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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64 views

Finite non-trivial extensions of local rings

I'm revising for exams and was looking over some old homework problems, and came across the following: Suppose $(A,M)$ and $(B,N)$ are local rings with $M\subseteq N$ and $A\subseteq B$, such that: $...
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68 views

Showing that injective limit is a local ring

Let $k$ be an algebraically closed field and let $(X,\mathcal{O}_X)$ be a space with functions(*) such that every stalk $\mathcal{O}_{X,x}$ is local. Furthermore let $Y\subset X$ be an irreducible ...
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Showing that $k$-algebra homomorphism between local rings is a local ring homomorphism

Let $k$ be a field. Then $(k,\text{id}_k)$ is a $k$-algebra. Let two further $k$-algebras $(A,\varphi_A)$ and $(B,\varphi_B)$, such that both are local as rings, and a $k$-algebra homomorphism $f:A\...
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$aX + bY$ is an element of $M^2$ if and only if the line $aX + bY = 0$ is tangent to $W$ at $(0, 0)$

Let $F(X, Y) = Y^2 - X^3 + X \in \mathbb{C}[X, Y]$, and let $a$ and $b$ be constants (elements of $\mathbb{C}$). Write $W = V(F)$, and let $P$ be the point $(0, 0)$ on $W$. Show that $aX + bY$ is an ...
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local ring of variety over not necessarily algebraically closed field

Let $V$ be an affine variety. The ideal of $V$ is $I(V) = \{f\in \bar K[X] \mid f(P)=0\;\forall P\in V \}$. If $I(V)$ is generated by elements in $K[X]$, the $V$ is said to be defined over $K$ and ...
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Is there a non-regular depth 1 noetherian local ring with this property?

Let $(R, \mathfrak{m})$ be a non-regular depth 1 noetherian local ring. Then if $x$ is any regular element of $R$ the module $R/(x)$ will have depth 0, and so it has $\mathfrak{m}$ as an associated ...
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Exact functors in local cohomology

In local cohomology the ideal transform functors with respect to a pair of ideals are defined by $D_{I,J}(-)=\underset{\textbf{a} \in \tilde{w‎‎} ‎(I,J)‎} {\varinjlim}\,\,\text D_{\textbf{a}}(-)$. ...
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Exponent = index in Brauer group over the quotient field of an Iwasawa algebra

Let $K$ be a field and $B(K)$ its Brauer group. We know that for all $[A]\in B(K)$, $$\exp[A]=\operatorname{ind}[A]$$ when $K$ is the quotient field of a complete DVR or when $K$ is a global field. (...
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Sufficient condition for a Weierstrass equation to be minimal when $\text{char}(K)\neq 2,3$

I'm reading about Minimal Weierstrass Equations in VII.1 of Silverman's Arithmetic of Elliptic Curves. Any elliptic curve $E/K$ can be represented by an affine cubic $$ Y^2+a_1XY+a_3Y=X^3+a_2X^2+a_4X+...
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When is the sequence $\{F_n\}$ is a Cauchy sequence in $m$-adic topology?

Here is a question (See question $9$). It says that we have the ring $K[[x,y]]$ and the ideal $I=\left\langle x,y \right\rangle$, where $K$ is a field. We to show that the sequence $\{s_n=\sum_{i=0}^n ...
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Zero dimensional quotient of a local Noetherian ring

Let $(R,m)$ be a local Noetherian integral domain. Let $I$ be an ideal in $R$, so $I \subseteq m$. Then $R/I$ is also a local Noetherian ring, see (and the image of any surjective ring homomorphism of ...
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Why the image of any $A$-regular sequence under $f$ is a $B$-regular sequence, $f: A \to B$ is flat.

The second answer to this question claims: "If $f: A \rightarrow B$ is flat, then obviously the image of any $A$-regular sequence under $f$ is a $B$-regular sequence. This can be seen by ...
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On a Krull-intersection type problem for certain two generated ideals in local rings

Let $(R, \mathfrak m)$ be a Noetherian local ring. Let $x,y\in \mathfrak m$ such that $y$ is not a zero-divisor on $R$. Then, is it true that $\cap_{n=1}^\infty (x,y^n) \subseteq (x)$ ? By Krull ...
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Question in a proof from Gathmann's notes on Algebraic Geometry: The tangent space.

