Questions tagged [local-rings]
The local-rings tag has no usage guidance.
211
questions
7
votes
1
answer
859
views
indecomposable module over a local ring
I ask this in mathematics for some days.it doesn't have an answer up to now. https://math.stackexchange.com/questions/2565828/indecomposable-module-over-a-local-ring
As we all know, for an arbitrary ...
- 496
9
votes
2
answers
686
views
The projective covers of Artinian module
The injective hull for a module always exists, however over certain rings modules may not have projective covers. I have a question.
If $A$ is an Artinian module on a Noetherian local ring $R$ then $...
2
votes
0
answers
275
views
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 ...
- 7,902
4
votes
2
answers
333
views
Canonical module of a semigroup ring
Let $S$ be a numerical semigroup and $k[S]$ is the associated semigroup ring. I would like to compute canonical module $\omega$ of $k[S].$
I want to show that $\omega=k[t^{-n}:n\in\mathbb Z\setminus ...
- 1,703
2
votes
1
answer
192
views
Length of a module and Frobenius map
Let $(R,m)$ be a regular local ring of dimension $d$ and char $p>0.$ Let $F^e:R\longrightarrow R$ defined by $r\longrightarrow r^{p^e}$be the Frobenius map.
How to compute $l(R/m^{[p^e]})?.$
I ...
- 1,703
2
votes
0
answers
75
views
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
1
vote
0
answers
274
views
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 ...
- 1,312
1
vote
1
answer
233
views
Localization of a maximal Cohen-Macaulay module
Let $(R,m)$ be a Cohen-Macaulay local ring of dimension $d\geq 2$ and $M$ an module with depth$M=d.$
Is there any example of $M$ such that
$(1)$ $M_p$ is not free for some $p\in Ass(R)$ and
$(2)$...
- 323
0
votes
1
answer
112
views
Adic filtration and integral closure
Let $(R,m)$ be a Noetherian local domain whose integral closure $S$ is too. Also assume that $S$ is module-finite over $R$.
Let $x\in m^k\setminus m^{k+1}$ and $u\in S^\times$ such that $ux \in R$. ...
- 3,031
2
votes
0
answers
217
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 ...
- 1,703
0
votes
1
answer
72
views
A question on infinite local rings which are not division ring
Is it true that if $(R,m)$ is a (not necessary commutative) local ring then $R$ and $m$ have the same cardinal ? (Exclude TWO trivial cases: when $R$ is finite and when $R$ is a division ring)
On ...
- 271
1
vote
0
answers
278
views
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.
- 1,703
0
votes
1
answer
249
views
Analytic spread of localization of an ideal
Let $J$ be an ideal in a Noetherian local ring $(R,m)$. It is well known that for any prime ideal $p\in Spec(R)$, $l(J_p)\leq l(J)$, where $l(J)$ is the analytic spread of $J$.
Q) Are there ...
- 1,703
5
votes
1
answer
486
views
how to pass from algebraic power series to the analytic ones
Fix a field of zero characteristic, $k$, e.g. $\Bbb{R}$ or $\Bbb{C}$. Suppose $k$ is normed (and complete for its norm). Consider the ring extensions: $k[x_1,..,x_n]\subset \ k<x_1,..,x_n> \ \...
- 2,212
1
vote
0
answers
112
views
Asymptotic stability of prime divisors
Suppose $I$ is an ideal in a formally equidimensional local ring $R.$ Let $A(I)$ and $\overline A(I)$ denote Ass$R/I^n$ and Ass$R/\overline{I^n}$ for all large $n$ respectively.
My question is
What ...
- 1,703
0
votes
1
answer
273
views
Behaviour of length function under faithfully flat extension
Let $(R,m)$ and $(S,n)$ be local Noetherian rings such that $S$ is a faithfully flat extension of $R$. Let $J\subsetneq I $ ideals of $R$.
Can we relate $l_R(I/J)$ and $l_S(IS/JS)$?
PS: Here $l(-)$ ...
- 1,703
1
vote
0
answers
133
views
Intersections of Noetherian regular local rings of finite Krull dimensions
Let us consider Noetherian regular local rings $R_i$ of finite Krull-dimensions for each $i \geq 1$ such that
\begin{equation*}
R_1 \supset R_2 \supset \cdots
\end{equation*}
Suppose each embedding $\...
- 781
0
votes
2
answers
445
views
Almost complete intersection ideal and $d$-sequence
In a Noetherian local ring $R$, an ideal $I$ is called an almost complete intersection ideal if $\mu(I)=\text{ht}(I)+1$.
