# Questions tagged [local-rings]

The local-rings tag has no usage guidance.

196
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### Examples of stretched artinian local ring

In Sally's Paper stretched artinian local ring is defined as :
Let $(R, \mathfrak{m})$ be an Artin local ring of length $\lambda.$ If $\nu$ is the embedding dimension of $R$, that is, $\nu$ is the ...

5
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1
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234
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### Alternative description of strict henselization

Let $R$ be a local ring with field of fractions $K$, maximal ideal $\mathfrak{m}$ and residue field $\kappa = R/\mathfrak{m}$. Let $R^\mathrm{sh}$ be a strict henselization of $R$, and let $L$ be the ...

4
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0
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### Complex representations of groups of invertible elements in finite local rings

Let $R$ be a finite local $\mathbb{F}_p$-algebra, and let $J$ be its Jacobson radical. Assume that $R/J\cong \mathbb{F}_p$, and assume that the socle of $R$ as an $R$-bimodule is one dimensional over $...

0
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### When a given set of primes of height 1 is a set associated primes of an element

Let $R$ be a Noetherian local ring of dimension $\geq 3$ and $\{p_1,\ldots , p_n\}$ be a collection of prime ideals of height $1$. Does there exist an element $x\in R$ such that $Ass(R/xR)=\{p_1,\...

1
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0
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101
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### 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) ...

3
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1
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124
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### Does Noetherianity imply division theorem?

I am trying to understand something which is probably basic for experts so I am sorry if this is not suited for this forum.
Let $\mathcal{O}_n$ denote the ring of germs at $0 \in \mathbb{R}^n$ of real-...

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0
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87
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### A question concerning cancellation of ideals

I am working on a number theory project, and at one stage, I encounter a commutative algebra problem. Vaguely speaking, my hope is to show that two ideals are equal. Now I shall explain the data I am ...

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0
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74
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### When does a local Cohen–Macaulay ring admit a non-zero finitely generated maximal Cohen–Macaulay module of finite injective dimension?

Let $(R,\mathfrak m)$ be a local Cohen–Macaulay ring. Then, it is well- nown that there exists a non-zero finitely generated $R$-module of finite injective dimension; for instance $\operatorname{Hom}...

0
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1
answer

192
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### Depth of almost complete intersection rings

Let $R$ be a regular local ring and let $I \subset R$ be an almost complete intersection ideal, that is, $\mu(I)=\text{ht}(I)+1$ where $\mu(I)$ is the number of minimal generators of $I$ and $ht(I)=\...

3
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1
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98
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### Analogue of Kock-Lawvere axiom for power series rings?

The Kock-Lawvere axiom for a topos $\mathcal{E}$ states that given a specified commutative ring object $R \in \mathcal{E}$, for all local Artinian $R$-algebra objects $A \in \mathcal{E}$, the morphism
...

4
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1
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197
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### For a local complete intersection ring $(R,\mathfrak m)$ with $\mathfrak m^3=0\ne \mathfrak m^2$, $\mathfrak m$ can be generated by two elements

Let $(R,\mathfrak m,k)$ be a local complete intersection ring with $\mathfrak m^3=0\ne \mathfrak m^2$. As $0\ne \mathfrak m^2 \subseteq \text{soc}(R)$ and $R$ is Gorenstein, so we get $\mathfrak m^2 =\...

0
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1
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90
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### On $\text{depth}_S\left(\dfrac {S}{JS+xS}\right)$, when $\text{depth}_R(R/J)=0$, and $R\to S$ is a certain flat map of local rings

Let $(R, \mathfrak m) \xrightarrow{\phi} (S,\mathfrak n) $ be a flat homomorphism of local rings such that $\mathfrak n=\mathfrak m S +xS$ for some $x\in \mathfrak n \setminus \mathfrak n^2$. Let $J$ ...

5
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1
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148
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### Equivalence of quadratic forms over $p$-adic integers vs over localisation at $p$

To discern whether two integral quadratic forms are equivalent over the $p$-adic integers, one can compute a Jordan decomposition at $p$ and read off some invariants. Restricting to $p\ne2$ for ...

