Questions tagged [local-rings]

In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or prime.

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If a finite sum is a unit, then it has a term that is a unit.

Source: Theorem 19.1 (A First Course in Noncommutative Rings by T.Y. Lam) Local Ring on Wikipedia Theorem 19.1 For any nonzero ring R, the following statements are equivalent: (1) $R$ has a unique ...
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Not-necessarily-unital von Neumann regular local commutative rings: are they fields?

In this question all rings are commutative, but don't necessarily have a multiplicative identity (so: commutative rngs). On Wikipedia there is the unsourced claim: It is well known that a local ring ...
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If Quotient ring is complete local is the ring complete local?

Let R be a Noetherian commutative ring and N the nil radical. Given R/N is a complete local ring, is R also a complete local ring? If R had two maximal ideal m and n then m+N and n+N are maximal ...
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Is this a complete local ring?

Is R = $\mathbb C [X]/(X^5)$ a complete local ring? I can see that $(X)$ is a maximal ideal (here I mean the ideal generated by image of X). And any prime ideal and hence maximal ideal will contain $...
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Completion of a polynomial ring over a complete ring

I'm learning about ring completions, and this question came to mind: If $R$ is a complete local ring with maximal ideal $\mathfrak{m}$ (e.g. $R = \mathbb{Z}_p$ or $R = k[[x]]$), is the completion of $...
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Finite dimensional local rings with infinitely many minimal prime ideals

Is there a finite dimensional local ring with infinitely many minimal prime ideals? Equivalent formulation: Is there a ring with a prime ideal $\mathfrak p$ of finite height such that the set of ...
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local versus graded free resolutions

I'm currently trying to learn about syzygies. Most material is written in the context of graded rings and/or graded modules but I'm interested in a specific question about local rings. Hence I need to ...
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Is $\mathbb{Q} [X,Y]/[x^{20},y^{20}]$ a local ring?

Is $\mathbb{Q}[X,Y]/[x^{20},y^{20}]$ is a local ring? My approach is to look for a maximal ideal, but got stuck how to find a single maximal ideal. Any feedback would be appreciated.
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Descending sequence for module with finite length

Let $(R,\mathfrak{m})$ be a Noetherian local ring and $M$ be a finite $R$-module with finite length. Then the descending sequence $$M\supseteq\mathfrak{m}M\supseteq\mathfrak{m}^2M\supseteq\cdots\...
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Flat extension of local rings with a specified extension of residue field [closed]

Let $(R, \mathfrak m_R, k)$ be a Noetherian local ring and $K$ be a field containing $k$. Then is it true that there is a Noetherian local ring $(S, \mathfrak m_S)$ and a flat ring homomorphism $f: ...
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An example of a local ring of $V$ at $p$ not ufd

Let be $O_{V,p}$ the local ring of an irreducible variety $V$ at point $p$, I would like to know an example where $O_{V,p}$ is not UFD.
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Localizations of $k[y,z]/(1-y^2+z^2)$ UFDs

Let $k$ be a non algebraically closed field with $i \not \in k$; equivalently the polynomial $T^2+1 \in k[T]$ is irreducible over $k$. How to prove or disprove that for the ring $R:=k[y,z]/(1-y^2+z^...
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Stalks of Plane Conic $C \cong \mathbb{P}^1$ are UFD

Assume $k$ is a alg closed field. Then it is easy to check that the morphism $\phi: \mathbb{P}^1 \to \mathbb{P}^2, (x_0:x_1) \mapsto (x_0^2: x_0x_1:x_1^2)$ induces an isomorphism between $\mathbb{P}^1$...
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Can commutative local rings have any non-zero zero divisors? [closed]

Can commutative local rings have any non-zero zero divisors? Is this possible?
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$R$ be local and suppose $x ∈ R$ satisfies $x^2 = x$

If we let $R$ be local and suppose $x ∈ R$ satisfies $x^2 = x$, then I have to show that $x = 0$ or $x = 1$. A commutative ring $R$ with $1$ is called local if $R − R^×$ is an ideal of $R$. If we ...
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Prove: $R$ is local $\iff$ $R$ has exactly one maximal ideal.

We have that a commutative ring $R$ with $1$ is called local if $R − R^×$ is an ideal of $R$. I have to proof the following: $R$ is local $\iff$ $R$ has exactly one maximal ideal. We have that every ...
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38 views

Is $(\mathbb{Z}/2\mathbb{Z})[T,T^{-1}]$ a DVR?

I want to check if $R = (\mathbb{Z}/2\mathbb{Z})[T,T^{-1}]$ is a DVR. What I tried is showing that/ checking if $R$ is a local noetherian integrally closed domain, with precisely two prime ideals, ...
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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 with residue field isomorphic to $F$ have an initial object? This is, for instance, true if $F=\mathbb{F}_{p}$ for some prime $p$: If $R$ is ...
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Is there an initial object in the category of local rings with given tangential space?

