Questions tagged [local-rings]
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218 questions
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A particular Isomorphism of graded algebras over a regular local ring
In Hartshorne's "Algebraic Geometry", the following statement is a weaker form of Theorem 8.21A (e), which he quotes from Matsumuura's book on commutative algebra:
Proposition. Let $R$ be a regular ...
- 4,902
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Can you suggest a good name for a local homomorphism φ:(R,m)->(S,n) of local rings with the property that φ(m)S is n-primary?
Can you suggest a good name for a local homomorphism $(R,\mathfrak{m})\stackrel{\varphi}{\rightarrow}(S,\mathfrak{n})$ of local rings with the property that $\varphi(m)S$ is $\mathfrak{n}$-primary?
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- 3,101
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Krull's intersection theorem for commutative local not necessarily noetherian rings
Is there a characterisation of those commutative local not necessarily noetherian rings that satisfy Krull's intersection theorem ? How can the intersection theorem be phrased in terms the module ...
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When a group ring is a local ring [closed]
Hi there, I'm stuck with my undergraduate thesis on the following proposition:
If $k$ is a field of characteristic $p > 0$ and $G$ is a finite $p$-group, then the group ring $kG$ is local.
In ...
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Local coordinate system under finite integral extension
Let $\varphi:(A,\mathfrak{m})\to(B,\mathfrak{n})$ be a local morphism of regular local $\mathbb{C}$-algebras (of the same dimension) which makes $B$ integral over $A$.
Let $\mathfrak{m}=(x_1,\ldots,...
- 4,902
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On the functoriality of scalar extensions of local rings (edited)
Note. I have edited my question to make it more transparent, following some very good comments that I received. I am sorry if it is a bit long.
A local homomorphism of local rings $(A,\mathfrak{m})\...
- 3,101
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Local Rings problem
$\newcommand{\End}{\operatorname{End}}$
let $R$ be a local ring, $\varphi\in \End(R_{R}^{2})$,
$\overline{\varphi}\in \End(\overline{R}_{\overline{R}}^{2})$,
$\overline{R} =R/J(R)$ , $J(R)$= ...
- 11
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maximal ideal in local subrings
Let $A,B$ be two local rings and put $\mathfrak{m}_A, \mathfrak{m}_B$ their maximal ideals. Now suppose that we have an injection $0 \to A \to B$ and put $\mathfrak{n} := A \cap \mathfrak{m}_B $. It ...
- 379
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Image of norm map for local field
Let $F$ be a finite extension of $Q_2$ (2-adic field) or $F_2((x))$ (function field). Let $E/F$ be a separable extension of degree $2$.
What is the image of the norm map $N_{E/F}$?
In particular - ...
- 899
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3
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on the relative conductor of curve singularity and quotient of ideals
Let $R$ be the local ring of a complex curve singularity. (Can assume the singularity planar, the ring locally analytic or formal.) Let $\bar{R}$ be the normalization, let $R\subset R'\subset \bar{R}$ ...
- 2,232
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covers of complete regular local rings
It is well-known that if one assumes algebraic closedness and characteristic 0 of the residue field then finite covers of complete DVRs are all of the form $A[x]/(x^m-a)$ for some $a \in A$ (direct ...
- 4,070
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prime ideals in regular local rings
Suppose $R$ is a regular local ring. Let $m$ be the maximal ideal. Then, if the dimension of $R$ is $n$, there is a regular sequence of size $n$, say $x_1,x_2,...,x_n$ s.t. $m=(x_1,x_2,...,x_n)R$. ...
- 225
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Etale cohomology of regular local rings
Let $R$ be a regular local ring (I am particularly interested in the case when $R$ is the local ring of a point on a smooth scheme of finite type over a field). Let $G$ be the etale fundamental group ...
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local Artin algebras
Given a commutative Artin algebra $A$ over an algebraically closed field $k$ one has a decomposition $A=A_1\oplus\ldots\oplus A_n$ into local Artin subalgebras, see for example Atiyah-McDonald, ...
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Algorithm for Weierstrass Preparation Theorem for Formal Power Series
The Weierstrass preparation theorem for formal power series rings guarantees that if a given formal series $f(z) = \sum a_k z^k \in R[[z]]$ where $R$ is a complete local ring with maximal ideal $M$ ...
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Are local, Noetherian rings with principal maximal ideal PIR?
A question asked by a friend. I believe it's false, but lack a decisive counterexample.
This question shows that it is true for valuation rings, but I know too little about them.
In the wider ...
- 545
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How much can we say about the number of nilpotents in a finite local commutative ring?
A commutative ring is local if it has a single maximal ideal. If the ring is finite, this implies that all elements are either units or nilpotents. Further, all finite local rings have prime power ...
- 1,793
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1
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Elementary proof wanted: every local principal ideal ring is a quotient of a PID
I am looking for a more elementary proof of the following result:
Theorem (Hungerford, 1968): Let $R$ be a principal ideal ring. Then $R \cong \prod_{i=1}^n R_i$, where each $R_i$ is a homomorphic ...
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