Questions tagged [local-rings]

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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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  • 695
1 vote
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116 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 $\...
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5 votes
1 answer
298 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 ...
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0 answers
51 views

Freeness of injective hull of finite module over Gorenstein Artinian ring

This is Lemma 4.1 in Brochard, Khare, Iyengar: Wiles defect and criteria for freeness. I have managed to understand the proof, except for the following part: Why is $F$, the injective hull of a finite ...
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2 votes
1 answer
75 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'm also interested in a more general form of this statement. Let $A$ be a commutative algebra (over $\mathbb{R}$ or $\mathbb{C}$), $I$ is ...
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3 votes
0 answers
79 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 $A$ ...
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2 votes
1 answer
158 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 ...
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3 votes
1 answer
166 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 ...
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  • 183
8 votes
0 answers
184 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 $...
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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 answers
316 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 \...
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2 votes
0 answers
50 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 $...
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2 votes
2 answers
246 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 ...
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4 votes
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129 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 ...
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201 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 ...
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  • 754
1 vote
1 answer
86 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)}=(...
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  • 1,643
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82 views

Complete local Noetherian ring whose normalized dualizing complex has depth equal to dimension of the ring

Let $(R, \mathfrak m,k)$ be a complete Noetherian local ring. Let $D$ be a normalized dualizing complex of $R$. If $\text{depth}_R D=\dim R$, then must $R$ be Cohen-Macaulay (i.e. must $D$ be actually ...
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2 votes
1 answer
162 views

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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4 votes
2 answers
284 views

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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5 votes
0 answers
206 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 ...
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1 vote
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164 views

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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  • 2,573
1 vote
0 answers
103 views

$A \to B$ 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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  • 2,573
1 vote
1 answer
180 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 ...
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  • 2,573
1 vote
1 answer
180 views

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 $...
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  • 2,573
5 votes
1 answer
223 views

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 ...
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0 votes
1 answer
231 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?
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1 vote
1 answer
135 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 ...
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6 votes
1 answer
173 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 ...
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5 votes
1 answer
267 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 ...
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  • 2,686
4 votes
0 answers
157 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/\...
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5 votes
1 answer
463 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.
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1 vote
0 answers
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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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  • 754
2 votes
1 answer
118 views

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:...
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  • 754
9 votes
2 answers
572 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 ...
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  • 183
8 votes
0 answers
204 views

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 ...
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6 votes
2 answers
323 views

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 ...
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3 votes
2 answers
310 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 ...
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  • 555
4 votes
1 answer
143 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 ...
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10 votes
2 answers
707 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 $\...
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  • 3,151
0 votes
0 answers
288 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}}...
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  • 1,614
1 vote
0 answers
60 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 ...
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1 vote
1 answer
176 views

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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  • 555
0 votes
0 answers
269 views

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 ...
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  • 555
1 vote
0 answers
147 views

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 $m_x$. ...
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  • 679
7 votes
1 answer
252 views

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?
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  • 155
0 votes
1 answer
166 views

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$\colon$ $$ f(T) = T^e + c_{e-1}T^{e-1} + \ldots + c_1T + c_0. $$ ...
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  • 555
3 votes
1 answer
155 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 $f=g \mod \mathfrak{m}_{A}^{n}$ ...
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  • 3,151
1 vote
0 answers
135 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,\...
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  • 555
5 votes
0 answers
375 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 ...
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  • 4,283
1 vote
0 answers
213 views

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 \...
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