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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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Valuation in the local ring of a curve

Let $k$ be an algebraically closed field and $A = k[X,Y]/(Y^2-X^3)$. Let $M$ be the ideal of $A$ generated by $X$, $Y$, and $A_M$ be the localization of $A$ at $M$. Is $A_M$ a discrete valuation ring?...
sunkist's user avatar
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Is $K[[X^2,X^3]]$ a local ring where $K$ is field? [closed]

I came a cross with this question "Is $K[[X^2,X^3]]$ a local ring where $K$ is field?" This is a ring of formal power series in which the terms are generated by $X^2$ and $X^3$ i.e. This ...
Elise9's user avatar
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Dimension of graded rings

Let $A=\oplus_{i \in N \cup \{0\}} A_i$ be a positively graded ring of dimension $d$ with $A_0=k$ and $k$ is a field. If $B$ is a Noetherian graded subring of $A$. Can we say dimension of $B$ cannot ...
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Module over local ring which is free after cutting down by a non-zero-divisor

Let $(R,\mathfrak m)$ be a commutative Noetherian local ring and $M$ be an $R$-module such that $M\neq \mathfrak m M$. Let $x\in \mathfrak m$ be a non-zero-divisor on both $R$ and $M$. If $M/xM$ is a ...
Alex's user avatar
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Does the uniformizer $\pi$ generate the extension $K/T$ where $T$ is the inertia field?

I'm trying to prove the equivalence of the statements (1) $\sigma(x) \equiv x$ mod $\mathfrak{p}^{i+1}$ (that is, $\sigma \in V_i$, the $i^{th}$ ramification group) for all $x\in \mathcal{O}_{K,\...
ljfirth's user avatar
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$K^{ab}=K^{ur}M$, $M$ maximal, abelian, totally ramified

I read this property $K^{ab}=K^{ur}M$ where $M$ is a maximal totally ramified abelian extension of $K$ (local field) as a corollary of the following Let $L/K$ abelian extension of a local field $K$. ...
noradan's user avatar
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Can Nakayama Lemma apply to complex

Let $R$ be a Noetherian commutative local ring with maximal ideal $\mathfrak{m}$. Consider a complex of finitly generated projective $R$-mod (so it is free) $$0\rightarrow P_1\rightarrow P_2\...
An Zhang's user avatar
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Minimal generator systems of a finitely generated

Let $A$ a local commutative ring with unity and $M$ an $A$-module. If $M$ is finitely generated, its minimal generator systems have the same cardinal. I am trying to prove this claim in order to use ...
Daniel García's user avatar
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How to compute $\operatorname{Length}(k[[x_1,\dots,x_n]]/I)$ for some ideal $I$

Let $k$ be an algebraically closed field and $\mathcal{O}_0=k[[x_1,\dots,x_n]]$ be the ring of formal power series, $I$ be an ideal of $\mathcal{O}_0$ such that $\operatorname{Spec}(\mathcal{O}_0/I)$ ...
Peter Wu's user avatar
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Cohen--Macaulay rings $R$ of dimension more than $1$ such that $R/\mathfrak p$ is also Cohen--Macaulay for every minimal prime ideal $\mathfrak p$

Let $R$ be a Cohen--Macaulay local ring. If $\dim R=1$, then $R/\mathfrak p$ is also a Cohen--Macaulay ring for every minimal prime ideal $\mathfrak p$ of $R$. It is known that for $\dim R\ge 2$, this ...
Alex's user avatar
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Is there a "minimal free resolution" for non-finitely generated modules?

Let $(R,\mathfrak m)$ be a commutative local ring and $M$ be an $R$-module such that $\mathfrak m^n M=0$ for some $n>0$. Then, does there exist a projective (necessarily free by Kaplansky's theorem)...
strat's user avatar
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Using Nakayama's lemma in non-local ring

Let $R$ be a Noetherian integral domain of dimension one and $\mathfrak{m}$ an ideal such that $\text{dim }\mathfrak{m}/\mathfrak{m}^2=1$ as an $R/\mathfrak{m}$-vector space. The localization of $R$ ...
Navid's user avatar
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Uniformizers in one-dimensional local rings

Suppose that $R$ is a Noetherian ring with unique maximal ideal $\mathfrak{m}$. Further suppose that $\mathfrak{m}/\mathfrak{m}^2$ is a one-dimensional $R/\mathfrak{m}$-vector space. By an application ...
Navid's user avatar
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Complete set of local orthogonal idempotents in $\mathbb{Z}_n$

