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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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{...
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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 ...
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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 ...
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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
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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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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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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 ...
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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 $R$. ...
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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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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 ...
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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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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
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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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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 ...
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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 ...
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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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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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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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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
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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 ...
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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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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
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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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A sort of "minimal presentation " for a local ring essentially of finite type over a field

Let $k$ be a field of characteristic $0$. Let $(R,\mathfrak m)$ be a local ring essentially of finite type over $k$ (https://stacks.math.columbia.edu/tag/07DR). Then, $R$ is the homomorphic image of ...
strat's user avatar
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Connection between heights of $(U:_R E)$ and $(ann(E):_R ann(U))$ for torsion-free modules $U\subseteq E$ of constant rank

Let $R$ be a Noetherian local ring. Let $E$ be a finitely generated torsion-free $R$-module of constant rank $e$. Let $s$ be an integer such that $s\geq e+1$. Let $U$ be an $R$-submodule of $E$ and ...
Snake Eyes's user avatar
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$1$-dimensional reduced local ring $R$ such that $(R:_{Q(R)} \overline R)$ is non-zero and consists of zero-divisors

This question is motivated by If $(R:_{Q(R) } S)$ is non-zero, then does $(R:_{Q(R) } S)$ contain a non-zero-divisor? (and a now deleted comment on it). Namely: Does there exist a one-dimensional ...
Muni's user avatar
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1 answer
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Determine whether a ring is local

Is the ring $$ \text{Span}\left\{\text{id}_{2x2},\begin{pmatrix} 0 & t \\ 1 & 0\end{pmatrix}\right\}$$ a local ring on $\mathbb{F}_{2}(t)$? I'm having some troubles with this...
Martin Gale's user avatar
4 votes
2 answers
91 views

Multiplicity of intersection of $y^2=x^3$ and $x^2=y^3$ at the origin

Those curves intersect at the origin with multiplicity 4, if I did everything correctly. In fact, parametrizing by $t \mapsto (t^2,t^3)$ the first curve and plugging into the second, yields $t^4=t^9$, ...
Harnak's user avatar
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Quotient of the polynomial ring over a local noetherian ring is finitely generated?

I am going through these lecture notes, and I am confused about the proof of Corollary 10.10 there. Corollary 10.10. Let $A$ be a local noetherian domain with maximal ideal $\mathfrak{p}$, let $g \in ...
Kyaw Shin Thant's user avatar
2 votes
1 answer
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If $M\cong N \oplus k^{\oplus a}$, where $a=\dim_k \dfrac{soc(M)}{soc(M)\cap \mathfrak m M }$, then $k$ is not a direct summand of $N$?

Let $M$ be a finitely generated module over a commutative Noetherian local ring $(R,\mathfrak m, k)$ such that $M\cong N \oplus k^{\oplus a}$, where $a=\dim_k \dfrac{soc(M)}{soc(M)\cap \mathfrak m M }$...
feder's user avatar
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1 answer
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When $R/I \cong S/J$, where $R$ is Cohen-Macaulay, $S$ is regular local and $ht(J)=\mu(J)$ [closed]

Let $(R,\mathfrak m)$ be a local Cohen-Macaulay ring. Let $I\subseteq \mathfrak m$ be an ideal of $R$. If $R/I \cong S/J$ for some regular local ring $S$ and ideal $J$ of $S$ such that $ht(J)=\mu(J)$, ...
feder's user avatar
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Lifting Idempotents in McDonald's Finite Rings w/ Identity Theorem XIV.3

Assume that $S$ and $R$ are finite commutative local rings with maximal ideals $M$ and $m$ respectively. I am working through the proof of Theorem XIV.3 in McDonald's Finite Rings with Identity The ...
dbossaller's user avatar
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1 answer
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stalk isomorphism + homeomorphism = isomorphism of varieties

I'm going over some old exercises in Chapter I of Hartshorne and am stuck on the reverse direction of Exercise I.3.3(b), which should follow straight from definitions. If $\varphi:X\to Y$ is a ...
mbert's user avatar
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1 answer
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Local ring of polynomials and intersection theory [duplicate]

I've been reading the Andreas Gathmann's notes https://agag-gathmann.math.rptu.de/class/curves-2018/curves-2018.pdf about plane algebraic curves, and I'm trying to solve this exercise: Exercise 2.17. (...
user34977's user avatar
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Definition of $k$-rational points on an algebraic set

