstein ideal of ${\Bbb F}_p[[X_1,\ldots,X_n]]$
Let $A \colon= {\Bbb F}_p[[X_1,\ldots,X_n]]$ be a $n$-variable power series ring over a finite field ${\Bbb F}_p$. We put ${\frak m}_A \colon= (X_1,\ldots,X_n)$.
Definition(Super-Gorenstein ideal): ...
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94 views
Universally catenary and all its formal fibers over minimal members are Cohen-Macaulay but it has a nonCohen-Macaulay formal fiber
Please help me to find a Noetherian local ring $R$ such that: $R$ is universally catenary and all its formal fibers over minimal members of $Spec(R)$ are Cohen-Macaulay but $R$ has a nonCohen-Macaulay ...
2
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1answer
68 views
Rings in which every J-matrix is non-singular
Let $R$ be a ring with identity. A matrix $A=[a_{ij}] \in M_n(R)$ is called a J-matrix if for any $i$, $a_{ii} \not \in J(R)$ but for any $i \not = j$, $a_{ij} \in J(R)$. Now suppose that every ...
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1answer
120 views
A question on local rings
Let $R$ be a finite local ring (with identity) with exactly one minimal left ideal. Is it necessarily true that $R$ has exactly one minimal right ideal !?
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Higher-dimensional generalization of Pink's theorem
Pink's theorem in the title of the question refers to the main theorem of Pink's paper "Compact Subgroups of Linear Algebraic Groups" that appeared in Journal of Algebra (206) in 1998. It essentially ...
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1answer
122 views
Local rings with simple radical
Is there a (finite) non-commutative local ring $R$ (containing identity) such that $J(R)$ is simple as a left module?
3
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0answers
222 views
PAC field : Algebraically closed field :: ? : Henselian local ring
I'm wondering if the following exists in the world as a definition. I'll use the word "pseudo-Henselian." I'll restrict to DVRs for simplicity.
I'd want to call a DVR $(R,\mathfrak{m})$ ...
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1answer
176 views
Characterization of non-commutative local rings of orders 64 and 128
I need the characterization (up to isomorphism) of non-commutative local rings (with identity) of orders 64 and 128. If you know the characterization or a reference, please let me know.
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1answer
480 views
Etale cohomology of the completion of a Henselian local ring
Let $\pi: R\to S$ be a local morphism of Henselian local rings. Let $f: R \to \hat{R}$ and $g: S \to \hat{S}$ be their completions. Let $\mathcal F$ be a constructible $l$-adic sheaf on $\operatorname ...
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109 views
Ring algebraically closed in its completion.
First I would like to be clear about the definition, which I am having trouble finding.
What does: The local ring $A$ is algebraically closed in $B\supset A$. (e.g. for $B:=\hat{A}$, the completion ...
1
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1answer
153 views
Is there a prime of height $i$ in support of $H^i_I(R)$?
$I$ is an ideal of a local Noetherian ring $R$ and $i>0$ .
Clearly the height of primes in support of $H^i_I(R)$ is at least $i$
The question is if it
contains a prime of height $i$, specially ...
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0answers
67 views
Example of a ring whose minimals are annihilators of idempotents?
I'm looking for examples† of rings with the property that for each
$P={\rm Ann}_R(a)\in{\rm Min}(R)$ then $a\in R$ is idempotent (ie $a^2=a$)
† other than domains!
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116 views
ideals of a noetherian ring $R$ Cohen-Macaulay as $R-$modules
When are (prime) ideals of a noetherian ring $R$ Cohen-Macaulay as $R-$modules?
That is, $depth_R(Ann_R(P))=dim_R(R/Ann_R(P)$ for each $P\in {\rm Spec}(R)$
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1answer
112 views
An example of a ring $R$ with the property that for each $P=Ann_R(r)\in {\rm Min}(R)$ we have $Ann_R(P)=Rr$.
I'm looking for an example of a commutative (preferably local) ring $R$ such that ${\rm dim}R>0$ and $R$ has the property that for each $P=Ann_R(r)\in {\rm Min}(R)$ we have $Ann_R(P)=Rr$.
This ...