I want to find a set of local orthogonal idempotents in $\mathbb{Z}_n$, i. e. such idempotents $e_1, \ldots e_n \in \mathbb{Z}_n$ that $1 = e_1 + \ldots + e_n$, $e_i e_j = e_j e_i = 0, (i \ne j)$, and ...
Nickeil's user avatar
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Proof that if $R \ne 0$ - commutative Artinian ring then $R \cong R_1 \times \ldots \times R_n$ [duplicate]

I'm trying to proof the following proposition. If $R \ne 0$ - commutative Artinian ring then $R \cong R_1 \times \ldots \times R_n$, where each $R_i$ is local Artian. We known that fact, if $R$ is ...
Nickeil's user avatar
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When is the initial form of a principal ideal generated by the initial form of the original ideal's generator?

$ \DeclareMathOperator{\init}{in} \DeclareMathOperator{\gr}{gr} \newcommand{\calO}{\mathcal{O}} $Let $(R,\mathfrak{m})$ be a Noetherian local ring and $\gr_{\mathfrak{m}}(R)=\bigoplus_{i=0}^\infty\...
mbert's user avatar
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Let $A$ be a commutative local noetherian ring and let $I$ be a proper ideal of $A$. Prove that $\bigcap_{n=1}^\infty I^n = 0$.

Let $A$ be a commutative local noetherian ring and let $I$ be a proper ideal of $A$. Prove that $\bigcap_{n=1}^\infty I^n = 0$. The first thing I tried was to see that $$\displaystyle\bigcap_{n = 1}^\...
Squirrel-Power's user avatar
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Possible inequality of krull dimension of local injection of Noetherian local domains

If $(A, \mathfrak{m}) \hookrightarrow (B, \mathfrak{n})$ is a local injection of Noetherian local domains, do we necessarily have $\dim B \geq \dim A$?
AprilGrimoire's user avatar
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Finiteness of the intersection multiplicity of plane algebraic curves

Hello guys i am trying to solve excerise 2.7 page 14 from Gathmann notes https://agag-gathmann.math.rptu.de/class/curves-2023/curves-2023-c2.pdf Definition About (a) : Stuck here.Not sure how to ...
oti nane's user avatar
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How to compute a minimal spanning set and the minimal spanning number of $\Bbb C[x,y]_{(x-1,y-1)}/(x^3-y^2)$?

Let $A=\frac{\mathbb C[X,Y]}{(X^3-Y^2)}$. I am asked to show that $\mathbf m=(\overline X-1,\overline Y-1)$ is a maximal ideal of $A$ which I have shown successfully. Now I am asked to compute the $\...
Kishalay Sarkar's user avatar
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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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How to understand a situation where one can use Nakayama's lemma even when the situation is not Tailor-made.

In commutative algebra we have the following version of Nakayama's Lemma(also calle NAK lemma): NAK Lemma: Let $R$ be a local ring and $\mathbf m$ be the unique maximal ideal of $R$.Let $M$ be a ...
Kishalay Sarkar's user avatar
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Algebraic properties of the ring of analytic functions on the complex plane

Let $r, R>0$ and $D_{R}=\{z\in\mathbb{C}: \|z\|<R\}$ and $D_{r,R}=\{z\in\mathbb{C}: r<\|z\|<R\}$. Consider the following $\mathcal{R}_{1}:=\mathcal{O}_{R}=\{f:D_{R}\to\mathbb{C}: f\hspace{...
user 987's user avatar
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Dominating rings: $\mathfrak{m}_A=A\cap \mathfrak{m}_B\Longleftrightarrow \mathfrak{m}_A\subset \mathfrak{m}_B$

Exercise $27$ from Atiyah and MacDonald states that if $A,B$ are two local rings, then $B$ is said to dominate $A$ iff $A$ is a subring of $B$ and the maximal ideal $\mathfrak{m}_A$ of $A$ is ...
kubo's user avatar
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Quotients of local rings [closed]

I have the following commutative algebra question. It came up at the time of reading local rings of schemes and their tangent spaces. Let $R$ be a local ring with unique maximal ideal $m$. Then is it ...
KAK's user avatar
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2 answers
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A boolean ring which is local must be isomorphic to $\Bbb{F}_2$.

