Questions tagged [local-rings]
The local-rings tag has no usage guidance.
66 questions with no upvoted or accepted answers
9
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When is $\lim_{n\rightarrow\infty}\:\mathrm{depth}\:R/\mathfrak{a}^n= \mathrm{depth}\:R/\mathfrak{a}$?
Suppose $(R,\mathfrak{m})$ is a noetherian local ring. I am interested in ideals $\mathfrak{a}$ of $R$ for which $$\lim_{n\rightarrow\infty}\:\mathrm{depth}\:R/\mathfrak{a}^n= \mathrm{depth}\:R/\...
- 3,101
8
votes
0
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293
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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 $...
- 149k
8
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0
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221
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Finitely generated commutative rings with the same profinite completion
Let $R_1$ and $R_2$ be two finitely generated commutative rings. Assume that their profinite completions are isomorphic: $\widehat{R_1}\cong \widehat{R_2}$.
Suppose that $R_1$ is a domain. Does ...
- 1,418
8
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0
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366
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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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6
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0
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399
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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 ...
- 15.9k
6
votes
0
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478
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The Zariski Riemann Space, but with Local Rings
The Zariski Riemann space, while an abandoned approach, has lead to later developments and generalizations, including $\text{Spv}$ (the space of valuations) and Huber's work. In studying it, I would ...
user30211
6
votes
0
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241
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Bezout theorem for germs of holomorphic functions
UPDATE.
It was pointed out by @Dmitri that two smooth curves given by $f=y$ and $g=y+x^k$ in $\mathbb C^2$ provide a simple counterexample.
Let $f_1, \ldots, f_p, g_1, \ldots, g_q$ be germs of ...
- 1,238
5
votes
0
answers
209
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Ext^1 for a local finite dimensional selfinjective algebra
Is there a nonprojective module $M$ over a finite dimensional local selfinjective algebra with $Ext^{1}(M,M)=0$? I asked this question also here:
http://arxiv.org/pdf/1609.00588.pdf.
There it is ...
- 26.5k
5
votes
0
answers
94
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How far finiteness dimension can be from edges? Example for $f_m(S/I)\ge depth S/I+2$
Let $ (R,m) $ be a commutative unital noetherian local ring (with $m$ as its maximal ideal), $ I $ an ideal of $ R $, and $ M $ a finite $R$-module with $\dim M\gt 0$. $f_I(M) = \inf\ \{i : H_I^i(M)\ ...
- 1,355
4
votes
0
answers
73
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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 $...
- 5,039
4
votes
0
answers
135
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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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4
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0
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265
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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/\...
- 1,590
4
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0
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177
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What kind of module is this?
Recall that, if $R$ is a commutative ring, then a suitably finite $R$-module $M$ is projective if and only if the localization $M_\mathfrak{m}$ is a direct sum of finitely many copies of $R_\mathfrak{...
- 985
4
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155
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rings with 'flat functions'
Let $(R,\mathfrak{m})$ be a local ring over a field. Suppose the ring has flat elements, i.e. $\mathfrak{m}^\infty\neq\{0\}$. (The prototype is of course $C^\infty(\Bbb{R}^p,0)$, or a quotient of it, ...
- 2,232
3
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0
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91
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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$ ...
- 541
3
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0
answers
69
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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 $...
- 429
3
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0
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1k
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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 ...
- 15.6k
2
votes
0
answers
66
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Projective cover (minimal) for (derived)complete modules over Noetherian local rings exist?
Let $(R,\mathfrak m)$ be a commutative Noetherian local ring. Let $M$ be an $R$-module which is $\mathfrak m$-adically derived complete. Then, does there exist a free $R$-module $F$ and a surjective $...
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2
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0
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275
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Can the completion of a local domain which is not a field be a field?
I would like to prove/disprove the following claim:
Let $A$ be an equicharacteristic local domain, and denote by $\widehat{A}$ its completion with respect to its maximal ideal. If $\widehat{A}$ is ...
- 293
2
votes
0
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111
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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}...