Let $F$ be a field and $V$ a vector space over $F$. Does the category $C_V$ of local rings with tangential space isomorphic to $V$ have an initial object? I know that the category $C_F$ of rings with ...
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Cancelling the canonical module in tensor products

Let $R$ be Cohen-Macaulay local ring with the canonical module $\omega_R$ and let $M$ and $N$ be two finitely generated $R$-modules. Assume that $$ \omega_R \otimes_R M= \omega_R \otimes_R N $$ Can ...
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Equality in proof of Theorem 14.9 from Harris's book

In the book Algebraic Geometry - A First Course by Harris, it is given a proof for the following theorem: Theorem 14.9. Let $\pi:X\rightarrow Y$ be a finite map of varieties. Then $\pi$ is an ...
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If in a commutative ring $R$ a maximal ideal is nilpotent, then $R$ is local [duplicate]

Let $R$ a commutative ring, and $M$ a maximal ideal of $R$. If there exists $n\in \mathbb{N}$ such that $M^n=0$, then $R$ is local. In general, I have proved that $R/M^n$ is local with a unique ...
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Dual of a module of finite projective dimension

Let $(R, {\frak m})$ be a Noehterian local ring and $M$ be finitely generated $R$-module of finite projective dimension. Is the projective dimension of ${\rm Hom}_R(M,R)$ finite?
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How much information about $R$ is encoded in its local rings OR in its tangent spaces if I know the topology of $\operatorname{Spec}(R)$?

Assume I know the topology of the spectrum $\operatorname{Spec}(R)=(X,\mathcal{O}_{X})$ of a reduced ring $R$, and I know what the local rings $\mathcal{O}_{X,x}$ look like for every $x\in X$. How ...
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57 views

Existence of system of parameters via prime avoidance

This is a question about the hint in exercise 11.3.I part (b) of Vakil's FOAG notes, which is to prove the existence of a system of parameters for a Noetherian local ring. The statement of the full ...
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Completion of finite local homomorphism.

Suppose that $(A,\mathfrak{m}_A)$ and $(B,\mathfrak{m}_B)$ are local rings and that $\varphi:A\rightarrow B$ is a local homomorphism (I am happy to assume that this morphism is quasifinite). Let $M$ ...
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Why $R[x,y,z,・・・] /(x^2,y^2,z^2,・・・)$ is $0$ dimmensional?

Let $R$ be a ring.Why $A=R[x,y,z・・・] /(x^2,y^2,z^2・・・)$ is $0$ dimmensional? I think if $R$ is algebraically closed, then there are bijection between $A$'s maximal ideal and $V(x^2,y^2,z^2・・・)$, so $A$...
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Ring appearing as a quotient of a local ring

I am working in $k[x,y]$ with two polynomials $F,G$ and the ideal $I = (x,y)$. I define $n$ and $m$ as the multiplicy of $F$ and $G$ at the point $(0,0)$. Let's denote $\mathcal{O}$ to be the ...
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On showing a set is Zariski open subset of $\mathfrak m/\mathfrak m^2$

Let $(R, \mathfrak m,k)$ be a Noetherian local ring such that the residue field $k$ is infinite. Let $n=\mu(\mathfrak m)$. Then $n=\dim_k(\mathfrak m/\mathfrak m^2)$ . By fixing $x_1,...,x_n \in \...
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Does a closed point of a scheme have an affine open environment with the same dimension?

Consider a scheme $X$, and an a closed point $x\in X$. I am wondering whether there is an affine open neighborhood $x\in U\subseteq X$ such that $$\dim \mathcal O_{X,x}=\dim U.$$ I tried the ...
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smooth morphisms and sheaf of differentials

Our course defined a smooth morphism in this way I'd like to know why this implies that the sheaf of differentials $\Omega_{X/S}^1$ is locally free of finite rank. Thanks in advance.
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Is the inclusion of a local $\mathbb R$-subalgebra of the algebra of continuous germs itself local?