I am studying chapter 10 from Gathmann's notes about algebraic geometry and there is something I don't understand in the following proof. I don't really understand the equality $\frac{g}{f} = c g$. ...
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Flat extension of local domains

Let $(R,m)$, $(S,n)$ be two local Noetherian domains, $R \subseteq S$ is flat, and $m \subseteq n$. Question 1: If $R$ is regular and $S$ is Cohen-Macaulay, is $S$ also regular? Question 2: If $R$ is ...
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Properties of $k[x(x-1)]_{\langle x(x-1) \rangle} \subseteq k[x]_{\langle x \rangle}$

Let $k$ be aa arbitrary field. Let $R=k[x(x-1)]_{\langle x(x-1) \rangle}$ and let $S=k[x]_{\langle x \rangle}$, $m=x(x-1)R$, $n=xS$, $k(m)=R/m$, $k(n)=S/n$. We have, $mS = n$ (since $x-1$ is ...
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Ideal $I$ with $\operatorname{depth}(I)=d$ in a local CM ring of dimension $d$

Let $(R,m)$ be a Noetherian Cohen-Macaulay local ring, having Krull dimension $d$ (by this, necessarily $d < \infty$). Let $I$ be an ideal of $R$ with $\operatorname{depth}(I)=d$, namely, $I$ ...
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Cohen-Macaulayness and regularity of $A/p$

This question claimed (and proved) that if $p$ is a prime ideal of $A=k[x_1,\ldots,x_n]$ with $\operatorname{ht}(p) \in \{0,1,n-1,n\}$, then $A/p$ is Cohen-Macaulay. Now, let $A$ be a (Noetherian) UFD ...
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Local Cohen-Macaulay rings

Case one: Let $(R,m)$ be a Noetherian local ring of Krull dimension $d$, $\dim(R)=d$. Let $I$ be an ideal of $R$. Assume that $\operatorname{depth}(I,R)=d$, namely, the maximal length of a regular ...
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Finitely generated module whose completion has finite length

Let $M$ be a finitely generated module over a Noetherian local ring $(R,\mathfrak m).$ If $n\ge 1$ is an integer such that $\widehat {\mathfrak m^n}\widehat M=0,$ then is it true that $\mathfrak m^n M=...
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Finite ring extension of local rings, revisited

This question says the following: Let $R$ and $S$ be local rings with the maximal ideals $M$ and $N$, respectively. Assume that $R\subset S$ and that $S$ is a finitely generated $R$-module. If there ...
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Powers of maximal ideal in local ring with a single prime ideal.

Let R be a non zero, local, Noetherian ring with $\mathfrak{m}$ the maximal ideal of R. If we assume that $\mathrm{Spec}R=\{\mathfrak{m}\}$, what can we say about powers of $\mathfrak{m}$ ? I have ...
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Flat morphism of local rings

Let $(A,m_A)$ and $(B,m_m)$ be two Noetherian local rings, $A \subseteq B$ and $B$ is a finitely generated $A$-algebra. Step 1: Assume that: (1) $A$ is regular. (2) $A \subseteq B$ is flat. Question ...
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Dimension of a scheme $X$ at a closed point $x$ and dimension of its local ring.

The dimension of an irreducible scheme $X$ at $x$, dim$_x(X)$ is defined as the smallest dimension among its open neighbourhoods and the dimension of its local ring dim$(\mathcal{O}_{X,x})$ is just ...
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Is a localization at a maximal ideal of a polynomial ring a perfect ring?

There are several equivalent definitions for a perfect ring $R$ (not necessarily a commutative ring), for example: Every flat left $R$-module is projective; see wikipedia. Also, there is the notion of ...
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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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Pullback on $\mathbb{A}^{n-1}$ of $\pi$ : $Z(F) \rightarrow \mathbb{A}^{n-1}$

I am self studying some introductory algebraic geometry and the author of lecture notes make the following claim without explaining the reason. Let $F\subset k[x_1,...,x_n]$, $Z(F)\subset \mathbb{A}^n$...
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When extension of a maximal ideal is maximal

Let $R \subseteq S$ be two commutative rings satisfying the following conditions: (1) $R$ and $S$ are $\mathbb{C}$-algebras. (2) $R$ and $S$ are integral domains. Denote their fields of fractions by $...
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A sandwich theorem for local rings

The following question seems natural to ask in view of this question and its comments/answers: Let $R \subseteq S$ be commutative Noetherian rings, let $q$ be a maximal ideal of $S$, $p$ a maximal of $...
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1answer
99 views

Definition of homomorphism of local rings

Let $(R,m_R)$, $(S,m_S)$ be two local rings. By definition, a local homomorphism of local rings is a ring homomorphism $f: R \to S$ such that the ideal generated by $f(m_R)$ in $S$ is contained in $...
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Conditions implying that a certain affine ring is a UFD

Let $R$ be a $\mathbb{C}$-algebra satisfying the following conditions: (i) $R \subset \mathbb{C}[x_1,\ldots,x_n]$. (ii) There exist $a_1,\ldots,a_l \in \mathbb{C}[x_1,\ldots,x_n]$ such that $R=\mathbb{...
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Flatness of localizations

Let $R \subseteq A$ be two $\mathbb{C}$-algebras, $P$ a prime ideal of $R$, $Q$ a prime ideal of $A$, and $Q \cap R = P$. Assume that $(A_Q,QA_Q)$ is flat over $(R_P,PR_P)$. 'When' $A$ is flat over $...
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A result concerning local rings

The following result appears in the book 'Commutative Algebra with a View Toward Algebraic Geometry' by David Eisenbud: Theorem 18.16* Let $(R,P)$ be a regular local ring, and let $(A,Q)$ be a local ...
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If $1+a$ is a unit for all non-units $a$ in a nonzero ring $R$, then $R-R^*$ is an ideal

Suppose that $R$ is a nonzero commutative ring with $1$. Suppose $1+a$ is a unit for all non-units $a$ in a nonzero ring $R$. I'd like to understand why this implies that $R-R^*$ is an ideal. In ...
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1answer
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Sufficient condition for complete local ring to be the completion of a certain subring

The context is two statements from this paper: A quasi-local ring $(R, M \cap R)$ contained in a complete local ring $(T,M)$ is Noetherian and has completion $T$ provided the map $R \to T/M^2$ is ...
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1answer
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Local rings of affine space in etale topology.

