# Questions tagged [regular-rings]

For questions regarding a commutative Noetherian ring whose localization at each prime ideal is a regular local ring.

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### Localization of module finite extension of regular ring

Let $R\subseteq S$ be an extension of Commutative Noetherian rings such that $R$ is a regular ring and $S$ is module finite over $R$. Let $P$ be a prime ideal of $S$, then $P\cap R$ is a prime ideal ...
1 vote
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### Dimension of $(A[[X]])_\mathfrak{m} \geq k+1$, regular ring, chain of prime ideals

Assume $A$ is a regular ring and $m$ a maximal ideal of $A$. We define $R= A[[X]]$.Then $\mathfrak{M}=mR + XR$ is a maximal ideal of $R$. Lets assume $h(mA_m) = k$. I want to show, that \begin{align*} ...
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### General method of finding dimension of a ring and determining regularity

Let $R=\mathbb{Z}[x,y]$, $A=R/(y^3-x^3-4)$ and $m=(x,y,2)$. Now I want to find out what the dimension is of $A_m$ and say whether it is regular or not. My prefered definition of the ring $A_m$ being ...
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### Ring of invariants of symmetric group acting on ring of formal power series

Let $k$ be a field and $R=k[[x_1, \dots, x_n]]$ be the ring of formal power series in $n$ variables. Let $S_n$ be the symmetric group of order $n!$. Then $S_n$ acts on $R$ by $k$-automorphisms by ...
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### motivation for regular schemes

Clearly regular schemes are like smooth varieties (in the sense of dimension of tangent spaces) and should be very important in algebraic geometry. Is there any big theorem focusing on regular schemes?...
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### Homological Methods in commutative Algebra Reference | lecture notes/ video lectures

I am studying Homological Methods in Commutative Algebra,TIFR Bombay pamphlet (this). Can anyone suggest any good reference/ notes/ video lectures for this? I am feeling lost. Thanks in advance.
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### Splitting of $k[[x_1,...,x_n]]\to \bar k [[x_1,...,x_n]]$

Let $k$ be a field, of positive characteristic, with algebraic closure $\bar k$. Then, is it true that for every integer $n\ge 1$, the inclusion map $k[[x_1,...,x_n]]\to \bar k [[x_1,...,x_n]]$ splits ...
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### Localization of a maximal ideal in a multivariate polynomial ring is regular

I want to show that if $R=K[x_1,\dots,x_n]$ and $m$ a maximal ideal, then $R_m$ is regular where $K$ is algebraically closed. The definition I have comes from Introduction to commutative algebra ...
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### Prove that if $M=f(M)+\ker(f)$ and $S=\text{End}_R(M)$ is a reduced ring, then $S$ is regular

Consider a right $R$-module $M$ with a reduced endomorphism ring $S=\text{End}_R(M)$. If $M=f(M)+\ker(f)$ for every $f\in S$, then prove that $S$ is a regular ring. I tried to prove that in the ...
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### Regular Noetherian Local Ring is Integral Domain

I am currently reading a proof of the fact that every regular Noetherian local ring $R$ is an integral domain. The proof argues by induction on $d=\operatorname{dim}R$. The base case $d=0$ is clear. ...
1 vote
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### Regularity vs smoothness in positive characteristic

It is well known that a scheme over a perfect field is smooth at $x$ if and only if it is regular at $x$, and that these two properties are not equivalent over non-perfect fields. What is an example ...
1 vote
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### Question on the Proof of Proposition 2.2.4 of Bruns and Herzog

Let $(R, \mathfrak m, k)$ be a regular local ring with unique maximal ideal $\mathfrak m$ and residue field $k = R / \mathfrak m.$ Let $\dim(R) = d.$ Let $I$ be a proper ideal of $R.$ Proposition 2.2....
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### Regularity of $\mathbb Z/ n \mathbb Z$.

For which values $n$ is the ring $\mathbb Z/ n\mathbb Z$ regular? We know that when $n$ is prime, this is regular. When $n$ is not square-free, it's not regular because it is not reduced. However, ...
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### UFD such that power series are not

In Matsumura's ''Commutative ring theory'' he proves the folowing: If $A$ is a regular UFD, the ring of formal series $A[\![ X ]\!]$ is a UFD (page $165$). Just below he says that we can't drop ...
1 vote
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### How to show this tensor product is regular?

Let $L,K$ be field extensions of a field $k$, and let $K$ be finitely generated ($K=k(x_1,\cdots,x_n)$). I need to prove $K\otimes_k L$ is regular in the following two cases: $K$ is separated. $L$ is ...
1 vote
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### How to prove this ring is regular?

Let $R$ be a DVR, $\pi$ be the uniformizer, and $n\geq 2$ an integer. I need to prove the ring $R[T_1,\dots,T_n]/(T_1\cdots T_n-\pi)$ is regular. I know a polynomial ring of a regular ring is regular,...
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### Not-necessarily-unital von Neumann regular local commutative rings: are they fields?

In this question all rings are commutative, but don't necessarily have a multiplicative identity (so: commutative rngs). On Wikipedia there is the unsourced claim: It is well known that a local ring ...
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### Depth of tensor product of finitely generated reflexive modules over regular local rings

Let $M,N$ be finitely generated reflexive modules over a regular local ring $(R, \mathfrak m)$ of dimension at least $3$ (Hence, $M,N$ both have depth at least $2$) . Then, is it necessarily true that ...
1 vote
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### Is $D^{-1}R$ regular?

If $D$ is a multiplicative subset of a regular ring $R$, then is $D^{-1}R$ also regular? I've been trying to think of an easy counterexample, but I'm stuck on figuring out how to either think of a ...
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### $\operatorname {Ext}$ vanishing and finitely generated reflexive modules over regular local rings

Let $M$ be a finitely generated reflexive module over a regular local ring $(R,\mathfrak m,k)$ such that $\operatorname {Ext}^1_R( \operatorname {Hom}_R(M,M),R)=0$. Then how to show that $M$ is a free ...
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### Is every idempotent semiring a completely regular?

A semiring $(S, +, \cdot)$ is said to be a completely regular semiring if for every $a\in S$, there exists some $x\in S$ satisfying the following conditions: (1) $a=a+x+a$ (2) $a+x=x+a$ and (...
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Let $\mathbb{K}$ be a field, $A=\mathbb{K}[x_1,\dots,x_n]$ and let $\mathcal{M}$ a maximal ideal in $A.$ I want to prove that the localisation $A_\mathcal{M}$ is a regular ring. I don’t know many ... ### On a special kind of local Gorenstein ring of dimension $2$
Let $(R, \mathfrak m,k)$ be a local Gorenstein ring of dimension $2$ such that $\mu (\mathfrak m^2)(=\dim_k \mathfrak m^2/\mathfrak m^3) =3$ . Then is it true that $R$ is regular ? Or at least is it ...