Recall: a Boolean ring is a (commutative) ring $R$ where $\forall x \in R: x^2=x$. I don't really know how to proceed. I have tried some things like If $x,y \ne 0$ in $R$ such that $x^2=x$ and $y^2=y$...
soggycornflakes's user avatar
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Free module over local ring $R$. [duplicate]

People often say that a module $M$ over a not necessarily neotherian local ring $R$ being projective is flat, and also free. However, some refer to the finitely-generatedness of $M$, i.e. $M$ being ...
Pierre MATSUMI's user avatar
3 votes
1 answer
95 views

Finitely generated torsion-free indecomposable modules over one-dimensional complete local domains are isomorphic to ideal?

Let $R$ be a complete local domain of dimension $1$. Let $M$ be a finitely generated torsion-free indecomposable $R$-module. Then, must $M$ be isomorphic to an ideal of $R$? Also clearly, $R$ embeds ...
uno's user avatar
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1 vote
1 answer
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Show that $A[T]/(P(T))$ is a local flat A-algebra where $(A,\mathfrak{m},k)$ is a local ring with $k[T]/(\tilde{P}(T))$ is a simple extension of $k$

Let $(A,\mathfrak{m},k)$ be a (Noetherian) local ring. Let $k[T]/(\tilde{P}(T))$ be a simple field extension of $k$ with $\tilde{P}(T)$ monic and irreducible (and separable). Lift $\tilde{P}(T)$ to ...
Z Wu's user avatar
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1 vote
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If $f\in \mathcal{O}_X(U)$ does not vanish in $U$ then it is a unit in $\mathcal{O}_X(U)$ [duplicate]

Let $X$ be an algebraic variety, $U\subset X$ open and $f\in \mathcal{O}_X(U)$ such that for all $P\in U$ we have that $f(P)\neq 0$. I want to prove that $f$ is a unit in $\mathcal{O}_X(U)$. My ...
kubo's user avatar
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1 vote
3 answers
122 views

A unital commutative ring having a square-zero maximal ideal is local

I have the following question in my homework. Let $R$ be a commutative ring with identity and $M$ be a maximal ideal of $R$ such that $M^2=\{0\}$. Show that $M$ is the unique maximal ideal of $R$. ...
Nothing special's user avatar
1 vote
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On the finiteness of certain Zariski closed subsets of the prime spectrum of commutative Noetherian local rings

Let $(R,\mathfrak m)$ be a Noetherian local ring. Let $S,T$ be Zariski closed subsets of $\text{Spec}(R)$ such that if $\mathfrak p\in S, \mathfrak q \in \text{Spec}(R)$ and $\mathfrak p \subsetneq \...
uno's user avatar
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5 votes
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On idempotents and local property of $\Lambda \otimes_R R_{\mathfrak p}$ where $\Lambda$ is module finite algebra over commutative local ring $R$

Let $R$ be a commutative Noetherian local ring. Let $\Lambda$ be a module finite associative $R$-algebra. Let $\mathfrak p$ be a prime ideal of $R$. I have the following two questions: (1) If $\...
uno's user avatar
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If $R$ is a local ring, the morphism of schemes $f: \operatorname{Spec} R \to X$ is determined by the image of the closed point? [duplicate]

I’m taking an introductory course on Scheme theory. In one of the proofs of the course, we were considering all morphisms of schemes $f: \operatorname{Spec} R \to X$, where $X$ is a scheme and $R$ is ...
Gokimo's user avatar
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2 answers
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If $f:\operatorname{Spec} R \to X$ is a map with $R$ local and with closed point landing in an open $U\subset X$, then $im(f)\subset U$ too.

I’m taking an introductory course on Scheme theory. In one of the proofs of the course, we were considering a situation where we have a morphism of schemes $f: \operatorname{Spec} R \to X$, where $X$ ...
Gokimo's user avatar
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6 votes
1 answer
215 views

Does this local ring have a name?

I came up with this: Let $S=\{(a_n)_{n\geq 1}\:|\:a_n\in \mathbb{C}, (a_n)_{n\geq 1}\text{ converges}\}$ be the set of convergent complex sequences. Then this set forms a ring under pointwise ...
semisimpleton's user avatar
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Proving two modules are free based on their direct sum [duplicate]

I have given two modules $M$ and $N$ over a local ring $R$. I also know that $M \oplus N \cong R^n$ for some $n\in \mathbb{N}$. I then have to prove that both $M$ and $N$ are free modules. Since $M \...
MarlonButBetter's user avatar
1 vote
1 answer
55 views

How to express elements of a DVR as power series of the uniformizer?