- 361
2
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1
answer
390
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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#...
- 480
2
votes
0
answers
292
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Local weak factorization
This is a follow-up to question Locally toric resolutions of compactifications, answered by Jason Starr.
In a series of papers (see https://arxiv.org/abs/math/9904076), Jaroslaw Wlodarczyk proves ...
- 7,972
2
votes
0
answers
77
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Analytic spread of an ideal after reduction
Let $(R,m)$ be a local ring and $I$ an ideal in $R.$ Let $l(I):=\dim \bigoplus_{n\geq 0}(I^n/mI^n)$ and $x\in R\setminus I.$
My question is
what is the relation between $l(I)$ anf $l(I+(x)/(x))?$
- 323
2
votes
0
answers
221
views
Meaning of the statement "$a\in I$ is a general element of $I$"
Suppose $I$ is an ideal in a Noetherian local ring $(R,m)$. In some papers I have seen the following statement:
"$a\in I$ is a general element of $I$".
What is the definition of general element ...
- 1,713
2
votes
0
answers
78
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Uniform Artin-Rees bound for annihilators in Noetherian local rings
Let $(A,\mathfrak{m})$ be a Noetherian local ring. If $I$ is an ideal of $A$, then by (a weak version of) the Artin-Rees lemma, there exists $j \geq 0$ such that for all $i \geq j$,
$$\mathfrak{m}^i \...
- 991
2
votes
0
answers
178
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Modern dictionary for "old" homological terms
I'm trying to build a little dictionary between old Homological algebra for local rings and the slightly more modern approach via derived functors.
Let $X = SpecA$ be a spectrum of a local ring $(A,...
- 7,799
2
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0
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145
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Question about Ext$^1$ in local commutative algebras
Given a local commutative (commutative only if needed...) selfinjective (non-semisimple) finite dimensional algebra $A$ over a field $K$ with enveloping algebra $A^e = A \otimes_K A^{op}$. Then $Ext_{...
- 26.5k
2
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140
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When does $R [x]/I $ have infinitely many idempotents in special case?
At < When does $R [x]/I $ has infinitely many idempotents? >, Er_Ro asked the following question.
Let $R $ be a commutative ring with identity and $R[x] $ its polynomial ring. I am looking for ...
- 21
2
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0
answers
327
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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 $(R,\mathfrak{m})$ pseudo-...
- 1,499
2
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0
answers
348
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How much can we say about the number of nilpotents in a finite local commutative ring?
A commutative ring is local if it has a single maximal ideal. If the ring is finite, this implies that all elements are either units or nilpotents. Further, all finite local rings have prime power ...
- 1,793
1
vote
0
answers
136
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Local cohomology and image of $1$ under the canonical map from Ext to local cohomology
Let $R$ be a commutative Noetherian local ring, and $S$ be an $R$-algebra. Let $x_1,\dots,x_t$ be elements, in the maximal ideal of $R$, which is a regular sequence on both $R$ and $S$, and let $I$ be ...
- 412
1
vote
0
answers
95
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References on the claims of moduli spaces of additive compactifications by Hassett-Tschinkel
I am considering the classification problem of Artinian local $\mathbb{C}$-algebras, and notice the paper
Hassett, Brendan; Tschinkel, Yuri, Geometry of equivariant compactifications of (\mathbb{G}^...
- 930
1
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0
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187
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Is it true that monomorphisms of local Artinian $\mathbb{R}$-algebras are regular?
A Weil algebra is a finite-dimensional real algebra, in which each element is the uniquely sum of a scalar and a nilpotent (so nilpotents constitute the only maximal ideal of codimension 1). In other ...
- 6,962
1
vote
0
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85
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Is $(I(R:_{Q(R)} I))^n$ generated by $(fI)^n$ as $f$ varies over $(R:_{Q(R)} I)$?
Let $(R, \mathfrak m)$ be a Noetherian local domain of dimension $1$ which is not a UFD. Let $Q(R)$ be the fraction field of $R$. If $I\subsetneq \mathfrak m$ is a non-zero, non-principal ideal of $R$ ...
