# Questions tagged [local-rings]

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### Is the following local map unramified?

Let $(R,m)$ and $(S,n)$ be two local rings, $R \subseteq S$, $R$ regular, $S$ a finitely generated and flat $R$-algebra, and $mS=n$. In comments to this question it was claimed that in such situation ...
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### $A \to B$ with $A$ regular imply that $B$ is CM

The answer to this question says the following: "The general statement is if $A \to B$ is finite and injective, and $A$ is noetherian and regular, then $B$ is CM if and only if $A \to B$ is flat. ...
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### Local rings $R \subsetneq S$ with $R$ regular and $S$ Cohen-Macaulay, non-regular

Let $R \subseteq S$ be local rings with maximal ideals $m_R$ and $m_S$. Assume that: (1) $R$ and $S$ are (Noetherian) integral domains. (2) $\dim(R)=\dim(S) < \infty$, where $\dim$ is the Krull ...
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### Do you know which is the minimal local ring that is not isomorphic to its opposite?

The most popular examples are non-local rings and minimal has 16 elements. I am interested in knowing examples of local rings not isomorphic to their opposite.
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### On the product in the power series ring

Let $A_n \colon= K[[X_1,\ldots,X_n,Y_1,\ldots,Y_n]]$ be a power series ring over a field $K$ in $2n$ variables and ${\frak m}_{A_n}$ be the unique maximal ideal of $A_n$. Suppose we have two ...
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### Structure of Complete Local Rings

Let $X$ be a proper $n$-dimensional $k$-scheme and $x \in X$ a closed point. Consider the stalk $\mathcal{O}_{X,x}$. We consider now it's completion $O_{X,x}^{\wedge}$ wrt it's maximal ideal $m_x$. ...
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### Additive group of local rings

Is there a theory or characterization for those finite $p$-groups that can be considered as the additive group of a finite local commutative ring with identity?
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### Finite extension of $K[[X]]$ and the norm

Let $R \colon= K[[X]]$ be a formal power series ring over a field $K$. We consider a monic polynomial $f(T) \in R[T]$ as follows$\colon$ $$f(T) = T^e + c_{e-1}T^{e-1} + \ldots + c_1T + c_0.$$ ...
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### Complete local rings, automorphisms and approximation

Consider two local morphisms $f,g: B\rightarrow A$ of noetherian complete local rings and $f$ surjective. Does there exist an integer $n\in\mathbb{N}$, such that if $f=g \mod \mathfrak{m}_{A}^{n}$ ...
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### Under which conditions on the homogeneous ideal $I$, the quotient ring $\mathbb{C} [X_0, \dots, X_n]/I$ is a regular ring?

If $I$ is a homogeneous ideal of the ring of homogeneous polynomials $\mathbb {C} [X_0, \dots, X_n]$ , under which conditions on the homogeneous ideal $I$, and particularly on $I_m$, the $m$ -...
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### Minimal resolution of local cohomology module

Let $(R,\mathfrak m)$ be a Noetherian local ring of dimension $d\geq 1.$ Suppose $\lambda(H_{\mathfrak m}^i(R))<\infty$ for some $0\leq i\leq d-1.$ Question Can we say anything about Betti numbers ...
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### Bezout theorem for germs of holomorphic functions

UPDATE. It was pointed out by @Dmitri that two smooth curves given by $f=y$ and $g=y+x^k$ in $\mathbb C^2$ provide a simple counterexample. Let $f_1, \ldots, f_p, g_1, \ldots, g_q$ be germs of ...
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### Reduced ring with all non-prime ideals finitely generated

Let $R$ be a reduced ring with all non-prime ideals finitely generated. Then is $R$ Noetherian ? If not, then is it true at least in the local case ? Without reduced assumption, it is not true even ...
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### Rings with all non-prime ideals finitely generated

Motivated by this question, I would like to ask: If all non-prime ideals in a ring are finitely generated, then is the ring Noetherian? Can we at least say anything in the local case? Note that ...
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### local ring all whose non-maximal ideals are finitely generated

Let $(R, \mathfrak m)$ be a commutative local ring such that every non-maximal ideal is finitely generated. Then, is $R$ Noetherian i.e. is $\mathfrak m$ finitely generated ideal ? It is easy to see ...
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### Valuation ring whose maximal ideal and every ideal of finite height are principal

Let $(R, \mathfrak m)$ be a valuation ring such that $\mathfrak m$ and every ideal of finite height is principal. Then is $R$ Noetherian , i.e. a discrete valuation ring ?
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### When is a zero dimensional local ring a chain ring?

A commutative ring with identity is called a chain ring if all its ideals form a chain under inclusion. I want to know is there any proof for the fact that a zero dimensional local ring is a chain ...
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### Is a filtered colimit of complete module complete?

This is probably a textbook question but i haven't been able to find a reference. Let $R$ be a complete commutative Noetherian local ring and $I$ its unique maximal ideal (I'm mostly interested in the ...
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### Given a non-field local domain $R$, finding a dominating Valuation ring whose residue field is algebraic/finite extension of the residue field of $R$

Let $(R, \mathfrak m)$ be a non-field local domain with fraction field $Q(R)$ . Let $k_{R}:=R/m$. We know that there is a Valuation ring $(V,\mathfrak m_V)$ such that $R \subseteq V \subsetneq Q(R)$ ...
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### a generalization of group (monoid with order-by-order invertible elements)

Fix a filtered monoid, $H=H_0\supsetneq H_1\supsetneq H_2\supsetneq\cdots$. Suppose for any $h\in H$ and any $n\in \Bbb{N}$ exists $h_n\in H$ such that $h\cdot h_n\in H_n$ and $h_n\cdot h\in H_n$. If ...
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### $K[[X_1,…]]$ is a UFD (Nishimura's Theorem)

Let us define the infinitely-many-variable formal power series ring $$K[[X_1,\ldots]] \colon= \underset{m \geq 1}{\varprojlim}\,K[[X_1,\ldots,X_m]].$$ $K[[X_1,\ldots]]$ is known to be a UFD by a ...