Q) Is it true that $I$ is generated by a $d$-sequence?
- 1,703
2
votes
1
answer
549
views
Example of a locally complete intersection ideal
Let $(R,\mathfrak m)$ be a Noetherian local ring.
Definition: $I$ is called locally complete intersection ideal if $I_p$ is a complete intersection for all $p\in V(I)$.
I want an example of an ideal ...
- 323
2
votes
0
answers
78
views
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 \...
- 941
1
vote
1
answer
288
views
Properties of d-sequence
Let $x_1,\ldots,x_n$ is a sequence in a Noetherian local ring $R$. We say $x_1,\ldots,x_n$ is a $d$-sequence if
1) $x_i\notin (x_1,\ldots,\hat{x_i},\ldots,x_n),$
2) for all $k\geq i+1$ and all $i\...
- 1,703
2
votes
0
answers
176
views
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,...
- 7,549
1
vote
1
answer
51
views
When is $rad(L)[x_1,\ldots]$ radical in $Ker(\varphi_\ast)$?
Suppose we have a local ring $L$ (not necessarily commutative) such that $L/rad(L)$ is a division algebra (here $rad(L)$ is the Jacobson radical of $L$). We clearly have the canonical surjection $\...
- 98
1
vote
1
answer
138
views
Is there a commonly used name and notation for $\beta(n)=\dim_{A/\mathbf{m}}\mathbf{m}^{n}/\mathbf{m}^{n+1}$, where $(A,\mathbf{m})$ is a local ring?
I've recently found myself doing some work on local rings,
and I found the following quantity keeps popping up-
Let $A$ be a local commutative unital ring, with maximal ideal $\newcommand{\mfr}{\...
- 993
3
votes
1
answer
331
views
uniqueness of uniformizers
Let R be a noetherian normal domain (if it makes any difference, I'm happy to assume R is also local).
If $p$ is a height one prime, then the localization $R_p$ is a dvr, hence the maximal ideal $...
- 1,869
2
votes
0
answers
145
views
Question about Ext$^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_{...
- 26k
5
votes
0
answers
206
views
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 ...
- 26k
1
vote
2
answers
733
views
Krull-dimension of local domain
Let $(R,{\frak m}_R)$ be a local domain (not necessarily Noetherian). That is, $R$ is integral and ${\frak m}_R$ is the unique maximal ideal of $R$. Suppose that ${\frak m}_R$ is finitely generated.
...
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4
votes
1
answer
367
views
Can K[[T_1,...,T_∞]] be embedded into K[[X,Y]]?
In the MathOverflow question about common false beliefs, the following answer teaches us that there is an embedding $\iota_n \colon K[[T_1,...,T_n]] \hookrightarrow K[[X,Y]]$. Now let us define the ...
- 781
2
votes
1
answer
105
views
Is there a characterization of r(M) by local cohomology instead of Ext
For a Noetherian local ring $(R,m)$, and a finite $R$-module $M$ with $\operatorname{depth} M=t,$ type of $M$ is defined to be $r(M):=dim_{R/m}Ext^t \ (R/m, M).$
Is there a characterization of $...
- 1,355
1
vote
0
answers
157
views
Popescu-Neron Desingularization for K[[T_1,...,T_∞]]
Let $K[[T_1,...,T_n]]$ be a finitely many variables formal power series ring over a field $K$.
Dorin Popescu proved that there are smooth algebras $A_{\lambda}$'s which are of finite type over $K$ ...
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3
votes
1
answer
275
views
Let $f:R\to S$ be a local finite monomorphism .If $M$ is an Artinian $S$-module, is it an Artinian $R$-module?
$(R,m)$ and $(S,n)$ are local rings (commutative Noetherian with 1).
Let $f:R\to S$ be a local homomorphism/monomorphism ($f(m)\subseteq n$), such that the natural induced homomorphism $R/m\to S/n$ ...
- 1,355
2
votes
1
answer
245
views
initial and terminal objects in local rings [closed]
in a class gave me the concept of initial and terminal object. I started to look for this objects in different categories. I already proved that $\mathbb{Z}$ is initial and the zero-ring is terminal ...
- 23
2
votes
1
answer
492
views
Structure theorem for non-Noetherian local rings
Is there a structure theorem (like Cohen 's structure theorem) for non-Noetherian local rings?
