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1
vote
1answer
106 views

what are the possible approximations for ideals

(Fix some local ring $(R,\mathfrak{m})$ over a field of zero characteristic.) Suppose an ideal $J$ is defined by some complicated formula/procedure. And there is no hope of computing it/or writing ...
21
votes
3answers
859 views

Two (other) rings…are they isomorphic?

Consider the local rings $$R = \mathbb{C}[[x,y,z,w]]/\langle xyz+xyw+xzw+yzw\rangle$$ and $$S = \mathbb{C}[[x,y,z,w]]/\langle xyz+xyw+xzw+yzw+xyzw\rangle.$$ Is $R$ isomorphic to $S$? Some ...
16
votes
1answer
905 views

Two rings…are they isomorphic?

Edit: I have reverted my question to its original version (which Bjorn Pooenen answered correctly) as requested in the comments. Consider the local rings $$R = \mathbb{C}[[x,y,z]]/\langle ...
2
votes
1answer
102 views

Projecting solutions of Hermitian forms over local rings

Let $R$ be a local ring (commutative and with $1$) with maximal ideal $M$, with an involution $\theta$. Let $h$ be a Hermitian form on $R^n$, i.e. $h:R^n\times R^n\rightarrow R$ such that $h$ is ...
7
votes
2answers
329 views

What is the probability that a random sequence of polynomials is regular?

Let $k$ be a finite field or a field with a height function, such as a number field. Consider the ring $k[[x_1,\dots, x_n]]$ and let $\mathfrak{m}$ be its maximal ideal. What is the asymptotic ...
5
votes
1answer
210 views

formally smooth functor

Let $p$ be a prime number, $\mathcal{O}$ the integers of a finite extension of $\mathbb{Q}_p$ with residue field $k$. Let $\mathcal{C}$ be the category of complete, local, noetherian ...
3
votes
1answer
178 views

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}, ...
1
vote
1answer
175 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 ...
2
votes
2answers
248 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 ...
1
vote
0answers
112 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 ...
5
votes
1answer
216 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
votes
1answer
301 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 ...
0
votes
0answers
87 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): ...
2
votes
0answers
100 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
votes
1answer
75 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 ...
3
votes
1answer
131 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 !?
7
votes
0answers
234 views

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 ...
0
votes
1answer
127 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?
2
votes
0answers
241 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 $(R,\mathfrak{m})$ ...
3
votes
1answer
199 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.
8
votes
1answer
547 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 ...
0
votes
0answers
120 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
vote
1answer
157 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 ...
0
votes
0answers
70 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!
0
votes
0answers
117 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 $P\in {\rm Spec}(R)$
1
vote
1answer
117 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 ...
2
votes
1answer
125 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 ...
3
votes
1answer
258 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 ...
3
votes
1answer
441 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
494 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
votes
2answers
309 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
147 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 ...
11
votes
2answers
549 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 ...
5
votes
1answer
411 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 ...
9
votes
1answer
801 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
votes
1answer
240 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. ...
1
vote
1answer
394 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
vote
1answer
256 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 ...
2
votes
3answers
316 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
vote
1answer
809 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
votes
1answer
179 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
votes
1answer
535 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? ...
1
vote
0answers
307 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
votes
4answers
1k views

When a group ring is a local ring [closed]

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 ...
3
votes
1answer
168 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 ...
5
votes
1answer
481 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 ...
1
vote
1answer
611 views

Local Rings problem

$\newcommand{\End}{\operatorname{End}}$ let $R$ be a local ring, $\varphi\in \End(R_{R}^{2})$, $\overline{\varphi}\in \End(\overline{R}_{\overline{R}}^{2})$, $\overline{R} =R/J(R)$ , $J(R)$= ...
0
votes
2answers
449 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 ...
4
votes
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 - ...
2
votes
2answers
385 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 $R\subset R'\subset \bar{R}$ ...