Questions tagged [local-rings]

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Hilbert–Samuel multiplicity under hypersurface sections

Let $\newcommand{\frakm}{\mathfrak{m}}(R,\frakm)$ be a reduced Noetherian local ring of dimension $d$ and $f\in\frakm^\alpha\setminus\frakm^{\alpha+1}$ a parameter of $R$, i.e. $\dim R/(f)=d-1$. Let $...
mbert's user avatar
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1 vote
0 answers
158 views

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 ...
Arshak Aivazian's user avatar
1 vote
0 answers
76 views

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 $R$ ...
Alex's user avatar
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2 votes
1 answer
140 views

The presentations of finite complete local rings

Suppose that $R$ is a commutative ring such that there is a surjection $ \pi:\mathbf{Z}_p[[T_1,\cdots,T_n]]\to R$ of rings where $\mathbf{Z}_p[[T_1,\cdots,T_n]]$ is the ring of formal power series ...
stupid boy's user avatar
5 votes
1 answer
176 views

Are module finite algebras over semiperfect rings again semiperfect?

Let $S$ be a Noetherian semiperfect ring (https://en.m.wikipedia.org/wiki/Perfect_ring). Let $R$ be a module finite associative $S$-algebra. Then, is $R$ also a semiperfect ring? (Clearly, $R$ is ...
uno's user avatar
  • 280
3 votes
1 answer
326 views

Diagonal morphism of henselization is an open immersion?

Let $(R,\mathfrak{m})$ be a local ring, denote by $R \rightarrow R^h$ its henselization. Write $S = \operatorname{Spec} R$ and $S^h = \operatorname{Spec} R^h$. Is it true that the diagonal morphism $\...
Hugo Zock's user avatar
2 votes
1 answer
133 views

Is $\mathrm{Hom}_R(M,R)\neq \mathfrak m \mathrm{Hom}_R(M,R)$ if $M \neq \mathfrak m M$ and $\mathrm{Hom}_R(M,R)\neq 0$?

Let $(R,\mathfrak m)$ be a commutative Noetherian local ring and $M$ be an $R$-module such that $M \neq \mathfrak m M$ and $\mathrm{Hom}_R(M,R)\neq 0$. Then, is it true that $\mathrm{Hom}_R(M,R)\neq \...
Snake Eyes's user avatar
3 votes
1 answer
222 views

Vanishing of $\operatorname{Ext}_R^{1}(M,R)$ when $R$ is a Gorenstein local ring of dimension $1$ and $M$ is not finitely generated

Let $(R,\mathfrak m)$ be a Gorenstein local ring of dimension $1$. Let $M$ be an $R$-module (not finitely generated) such that $M\neq \mathfrak m M$ and there exists a non-zero-divisor $x\in \mathfrak ...
Snake Eyes's user avatar
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1 answer
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Is the integral closure of a henselian local domain of dimension $1$ again local?

Let $(R,\mathfrak m)$ be a local domain of dimension $1$. Let $\overline R$ be the integral closure of $R$ in the field of fractions $Q(R)$. If $R$ is henselian, then is $\overline R$ also a local ...
uno's user avatar
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0 answers
174 views

Finite monomorphism $A \to B$ with reduced $A$ and special fiber implies $B$ reduced

I have a question about correctness of following statement claimed here in $\boxed{2} \ $: Let $k$ arbitrary field, let $f : X \longrightarrow Y$ be a finite dominant morphism between finite type $k$-...
user267839's user avatar
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4 votes
1 answer
124 views

DG algebra structure on minimal free resolution of modules over regular local ring

Let $(Q, \mathfrak n, k)$ be a regular local ring. Let $I\subseteq \mathfrak n^2$ be an ideal, and fix a minimal generating set $\mathbb f= f_1,\cdots, f_n$ of $I$. The Koszul complex $E:= Q[e_1,...,...
uno's user avatar
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2 votes
1 answer
109 views