- 480
1
vote
0
answers
159
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On certain definition of arithmetical ring
The definition of an arithmetical ring states that
A ring $R$ is arithmetical if the ideal lattice is distributive or equivalently $R$ is locally a valuation ring.
I was reading a paper where ...
- 19
1
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0
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136
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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) ...
- 413
1
vote
0
answers
132
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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 ...
- 11
1
vote
0
answers
87
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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 ...
- 5,039
1
vote
0
answers
217
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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 $\...
1
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0
answers
194
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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 ...
- 2,837
1
vote
0
answers
117
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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.
...
- 2,837
1
vote
0
answers
94
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What would be the quotient groups $U_{\mathrm{gen}}/U_{\mathrm{gen}}^{(n)}$ and $U_{\mathrm{gen}}^{(n)}/U_{\mathrm{gen}}^{(n+1)}$?
Let $K \supseteq \mathbb{Q}_p$ be a $p$-adic field with ring of integer $O$ and maximal ideal $m$. Let $O^*$ be the group of units in $O$.
Consider the group of units $U^{(0)}=U=O^*$ and $U^{(n)}=1+m^...
- 930
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vote
0
answers
94
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Generators for Ideals in ring of multivariate Laurent Polynomials
Consider the following problem:
Find an ideal $I \subset \mathbb{Q}[x^{\pm}_1,x^{\pm}_2,x^{\pm}_3]$ such that $I_{aff} \subset \mathbb{Q}[x_1, x_2, x_3] = I \cap k[x_1, x_2, x_3]$ requires more ...
- 11
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vote
0
answers
165
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Structure of Complete Local Rings
Let $X$ be a proper $n$-dimensional $k$-scheme and $x \in X$ a closed point. Consider the stalk $\mathcal{O}_{X,x}$. We consider now it's completion $O_{X,x}^{\wedge}$ wrt it's maximal ideal $m_x$.
...
- 5,966
1
vote
0
answers
138
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Power series ring $R[[X_1,\ldots,X_d]]$ over a domain $R$
Let $R$ be a domain and
\begin{align*}
T \,\colon= R[[X_1,\ldots,X_d]].
\end{align*}
Suppose that we have $d$ elements $f_1,\ldots,f_d \in T$ and let us consider an ideal $J$ of $T$ such that $(f_1,\...
- 563
1
vote
0
answers
310
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Primes of the power series rings
Let $A_n \colon= K[[X_1,\ldots,X_n]]$ be a $n$-variable formal power series ring. By setting $X_n \mapsto 0$, we obtain a natural surjection
\begin{equation*}
\psi_{n,n-1} \colon A_n \...
- 563
1
vote
0
answers
78
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Relation between lifts of simple roots and lifts of idempotents (Henselian property)
Let $f:A\to B$ be a morphism of commutative rings. Given a monic $\varphi\in A[x]$ write $Z(A,\varphi)$ for the set of simple roots of $\varphi$ in $A$. Consider the following properties of $f:A\to B$....
- 10.5k
1
vote
0
answers
46
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Integral closure of lexsegment ideal
Let $R=k[x_1,\ldots,x_d]$ where $k$ is a field and $I$ be a lexsegment ideal of $R$ and $l(I)=d$ (where $l(I)$ is analytic spread of $I$).
Is $I$ integrally closed?
If I is generated by elements ...
- 1,713
1
vote
0
answers
294
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Is it true that the functor of completion of a module over a local ring is injective on isomorphism classes?
Let $A$ be a commutative Noetherian local ring and $\hat A$ be its completion. Then we have the functor of completion from the category of finitely generated $A$-modules to the category of finitely ...
- 1,484
1
vote
0
answers
284
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Analytic spread of an ideal
How to calculate analytic spread of the ideal $I=\left<xyw^2,xyz^2,xw^2+yz^2\right>$ in $\mathbb Q[x,y,z,w]?$
I think it is 3.
- 1,713