I am learning introductory algebraic geometry by myself. Probably I am misunderstanding something. Could you point out where I mistake? Let $k$ be a field that is not necessarily an algebraically ...
Kazune Takahashi's user avatar
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On finding certain minimal generating set of the maximal ideal in a local ring of positive depth

Let $(R,\mathfrak m)$ be a local ring with depth $R=2$ and $\mu(\mathfrak m)=4$. Then, does there exist $x_1,x_2, x_3, x_3\in \mathfrak m$ such that $\mathfrak m=(x_1,x_2,x_3,x_4)$ and $x_i,x_j$ is an ...
Alex's user avatar
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0 answers
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Examples of Gorenstein local unique factorization domain of dimension $2$ and embedding dimension $4$

I am looking for an example of a local UFD $(R,\mathfrak m)$ of dimension $2$ which is also Gorenstein and $\mathfrak m$ is minimally generated by four elements. Does there exist any such examples?
Snake Eyes's user avatar
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Prove that if $P ∈ C(\overline{K})$ is a pole of $f ∈ K(C)$, then $v_P(f) < 0$ and $P$ is a zero of $1/f$.

Let $C$ is a genus-2 hyperelliptic curve defined over a finite field $K$. As $P$ is pole of $f$, $f \in K(C)$ such that $f\equiv \frac{f_1}{f_2}$, then $f_2(P)=0$, so $\left(\frac{1}{f}\right)(P)=\...
George's user avatar
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1 vote
1 answer
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non-free module of finite projective dimension and locally free on punctured spectrum on a local non-Cohen-macaulay ring

Consider the local ring $R=\mathbb C [[x,y,z]]/(xy, yz)$. My question is: Does there exist a non-free finitely generated $R$-module $M$ of finite projective dimension such that $M_P$ is $R_P$-free for ...
Alex's user avatar
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2 votes
1 answer
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On local rings $(R, \mathfrak m)$ such that $\text{Spec}(R)$ is disjoint union of $\text{Ass}(R)$ and $ \{\mathfrak m \}$

Let $(R, \mathfrak m)$ be a Noetherian local ring such that $\mathfrak m \notin \text{Ass}(R)$ and $\text{Spec}(R)=\text{Ass}(R)\cup \{\mathfrak m \}$. Then, must $R$ be Cohen-Macaulay? Of course the ...
Alex's user avatar
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0 answers
36 views

left bounded chain complex $Z$ such that $Z \otimes_R^{\mathbf L} (R_P/PR_P)$ is uniformly right bounded as $P$ varies over prime ideals of $R$

Let $(R,\mathfrak m)$ be a Noetherian local ring. Let $Z$ be a left bounded chain complex of finitely generated $R$-modules. If there exists an integer $n$ such that for every prime ideal $P$ of $R$, ...
Snake Eyes's user avatar
1 vote
1 answer
95 views

Ideal of definition of local ring

Definition. Let $R$ be a Noetherian local ring with maximal ideal $\mathfrak{m}$. An ideal $I$ is called an ideal of definition of $R$ if $\mathfrak{m}^n \subset I \subset \mathfrak{m}$ for some $n\...
A  Narode's user avatar
1 vote
0 answers
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When is the intersection of every non-zero ideal with socle again non-zero?

Let $(R,\mathfrak m)$ be a Noetherian local ring such that $(0:_R \mathfrak m)\neq 0$. Then, is it true that for every non-zero ideal $I$ of $R$, one also has $I\cap (0:_R \mathfrak m)\neq 0$ ? I know ...
feder's user avatar
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2 votes
1 answer
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Freeness of $I/I^2$ implies that of $\dfrac{I}{I^2+xR}$ over $R/I$?

Let $(R,\mathfrak m)$ be a Noetherian local and $I$ be an ideal of $R$ such that $\sqrt{I}=\mathfrak m$. Let $x\in I\setminus I \mathfrak m$ be such that $x$ is a non-zerodivisor on $R$. If $I/I^2$ is ...
Snake Eyes's user avatar
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1 answer
152 views

Residue field of DVR

Let $A$ be a local ring with maximal ideal $\mathfrak{m}$ and residue field $K = A/\mathfrak{m}$ and suppose that $A$ is also a $k$-algebra. Under what conditions can I say that $K/k$ is a finite ...
Joseph Harrison's user avatar

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