Questions tagged [local-rings]
The local-rings tag has no usage guidance.
218 questions
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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
2
votes
0
answers
87
views
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=...
user341790
3
votes
3
answers
486
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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 \...
- 361
3
votes
0
answers
69
views
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
2
votes
2
answers
622
views
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 ...
- 31
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 ...
- 259
0
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0
answers
202
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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 ...
- 930
1
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1
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118
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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)}=(...
- 1,713
2
votes
1
answer
348
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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 $\...
- 63.1k
4
votes
2
answers
458
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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 ...
- 279
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votes
0
answers
399
views
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
1
vote
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
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0
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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
1
answer
235
views
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 ...
- 2,837
1
vote
1
answer
264
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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 $...
- 2,837
6
votes
1
answer
318
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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 ...
- 63.1k
0
votes
1
answer
248
views
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
vote
1
answer
147
views
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 ...
6
votes
1
answer
219
views
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 ...
- 435
5
votes
1
answer
395
views
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 ...
- 2,834
4
votes
0
answers
265
views
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
6
votes
1
answer
499
views
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.
1
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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^...
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2
votes
1
answer
152
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When can we decompose a multivariable p-adic power series into product of single variable power series?
Is there any known result of decomposing multivariable power series over $p$-adic field into product of single variable power series ?
For example, consider the following power series in $n$ variables:...
- 930
9
votes
2
answers
815
views
Ideal norm in orders
Let $\overline{T}$ be a Dedekind ring such that $\overline{T}/\overline{I}$ is finite for every nonzero ideal $\overline{I}$ of $\overline{T}$. Let $T$ be a subring of $\overline{T}$ with the same ...
- 328
8
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0
answers
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
6
votes
2
answers
421
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Does the category of local rings with residue field $F$ have an initial object?
Let $F$ be a field. Does the category $C_F$ of local rings $R$ equipped with a surjective morphism $R\longrightarrow F$ have an initial object?
This is, for instance, true if $F=\mathbb{F}_{p}$ for ...
- 1,805
3
votes
2
answers
338
views
Isomorphism between finite algebras over ${\Bbb Z}_p$
Let $\pi \colon R \twoheadrightarrow {\Bbb T}$ be a surjective ring homomorphism between finite algebras over ${\Bbb Z}_p$. Further, we suppose the following three conditions$\colon$
$R$ is a ...
- 563
4
votes
1
answer
158
views
injective hulls in mixed characteristic
Let $R=\underleftarrow\lim (R/\mathfrak m^i)$ be a complete local ring, with residue field $k=R/\mathfrak m$,
and let's assume that $R$ is Noetherian.
If $R$ is a $k$-algebra, then
I believe that ...
- 43.2k
10
votes
2
answers
1k
views
Krull dimension of a local ring and completion
Let $A$ be a local ring (not noetherian) of finite Krull dimension such that its maximal ideal $\mathfrak{m}$ is of finite type.
Let $\hat{A}$ be its $\mathfrak{m}$-adic completion.
Do we have that $\...
- 3,472
0
votes
0
answers
440
views
A question about strict henselian local rings
Let $f: X\to Y$ be a finite morphism of schemes, let $y\in Y, Y(\bar{y}):={\rm Spec}(\mathscr{O}_{Y, \bar{y}}), X_{\bar{y}}:=X\times_Y{\rm Spec}(\kappa_y^s)=X_y\otimes_{\kappa_y}\kappa_y^s, X({\bar{y}}...
- 1,906
1
vote
0
answers
94
views
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
1
vote
1
answer
224
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Power series ring $\Theta[[X_1,\ldots,X_d]]$ and prime ideals
Let $\Theta$ be a domain. We shall choose $d$ elements $\theta_1,\ldots,\theta_d \in \Theta$ such that any chosen $j$ elements $\theta_{i_1},\ldots,\theta_{i_j}$ form a prime ideal $(\theta_{i_1},\...
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0
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287
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On the product in the power series ring
Let $A_n \colon= K[[X_1,\ldots,X_n,Y_1,\ldots,Y_n]]$ be a power series ring over a field $K$ in $2n$ variables and ${\frak m}_{A_n}$ be the unique maximal ideal of $A_n$.
Suppose we have two ...
- 563
1
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
7
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1
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268
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Additive group of local rings
Is there a theory or characterization for those finite $p$-groups that can be considered as the additive group of a finite local commutative ring with identity?
- 155
-1
votes
1
answer
231
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Finite extension of $K[[X]]$ and the norm
Let $R \colon= K[[X]]$ be a formal power series ring over a field $K$. We consider a monic polynomial $f(T) \in R[T]$ as follows$\colon$
$$
f(T) = T^e + c_{e-1}T^{e-1} + \ldots + c_1T + c_0.
$$
...
- 563
3
votes
1
answer
193
views
Complete local rings, automorphisms and approximation
Consider two local morphisms $f,g: B\rightarrow A$ of noetherian complete local rings and $f$ surjective.
Does there exist an integer $n\in\mathbb{N}$, such that if $f=g \mod \mathfrak{m}_{A}^{n}$ ...
- 3,472
1
vote
0
answers
138
views
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
6
votes
0
answers
478
views
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
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
views
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
3
votes
1
answer
239
views
commutative ring satisfying descending chain condition on radical ideals
Let $R$ be a commutative ring with unity which satisfies d.c.c. on radical ideals. then does $R$ satisfy a.c.c. on radical ideals ? If this is not true in general, then what happens if we also assume $...
- 1,209
0
votes
1
answer
339
views
Under which conditions on the homogeneous ideal $ I $, the quotient ring $ \mathbb{C} [X_0, \dots, X_n]/I $ is a regular ring?
If $ I $ is a homogeneous ideal of the ring of homogeneous polynomials $ \mathbb {C} [X_0, \dots, X_n] $ , under which conditions on the homogeneous ideal $ I $, and particularly on $ I_m $, the $m$ -...
- 325
8
votes
1
answer
257
views
Minimal resolution of local cohomology module
Let $(R,\mathfrak m)$ be a Noetherian local ring of dimension $d\geq 1.$ Suppose $\lambda(H_{\mathfrak m}^i(R))<\infty$ for some $0\leq i\leq d-1.$
Question Can we say anything about Betti numbers ...
- 1,713
6
votes
0
answers
241
views
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
-2
votes
2
answers
764
views
Reduced ring with all non-prime ideals finitely generated
Let $R$ be a reduced ring with all non-prime ideals finitely generated. Then is $R$ Noetherian ? If not, then is it true at least in the local case ?
Without reduced assumption, it is not true even ...
user111492
9
votes
1
answer
444
views
Rings with all non-prime ideals finitely generated
Motivated by this question, I would like to ask:
If all non-prime ideals in a ring are finitely generated, then is the ring Noetherian? Can we at least say anything in the local case?
Note that ...
user111492
5
votes
1
answer
1k
views
local ring all whose non-maximal ideals are finitely generated
Let $(R, \mathfrak m)$ be a commutative local ring such that every non-maximal ideal is finitely generated. Then, is $R$ Noetherian i.e. is $\mathfrak m$ finitely generated ideal ?
It is easy to see ...
user111524
1
vote
1
answer
465
views
Valuation ring whose maximal ideal and every ideal of finite height are principal
Let $(R, \mathfrak m)$ be a valuation ring such that $\mathfrak m$ and every ideal of finite height is principal. Then is $R$ Noetherian , i.e. a discrete valuation ring ?
user111492