This is probably a very straightforward question. What does the etale local rings of $\mathbb{A}_{\mathbb{F}_q}^n$ look like? (In other words the strict henselization of local rings at closed points)
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Doubt in my proof of Hartshorne I (4.7).

Let $X,Y$ be varieties, and choose points $P$ and $Q$ so that there exists an isomorphism of k-algebras $f^*\colon \mathcal{O}_{P,X} \to \mathcal{O}_{Q,Y}$. Then show that there exists open sets $P \...
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A problem on induced local rings

Let $X,Y$ be varieties. Show that the pullback map $\phi^{*}\colon O(Y)\rightarrow O(Y)$ of $\phi\colon X\rightarrow Y$, induces the map $\phi^{*}_P\colon O_{P,X}\rightarrow O_{\phi(P),Y}$ of local ...
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What exactly is sheafification?

I have recently learned about the very BASICS of sheaves, but I was wondering is there an easier definition for sheafification? I could not find anywhere an easier definition for sheafification. I ...
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Isomorphic as $R$-modules v.s. Isomorphic as abelian groups

Let $(R,\mathfrak m, \mathbb Q)$ be a Noetherian local ring. Let $M$ be a finitely generated $R$-module such that for some integer $n\ge 0$, there is an isomorphism of abelian groups $M \cong \mathbb ...
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1answer
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Why is the set of elements in a valuation ring $R$ with positive valuation the unique maximal ideal of $R$?

Definition. Let $K$ be a field and let $G$ be a totally ordered abelian group. A valuation of $K$ with values in $G$ is a map $v: K-\{0\} \rightarrow G$ such that for all $x, y \in K, x, y \neq 0,$ we ...
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1answer
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Definition of local ring (unique maximal left/right ideals)

Wikipedia lists a few equivalent definitions of local rings, the first two of which are $R$ has a unique maximal left ideal. $R$ has a unique maximal right ideal. However, it does not list this ...
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Unique maximal ideal and ring epimorphism kernel with prime numbers equivalence

Let $A$ be a ring. Let $f: \mathbf{Z} \to A$ be a surjective ring homomorphism. Prove that $A$ has a unique maximal ideal iff there exists $n\in \mathbf{N}$ and $p\in\mathbf{N}$ a prime number such ...
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Localization of Dedekind domain at a prime ideal is a P.I.D

Let $A$ be a Dedekind domain and $\mathfrak{p}\subset A$ be a prime ideal. Then the localization $A_\mathfrak{p}$ is also a Dedekind domain. I can show it has a unique maximal ideal $\mathfrak{p}':=\...
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idempotent matrix over local ring is similar to a diagonal matrix with elements $0$ and $1$

I want to know why any idempotent matrix $P$ over local ring is similar to a diagonal matrix with elements $0$ and $1$. I saw a proof in lemma 3.3 of ncatlab, but I don't understand why $P$ is similar ...
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Ring structure on a space of functions between vector spaces?

In this Wikipedia article about jets, in the section about rigorous definitions, for the algebro-geometric definition, they take the vector space $C^\infty_p(\mathbb R^n,\mathbb R^m)$ of germs of ...
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Every element of a local ring is a sum of two units iff the cardinality of the ring is less than twice the cardinality of units

Is it true that every element of a local ring with unity can be written as a sum of two units iff the cardinality of the ring is strictly less than twice the cardinality of units of the ring? It is ...
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165 views

Local ring of a generic point on an integral scheme is a field

Let $X$ be an integral scheme and let $\eta \in X$ be its generic point. Then the local ring $K(X) := \mathcal{O}_{X, \eta}$ is a field. Moreover, if $U = \text{Spec} A$ is any open affine subset of $...
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1answer
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Complete Noetherian local ring with residue field $\mathbb{F}_p$ is an algebra over the $p$-adic integers $\mathbb{Z}_p$

Let $R$ be a complete, Noetherian, local ring with maximal ideal $\mathfrak{m}$ and residue field $R/\mathfrak{m} \cong \mathbb{F}_p$. Note that by "complete", we mean that $$ R \cong \...
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1answer
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My trial for showing that $K[[x]]$ over a field is a local ring.

Here is the question I want to answer letter $(b)$ in it: A commutative ring $R$ is local if it has a unique maximal ideal $\mathfrak{m}.$ In this case, we say $(R, \mathfrak{m})$ is a local ring. For ...

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