I am adding what I am looking for as someone asked in the comment.
If $R$ is a local domain (not ...
- 281
2
votes
1
answer
207
views
when there is an injection $0 \to R \to K_R$?
Let $(R,m)$ be a Cohen-Macaulay local ring which possesses the canonical module $K_R$. Then $R$ is said to be an almost Gorenstein local ring, if there is an exact sequence $0 \to R \to K_R \to C \to ...
- 1,355
7
votes
2
answers
699
views
invariants that can be measured by Local Cohomology
What invariants can be measured by Local Cohomology (and what application it has)?
As an example of what I mean:
Local Cohomology can measure invariants like depth and dim. So in some cases Local ...
- 1,355
10
votes
1
answer
793
views
Is $k(\!(x,y)\!)$ a topological field?
More generally, let $(R,m)$ be a Noetherian local domain with fraction field $K$. The $m$-adic topology turns $R$ into a topological ring. When $R$ is a discrete valuation ring, this topology extends ...
- 19.3k
4
votes
1
answer
153
views
Reference on the classification of (low rank) Gorenstein rings over $\mathbb{C}$
I am interested in the question of the classification of (low rank) Gorenstein rings over $\mathbb{C}$. The socle of a local algebra is the annihilator of its maximal ideal. A commutative local ring ...
- 41
3
votes
1
answer
120
views
Derivations annihilated by powers of the augmentation ideal
Consider an augmented commutative ring $R$, with augmentation ideal $\varpi$. Let $\delta$ be a derivation of $R$. The example I have in mind is $R=\mathbb F_p[x]/(x^{p^i})$ and $\delta=d/dx$, though ...
- 2,489
2
votes
0
answers
140
views
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
4
votes
0
answers
153
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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, ...
- 2,212
0
votes
1
answer
109
views
How to find ideals of finite length in a power series ring with special properties?
Let $A$ be the power series ring $\mathbb{C}[[x,y]]$.
Assume we are given two ideals $I,J$ of finite length in $A$ such that:
$xJ\subseteq I\subseteq J$
Is it possible to find ideals of finite ...
- 1,015
3
votes
1
answer
437
views
Castelnuovo-Mumford regularity in multigraded case
Let $R=\oplus_{n\geq 0}R_n$ be a standard Noetherian commuative graded ring over a local ring $(A,m)$ where $R_0=A.$ Put $R_+=\oplus_{n\geq 1}R_n.$ Let $M$ be a finitely generated $\mathbb Z$-graded $...
- 1,703
2
votes
1
answer
250
views
Are the fibers of this morphism geometrically regular?
Let $A\rightarrow B$ be a local morphism of complete noetherian rings making $B$ a formally smooth $A$-algebra. Does the induced morphism $\textrm{Spec}(B)\to\textrm{Spec}(A)$ have geometrically ...
- 255
2
votes
2
answers
241
views
if $R$ is Noetherian local with a finite module of finite injective dimension and if "?" , then $R$ is "Gorenstein"
I know that if $R$ is Noetherian local with a finite module of finite injective dimension, then $R$ is Cohen-Macaulay.
Can one add assumptions on $M$, so that $R$ be Gorenstein or Complete ...
- 1,355
5
votes
0
answers
94
views
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
1
vote
1
answer
499
views
Reducedness of a ring with prime nilradical
Let $A$ be a regular ring and $\mathfrak q$ be an ideal, such that $\sqrt{\mathfrak q}$ is prime. Further assume that $\mathfrak q$ is locally principal (i.e. $\mathfrak q$ is an irreducible divisor ...
- 33
2
votes
1
answer
284
views
finiteness dimension
$R$ is a local Noetherian ring. $f_I(M)$, the finiteness dimension of a module $M$ relative to $I$, is defined in ...
- 1,355
7
votes
1
answer
614
views
automorphisms of local rings vs local change of coordinates
Let $R$ be a local (commutative, associative) ring over a field of zero characteristic. (My typical examples are $k[[x_1,..,x_p]]/I$, $k\{x_1,.,x_p\}/I$, $C^\infty(\Bbb{R}^p,0)$. If it helps one can ...
- 2,212
17
votes
4
answers
4k
views
Completion of a local ring of a curve
Let $X$ be a smooth projective irreducible curve defined over an algebraically closed field $\mathbb{K}$ (of arbitrary characteristic), and let $p\in X$ be a closed point. Denote by $\mathcal{O}_p(X)$ ...
- 7,652