2
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1
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192
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### When is Hilbert-Samuel multiplicity of a local ring non-increasing along localization at prime ideals?

For Noetherian local ring $(R,\mathfrak m)$, let $e(R)$ denote the Hilbert-Samuel multiplicity of $R$ with respect to $\mathfrak m$ (https://en.m.wikipedia.org/wiki/Hilbert%E2%80%93Samuel_function#...

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1
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182
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### On image of map $\text{Ext}^1_R(X,F)\to \text{Ext}^1_R(X,G)$ induced by $R$-linear map of free modules $F\to G$ with entries in the maximal ideal

$\DeclareMathOperator\Ext{Ext}$Let $(R,\mathfrak m)$ be a Noetherian local ring. Let $F,G$ be finitely generated free $R$-modules and $f:F\to G$ be an $R$-linear map such that $f(F)\subseteq \mathfrak ...

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0
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77
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### Abelianization of the group of invertible elements in a finite local ring

Let $R$ be a finite local $\mathbb{F}_q$-algebra. Assume that $R\cong R^*$ as left $R$-modules. Are there any known results about the abelianization $(R^{\times})_{\mathrm{ab}}$?
(We can factor $R$ be ...

7
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1
answer

225
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### Is there a "spherical building" for a reductive group over a Henselian local ring?

Let $A$ be a Henselian local ring and let $G$ be a split reductive $A$-group. I'm interested in some notion of a "building of parabolic subgroups" for the group scheme $G$.
In my specific ...

21
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1
answer

991
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### A Krull-like Theorem and its possible equivalence to AC

A well known equivalent of the Axiom of Choice is Krull's Maximal Ideal Theorem (1929): if $I$ is a proper ideal of a ring $R$ (with unity), then $R$ has a maximal ideal containing $I$. The proof is ...

4
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1
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257
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### When is it possible to localize a scheme along a closed subscheme?

If we have $Z\subset X$ a closed irreducible subscheme of an integral scheme $X$ (which you can take to have various further niceness properties if you want), one can take its generic point $\eta_Z$ ...

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203
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### An example of a local integral domain with special spectrum

I am looking for a local integral domain $(D, m)$ with $Spec(D)=\{0,m\}\cup\{ P_i\}_i$ such that $P_i's$ are incomparable (that is, $P_i\not\subseteq P_j$ and $P_j\not\subseteq P_i$ for $i\not= j$) ...

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4
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536
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### $R$ is a UFD iff $R_{\frak{m}}$ is a UFD?

Let $R$ be a graded ring such that $R_0$ is a field and let $\frak{m}$ be the maximal ideal generated by all the elements of positive degree. Then, is it true that $R$ is a UFD iff $R_{\frak{m}}$ is a ...

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71
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### Which power series in $\mathbb{Z}_p[[T]]$ are rational functions? [duplicate]

Consider the power series ring $\mathbf{Z}_p[[T]]$, where $\mathbf{Z}_p$ denotes the $p$-adic integers. I'll call a function $f(T) \in \mathbf{Z}_p[[T]]$ a rational function if I can write it as:
$$f(...

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0
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153
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### Artin-Winters proof of semi-stable reduction theorem: details

I've been reading through Artin-Winters proof of the semi-stable reduction theorem (Degenerate fibers and stable reduction of curves) and found myself confused about the following detail—
Let $\...

5
votes

1
answer

323
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### About the structure of unit groups appearing in number theory

I think the following statement is not true in the general situations, but consider it:
$R$ is a ring, $\mathfrak{p}$ is a prime ideal, then the unit group of $\dfrac{R}{\mathfrak{p}^nR}$ is ...

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2
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148
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### The quotient of an algebra with an ideal whose generators are decomposed as the product of irreducible elements

I would like to find reference for the following statement.
I need it only in the particular case when $A=\mathcal{O}_{(\mathbb{C}^n, 0)}$ is the local algebra of holomorphic germs $(\mathbb{C}^n, 0) \...