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1answer
123 views
Open idempotents in modules over a local ring
Let $R$ be a local ring. By an open idempotent I mean an $R$-module $F$ equipped with a homomorphism $e : F \to R$ such that $e \otimes F = F \otimes e$ is an isomorphism $F \otimes F \cong F$ (this ...
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1answer
224 views
Artin approximation theorems over non-regular rings/non-Noetherian rings
In Artin1968 he considers $\underline{analytic}$ equations, but over the ring $R=k\{x_1,..,x_n\}$. In Artin1969 he works with $R=k\{x_1,..,x_n\}/I$, not necessarily regular, but considers ...
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0answers
287 views
working with local rings: “abstract” vs “geometric” proofs
Let $R$ be a local ring (commutative, Noetherian, over an algebraically closed field; if needed Henselian). Suppose one wants to prove some statement.
Suppose $R$ happens to be the ring of ...
0
votes
1answer
490 views
“thematic” algebras
I scoured what I could in the literature but I have yet to find the information that should be out there. Consider the property
(P1) Every local subalgebra can be embedded in a local ideal ...
3
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2answers
305 views
Can a zerodivisor reduce both the depth and the dimension?
In this question $R$ is a commutative noetherian local ring with unity.
One can construct examples of rings $R$ and zerodivisors $z$ such that $\dim R/(z)=\dim R-1$, e.g., $S\colon=k[a,b,c],\ ...
2
votes
1answer
143 views
Depth zero, high dimension
$\textbf{Question: }$We know that the depth of a noetherian local ring is at most the dimension. Do there exist noetherian local rings with high dimension but zero depth? If not, what's the smallest ...
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2answers
502 views
A graded ring $R$ is graded-local iff $R_0$ is a local ring?
I asked this question some months ago on math.stackexchange.com:
http://math.stackexchange.com/questions/126810/a-graded-ring-r-is-graded-local-iff-r-0-is-a-local-ring
It would be great (for me) to ...
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1answer
374 views
Left ideals vs right ideals
By default, let all algebras be complex and unital. I am concerned with the non-commutative algebras. I am wondering if the following might be true (at least for some classes of algebras, like ...
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1answer
789 views
Lengths over a local ring
Let $A$ be a noetherian domain, $\mathfrak{m}$ а maximal ideal, $s$ a non-zero element of $\mathfrak{m}$, $d= \dim A_\mathfrak{m}$.
Is the following claim true?
Claim:
For any $\epsilon>0$, there ...
2
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1answer
224 views
Modules with small support have big depth - reference wanted
Hello,
I would appreciate an exact reference / proof of the following fact, which I am almost able to prove, but not really:
Let $A$ be a regular Noetherian comm. ring, of finite Krull dimension. ...
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1answer
372 views
Norm map and units in local rings
Let
$$
L=\mathbb{Q}(\sqrt{-1})\otimes_\mathbb{Q} \mathbb{Q}_3
$$
where $\mathbb{Q}_3$ denotes de $3$-adic rational numbers.
Then $L$ is a quadratic extension of the local field $\mathbb{Q}_3$.
...
1
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1answer
210 views
Minimal generating set of a free module over local ring
Greetings,
in my studies I went into a statement "minimal generating set of a free module over a local ring is a free basis". The statement came without a proof, just with a reference to Kaplansky's ...
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3answers
307 views
On the comparison of linear topologies on a local ring
Let $R$ be a local ring, $a_{\lambda}$ be a decreasing net of ideals, indexed by a directed set, such that each $a_{\lambda}$ is contained in the nilradical ideal and $\bigcap a_{\lambda}=(0)$. Then ...
1
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1answer
737 views
On the Completion of a complete local ring
Let $(R,\mathfrak{m})$ be a complete local ring, $a_{\lambda}$ be a decreasing net of ideals in $R$, indexed by a directed set. Consider the completion under $a_{\lambda}$-topology ...
2
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1answer
178 views
A particular Isomorphism of graded algebras over a regular local ring
In Hartshorne's "Algebraic Geometry", the following statement is a weaker form of Theorem 8.21A (e), which he quotes from Matsumuura's book on commutative algebra:
Proposition. Let $R$ be a ...