Local rings whose the endomorphism rings of E(R/J) is division ring

Let $R$ be a local ring with maximal ideal $J$. Assume that ${\rm End}_{R}({\rm E}(R/J))$ is a division ring (${\rm E}(R/J)$ means the injective envelope of $R/J$). Does $R/J$ is injective?
Najmeh Dehghani's user avatar
1 vote
0 answers
145 views

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 ...
Amit Phogat's user avatar
6 votes
1 answer
401 views

Ring in which $x^n-x$ is central for every $x$

Let $R$ be a ring , $n \gt 1$, such that for all $x \in R$: $x^n-x \in Z(R)$, the center of $R$. Does it follow that $R$ is commutative? For $n=2,3$ this is pretty straightforward to prove. But what ...
Nicky Hekster's user avatar
2 votes
2 answers
238 views

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 ...
SKS's user avatar
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5 votes
1 answer
371 views

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 ...
Jens Hemelaer's user avatar
4 votes
0 answers
72 views

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 $...
Ehud Meir's user avatar
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0 votes
0 answers
48 views

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,\...
Cusp's user avatar
  • 1,703
1 vote
0 answers
122 views

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) ...
Snake Eyes's user avatar
3 votes
1 answer
131 views

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-...
cs89's user avatar
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1 vote
0 answers
118 views

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 ...
BenjaminY's user avatar
2 votes
0 answers
95 views

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}...
strat's user avatar
  • 301
0 votes
1 answer
264 views

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)=\...
Serge the Toaster's user avatar
3 votes
1 answer
110 views

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 ...
Madeleine Birchfield's user avatar
4 votes
1 answer
214 views

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 =\...
feder's user avatar
  • 63
0 votes
1 answer
105 views

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 $J$ ...
feder's user avatar
  • 63
5 votes
1 answer
229 views

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 ...
a196884's user avatar
  • 323
2 votes
1 answer
297 views

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#...
Alex's user avatar
  • 343
1 vote
1 answer
213 views

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 ...
uno's user avatar
  • 280
1 vote
0 answers
82 views

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 ...
Ehud Meir's user avatar
  • 4,979
7 votes
1 answer
274 views

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 ...
David Schwein's user avatar
21 votes
1 answer
1k views

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 ...
Michael Kinyon's user avatar
4 votes
1 answer
420 views

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 $\eta_Z$ ...
xir's user avatar
  • 1,964
5 votes
2 answers
218 views

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 $i\not= j$) ...
Antony's user avatar
  • 147
3 votes
4 answers
726 views

$R$ 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 ...
It'sMe's user avatar
  • 767
0 votes
0 answers
71 views

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(...
Adithya Chakravarthy's user avatar
1 vote
0 answers
181 views

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 $\...
BelowAverageIntelligence's user avatar
5 votes
1 answer
329 views

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 ...
Tireless and hardworking's user avatar
1 vote
2 answers
174 views

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) \...
Pintér Gergő's user avatar
3 votes
0 answers
90 views

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 $A$ ...
Fernando Peña Vázquez's user avatar
2 votes
1 answer
231 views

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 ...
babu_babu's user avatar
  • 229
3 votes
1 answer
228 views

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 ...
Rafael's user avatar
  • 183
8 votes
0 answers
275 views

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 $...
Will Sawin's user avatar
  • 137k
2 votes
0 answers
83 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=...
user avatar
3 votes
3 answers
438 views

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 \...
strat's user avatar
  • 301
2 votes
0 answers
63 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 $...
Ahmed Matar's user avatar
2 votes
2 answers
534 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 ...
Danimenru's user avatar
4 votes
0 answers
135 views

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 ...
Liddo's user avatar
  • 259
0 votes
0 answers
202 views

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 ...
MAS's user avatar
  • 870
1 vote
1 answer
111 views

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)}=(...
Cusp's user avatar
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