Let $X$ be a topological space and $C^0_{X,x}$ be the local $\mathbb R$-algebra of germs at $x$ of real-valued continuous maps. The maximal ideal consists of germs vanishing at $x$. Let $A\leq C^0_{X,...
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Polynomial ring over an analytically unramified ring is locally analytically unramified

Let us call a Noetherian local ring $(R, \mathfrak m)$ to be analytically unramified if the $\mathfrak m$-adic completion of $R$ is reduced i.e. has no non-zero nilpotent. https://en.m.wikipedia.org/...
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On some basic property of a valuation ring

Let $R$ be a valuation ring. Suppose its field of fractions $k$ is algebraically closed. Let $(a_0, \ldots, a_n)$ be a tuple of elements in $k$. Then I would like to deduce that there exists some $b \...
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modules finitely generated over a local commutative ring

Let $R$ be a local commutative ring, that is, $R$ has a unique ideal maximal $J$. Let $M$ a finitely generated $R-$module such that $MJ = M$. Show that $M = 0$. if assume by contradiction that $M\neq ...
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The endomorphism ring of a uniserial module over a right Noetherian ring or a commutative ring is local

I found this example (Example 2.3 in Facchini's paper Krull-Schmidt Fails for Serial Modules) and couldn't quite understand the proof. The statement goes like this: Let $U$ be non-zero uniserial ...
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What is the local ring of a scheme $X$ along a subvariety $V$: $\mathcal{O}_{V,X}$?

I am trying to learn about scheme theoretic algebraic geometry, because I actually want to study the basics of interseciton theory. I stumbled across the term "local ring $\mathcal{O}_{V,X}$ of a ...
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Bijection Between Morphisms of Schemes and Local Ring Homomorphisms

Let $R$ be a local ring with maximal ideal $\mathfrak{m}$ and let $(X, \mathcal{O}_X)$ be a scheme. I want to show that there is a bijection: $$ \begin{align} & \quad \{ (f, f^\#) \in \mathrm{...
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$A/m_a \simeq B/m_b$? Where $A,B$ are local rings, $A\subset B$ and $m_a\subset m_b$

I'm currently reading a math book in french so I'm translating everything as I go and also proving the remarks made throughout. One remark that I haven't been able to prove is: if $A$ and $B$ are ...
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On the prime spectrum of completion of local rings

Let $(R, \mathfrak m)$ be the henselization of the local ring $\mathbb C[x,y]_{(x,y)}$ . Let $\hat R$ be the $\mathfrak m$-adic completion of $R$. Then there is a natural map $R \to \hat R$ which ...
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Local rings in the etale topology are henselizations?

I am reading Milne's lecture notes on etale cohomology and am now in the chapter on local rings in the etale topology. Let $X$ be some algebraic variety over an algebraically closed field $k$. Milne ...
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Prove that for local rings $R,S$, $M_m(R)\cong M_n(S)\implies R\cong S$ and $m=n$

Prove that for local rings (not necessarily commutative, and local means the nonunits form a two-sided ideal) $R,S$, $M_m(R)\cong M_n(S)\implies R\cong S$ and $m=n$. My first attempt was to consider ...
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Dimension of a certain finitely generated quotient module over a local ring.

I've been stuck on the following question from dimension theory in commutative algebra. Let $(A,m)$ be a local ring and $M$ a finitely generated $A$-module. Given $x_1,...,x_r \in \mathfrak{m}$, ...
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78 views

On reduction of ideals

Let $(R, \mathfrak m)$ be a Noetherian local ring of dimension $d>0$. Let $I$ be an $\mathfrak m$-primary ideal of $R$ i.e. $\sqrt I =\mathfrak m$ . How to show that there exists $x_1,...,x_d \in ...
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117 views

Quotient of a local Cohen-Macaulay ring by a minimal prime

Let $R$ be a local Cohen-Macaulay ring. Let $P$ be a minimal prime ideal of $R$. Is it true that $\operatorname {depth} R/P=\operatorname {depth} R$ ? Notice that since $R$ is local Cohen-Macaulay, ...
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The dimension of a valuations ring vs the dimension of its center

Question Let $A$ be a local domain and $\mathcal{O}$ a valuation ring of $\mathrm{Frac}\,A$ such that $A\subseteq \mathcal{O}$ and $\mathfrak{m}_A\subseteq \mathfrak{m}_{\mathcal{O}}$. That is, $\...
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Image of a local map is closed in $m$-adic topology

I'm reading rigid and formal geometry book of Bosch and in the proof of the proposition 4.2.3 it claims that if $f:A\to B$ is a finite local morphism between two local Noetherian rings such that the $...
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What is the local ring $\mathscr{O}_P(\mathbb A ^2)$?

I'm currently studying intersection numbers (which I don't really understand) as described in Fulton's Algebraic Curves. He proves that the intersection number (which he denotes $I(P, F \cap G)$), ...
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Regular local rings are normal

Let $(A, \mathfrak{m}, k)$ be a local ring, and let $\mathrm{gr}_{\mathfrak{m}}(A)$ be its associated graded ring. On page 123 of Atiyah-MacDonald, the following is stated without proof, and not given ...
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157 views

Counterexample: Local Artinian ring is not a PID

Specifically, I am looking for a commutative ring with unity, which is local and Artinian, but not a PID. I'm not too sure how to figure it out. I know the power series $\mathbb{F}\lbrack\lbrack x \...

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