2
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1answer
528 views
Can you suggest a good name for a local homomorphism φ:(R,m)->(S,n) of local rings with the property that φ(m)S is n-primary?
Can you suggest a good name for a local homomorphism $(R,\mathfrak{m})\stackrel{\varphi}{\rightarrow}(S,\mathfrak{n})$ of local rings with the property that $\varphi(m)S$ is $\mathfrak{n}$-primary?
...
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297 views
Krull's intersection theorem for commutative local not necessarily noetherian rings
Is there a characterisation of those commutative local not necessarily noetherian rings that satisfy Krull's intersection theorem ? How can the intersection theorem be phrased in terms the module ...
2
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4answers
979 views
When a group ring is a local ring
Hi there, I'm stuck with my undergraduate thesis on the following proposition:
If $k$ is a field of characteristic $p > 0$ and $G$ is a finite $p$-group, then the group ring $kG$ is local.
In ...
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1answer
164 views
Local coordinate system under finite integral extension
Let $\varphi:(A,\mathfrak{m})\to(B,\mathfrak{n})$ be a local morphism of regular local $\mathbb{C}$-algebras (of the same dimension) which makes $B$ integral over $A$.
Let ...
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1answer
469 views
On the functoriality of scalar extensions of local rings (edited)
Note. I have edited my question to make it more transparent, following some very good comments that I received. I am sorry if it is a bit long.
A local homomorphism of local rings ...
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1answer
606 views
Local Rings problem
$\newcommand{\End}{\operatorname{End}}$
let $R$ be a local ring, $\varphi\in \End(R_{R}^{2})$,
$\overline{\varphi}\in \End(\overline{R}_{\overline{R}}^{2})$,
$\overline{R} =R/J(R)$ , $J(R)$= ...
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2answers
404 views
maximal ideal in local subrings
Let $A,B$ be two local rings and put $\mathfrak{m}_A, \mathfrak{m}_B$ their maximal ideals. Now suppose that we have an injection $0 \to A \to B$ and put $\mathfrak{n} := A \cap \mathfrak{m}_B $. It ...
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2answers
1k views
Image of norm map for local field
Let $F$ be a finite extension of $Q_2$ (2-adic field) or $F_2((x))$ (function field). Let $E/F$ be a separable extension of degree $2$.
What is the image of the norm map $N_{E/F}$?
In particular - ...
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2answers
355 views
on the relative conductor of curve singularity and quotient of ideals
Let $R$ be the local ring of a complex curve singularity. (Can assume the singularity planar, the ring locally analytic or formal.) Let $\bar{R}$ be the normalization, let $R\subset R'\subset \bar{R}$ ...
1
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1answer
232 views
covers of complete regular local rings
It is well-known that if one assumes algebraic closedness and characteristic 0 of the residue field then finite covers of complete DVRs are all of the form $A[x]/(x^m-a)$ for some $a \in A$ (direct ...
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2answers
909 views
prime ideals in regular local rings
Suppose $R$ is a regular local ring. Let $m$ be the maximal ideal. Then, if the dimension of $R$ is $n$, there is a regular sequence of size $n$, say $x_1,x_2,...,x_n$ s.t. $m=(x_1,x_2,...,x_n)R$. ...
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253 views
Ring Theory (indecomposable) [closed]
Show that a ring $R$ is local if and only if every principal left $R$-module is indecomposable.
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759 views
Etale cohomology of regular local rings
Let $R$ be a regular local ring (I am particularly interested in the case when $R$ is the local ring of a point on a smooth scheme of finite type over a field). Let $G$ be the etale fundamental group ...
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3answers
429 views
local Artin algebras
Given a commutative Artin algebra $A$ over an algebraically closed field $k$ one has a decomposition $A=A_1\oplus\ldots\oplus A_n$ into local Artin subalgebras, see for example Atiyah-McDonald, ...
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2answers
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Algorithm for Weierstrass Preparation Theorem for Formal Power Series
The Weierstrass preparation theorem for formal power series rings guarantees that if a given formal series $f(z) = \sum a_k z^k \in R[[z]]$ where $R$ is a complete local ring with maximal ideal $M$ ...