3
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0
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86
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### On the descent of noetherianess along completion

Let $A$ be a commutative local ring with maximal ideal $m$ and $\hat{A}$ be its $m$-adic completion. Are there any non-trivial conditions on $A$, under which $\hat{A}$ noetherian implies $A$ ...

2
votes

1
answer

201
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### Strict henselianization and branches of explicit curve at singularity

Let $A$ be a local ring, which we can assume is reduced. Let $k$ be the residue field of $A$.
In the Stacks project (https://stacks.math.columbia.edu/tag/06DT), I have learned some notion of the ...

3
votes

1
answer

196
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### Vanishing of $\operatorname{Ext}_R(\operatorname{Tr} M,N)$ and freeness criteria

$\DeclareMathOperator\Ext{Ext}\DeclareMathOperator\Tr{Tr}\DeclareMathOperator\coker{coker}\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Tor{Tor}$I am investigating the interplay between freeness ...

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257
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### Image of multiplication map in tensor powers of finite-dimensional ring

Let $R$ be a (commutative, unital) ring of dimension $n$ over a field $k$. Assume the characteristic of $k$ is greater than $n$.
Then $R^{\otimes n}$ has a natural ring structure, together with an $...

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0
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79
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### Order of the symplectic group over $\mathbb{Z}/4\mathbb{Z}$ [duplicate]

Let $p$ be a prime number and $q$ some power of it. It is well-known that the order of the symplectic group $\text{Sp}_{2g}(\mathbb{F}_q)$ over the finite field $\mathbb{F}_q$ equals $q^{g^2}\prod_{i=...

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3
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### On the map $\Phi_M: M\otimes_RM^*\xrightarrow{x\otimes y\mapsto \left\{f\mapsto f(x)y\right\}}\text{Hom}_R(M^*,M^*) $

$\DeclareMathOperator\Hom{Hom}$Let $M$ be a finitely generated module over a Noetherian local ring $(R,\mathfrak m)$. Denote $(\_)^*:=\Hom_R(\_,R)$. There is a natural map
\begin{align} \Phi_M: M \...

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0
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### Division algorithm for multivariable power series

Let $\mathbb{Z}_p$ be the ring of $p$-adic integers. Consider the ring $R=\mathbb{Z}_p[[T]]$. Let $f,g \in R$ and assume that $f=a_0+a_1T+...$ with $a_i \in p\mathbb{Z}_p$ for $0\le i \le n-1$, but $...

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2
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### Why is $M$ torsion-free?

I am studying the following article
https://www.math.nagoya-u.ac.jp/~takahashi/tc9.pdf
The main theorem is the Theorem 3.3. Howewer, I have the following questions about the proof:
How does it help ...

4
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0
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### Structure of $\bigwedge^{2}_{\mathbb{Z}}(A)$ with $A$ a local integral domain

I am trying to see the structure of $\bigwedge^{2}_{\mathbb{Z}}(A)$ where $A$ is a local integral domain with small residue field.
Let $A$ be a local integral domain with maximal ideal $M$, residue ...

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0
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201
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### When is $u \circ v=v \circ u$ for $p$-adic power series $u$ and $v$ in two power series rings $A$ and $B$ respectively?

Let $K \supset \mathbb{Q}_p$ be the $p$-adic field with ring of integers $O_K$ and maximal ideal $m_K$. Let $\bar K$ be the algebraic closure and $\bar{m}_K$ be the integral closure of $m_K$ with ...

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1
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### Symbolic power of an ideal associated to non-singular algebraic set

Let $Z\subset \mathbb P^n$ be a reduced non-singular algebraic set and $I$ denote the saturated homogeneous ideal of $Z$. I have seen the following result without proof:
For all $ n\geq 1$, $I^{(n)}=(...

2
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1
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223
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### Good prime ideals in tensor products of local rings

Let $L/K$ be a field extension.
Let $R,S$ be two local commutative $K$-algebras and let $\varphi : R \to S$ be a homomorphism of $K$-algebras, not assumed to be local. Let's call a prime ideal $\...