Let $R$ be a DVR with a fixed uniformizer $t$. My main considering example is the local ring of an algebraic curve over an algebraically closed field $k$, so let's assume $R$ is a $k$-algebra and has ...
S.Gau at Math's user avatar
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1 answer
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Is a flat extension of DVRs always finite?

Is it true that if $S/\mathbb{Z}_p$ is a flat extension of discrete valuation rings then $S$ is finite over $\mathbb{Z}_p$?
Fraz's user avatar
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Module over a commutative local ring has nonzero homomorphism

The question is from a Persian text book: Let $R$ be a commutative local ring with $1$, and $\mathfrak m$ its maximal ideal. If $M$ is a non-zero module over $R$ prove that $\text{Hom}_R(M,R/\mathfrak ...
Amir Mg's user avatar
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3 votes
1 answer
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Serre conditions and local isomorphisms

I encountered the following in the introduction of the paper "Duality for Koszul Homology over Gorenstein Rings": I assume it's easy, but why is this fact true? I played with modding out by ...
user2154420's user avatar
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-2 votes
1 answer
98 views

How can I show that if $x$ is a nonunit then $1-x$ is a unit? [duplicate]

Let $R$ be a commutative local ring. I want to show that then $x\in R$ is not a unit implies $1-x$ is a unit. My idea was the following: Since $R$ is a local ring we can chose $\mathfrak{m}$ to be ...
Summerday's user avatar
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1 vote
1 answer
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On pushforward functor and freeness of modules

Let $R$ be a commutative Noetherian ring. Given an $R$-module $M$ and a ring homomorphism $\phi:R\to R$, let $\phi^* M$ be the $R$-module whose underlying abelian group is the same as $M$ but the $R$-...
Alex's user avatar
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1 vote
1 answer
63 views

Equivalent definitions of a local algebra

Let $A$ be a unital, associative commutative algebra over a field $k$. In particular, $A$ is a ring and we call $A$ local if it is local as a ring. Namely, call $A$ local if $A$ has a unique maximal ...
Margaret's user avatar
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Betti numbers of associated graded ring

Let $(R,\mathfrak{m})$ be a (not necessarily commutative) local ring with commutative residue field $k$. Define it's betti numbers as $$\beta_i(R):=\dim_k\mathrm{Tor}^i_R(k,k).$$ The associated graded ...
Ben's user avatar
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3 votes
0 answers
109 views

Is the ring of formal power series a localization of some non local ring at prime ideals?

Consider $F = K[[x]],$ the formal series over field $K.$ We know that $K$ is a local ring with maximal ideal $(x).$ Does there exist a non local ring $R$ and prime ideal $p$ such that $R_p = K[[x]]?$ ...
Eloon_Mask_P's user avatar
1 vote
1 answer
51 views

Do we have $\bigcap_{1\leq i\leq r}(\mathfrak{m}^n+\mathfrak{p}_i)=\mathfrak{m}^n$ in a local ring, where $\mathfrak{p_i}$ are the minimal primes?

$\newcommand{\cO}{\mathcal{O}}\newcommand{\fm}{\mathfrak{m}}\newcommand{\fp}{\mathfrak{p}}$ Let $(\cO,\fm)$ be a Noetherian reduced local ring, essentially of finite type over an algebraically closed ...
imtrying46's user avatar
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0 votes
0 answers
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On Elkik's Theorem of Henselian pair.

As far as I know, the celebrated Elkik's result often quoted boil down as follows$\colon$ Let $A$ be a heneselian local ring with its maximal ideal ${\frak m}$ and choose an arbitrary ideal $I \subset ...
Pierre MATSUMI's user avatar
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1 answer
36 views

Prolongation valuation of non-complete valued field

Let $(K,v)$ be an algebraically closed complete valuation field. Let $(R,v|_R)$ be a valuation subring of $K$. Denote by $F^{\operatorname{sep}}$ the separable closure of $\operatorname{Frac}R$ in $K$....
Yijun Yuan's user avatar
1 vote
0 answers
69 views

Finiteness of the homology of the Koszul complex

Let $A$ be a local noetherian commutative ring, let $x_1, \dots, x_r$ be elements of the maximal ideal of $A$, $I$ the ideal that they generate, $M$ a $A$-module of finite type such that $M/IM$ is of ...
Plafonddeplatre's user avatar

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