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2
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330
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### Ideals in Artinian Gorenstein local ring $(R,\mathfrak m)$ with $\mu(\mathfrak m)=2, \mathfrak m^2\ne 0$ and $\mathfrak m^3=0$

Let $(R,\mathfrak m,k)$ be an Artinian Gorenstein local ring such that $$\mu(\mathfrak m)=2, \quad\mathfrak m^2\ne 0,\quad\text{and}\quad \mathfrak m^3=0.$$
Then, is it true that every non-maximal ...

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0
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279
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### Unbounded derived Nakayama lemma

Let $R$ be a (commutative) local ring, which I don't assume to be noetherian. Let $m$ be its maximal ideal, and $k$ its residue field.
Let $X$ be a complex of $R$-modules with finitely generated ...

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0
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174
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### Is the following local map unramified?

Let $(R,m)$ and $(S,n)$ be two local rings, $R \subseteq S$, $R$ regular, $S$ a finitely generated and flat $R$-algebra, and $mS=n$.
In comments to this question it was claimed that in such situation ...

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0
answers

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### $A \to B$ with $A$ regular imply that $B$ is CM

The answer to this question says the following:
"The general statement is if $A \to B$ is finite and injective, and $A$ is noetherian and regular, then $B$ is CM if and only if $A \to B$ is flat.
...

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1
answer

199
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### Local rings $R \subsetneq S$ with $R$ regular and $S$ Cohen-Macaulay, non-regular

Let $R \subseteq S$ be local rings with maximal ideals $m_R$ and $m_S$.
Assume that:
(1) $R$ and $S$ are (Noetherian) integral domains.
(2) $\dim(R)=\dim(S) < \infty$, where $\dim$ is the Krull ...

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1
answer

218
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### 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 $...

5
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1
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254
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### Hakim's definition of a locally ringed topos

In Hakim's book "Topos annelés et schémas relatifs", Chap. III, Def. 2.3 states that a ringed topos $(X,A)$ is a locally ringed topos when two equivalent conditions are satisfied:
(i) For ...

0
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1
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245
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### Given a unitary commutative ring $R$, what are the rings $R\langle x,y\rangle/(x^2-A,y^2-B,yx-a-bx-cy-dxy)$ called

We are studying the rings
$$
R \langle x, \, y \rangle\,\big/\left(x^2-A, \, y^2-B, \, yx-a-bx-cy-dxy \right)
$$
Do you know if they have a name?

1
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1
answer

141
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### Do you know a finite unitary reversible ring that is not isomorphic to its opposite? And the minimal with that property?

Do you know a finite unitary reversible ring that is not isomorphic to its opposite? And the minimal with that property?
The examples of rings not isomorphic to their opposite that I know of are not ...

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votes

1
answer

190
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### Cohomology of finite $p$-groups over integers in local fields

Let $p$ be a prime, $G$ be a finite group of order $p^a$. Let $M$ be a $\mathbb{Z}[G]$-module. Then $H^n(G, M)$ is annihilated by $p^a$ for all $n \geq 1$ (see e.g. Brown, Corollary III.10.2).
In ...

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1
answer

299
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### Kähler differentials on an Artinian local ring

Suppose $R$ is a commutative Artinian local ring over an algebraically closed characteristic 0 field $k$. Suppose $f\in R$ is such that $df=0$ (in the sense that the element $df$ vanishes in the ...

4
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0
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206
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### Sections of smooth morphisms over henselian rings

Let $(A,\mathfrak m)$ be a henselian local ring. Let $R$ and $S$ be $A$-algebras of finite type and $f\colon R\to S$ be a smooth morphism. Assume that the induced morphism $R/\mathfrak m R\to S/\...

6
votes

1
answer

476
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### Do you know which is the minimal local ring that is not isomorphic to its opposite?

The most popular examples are non-local rings and minimal has 16 elements. I am interested in knowing examples of local rings not isomorphic to their opposite.