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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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Singular locus of $\mathbb{F}_p[x,y,z]/(xy-z^2)$

How would I go about determining the singular locus of the hypersurface ring $R=\mathbb{F}_p[x,y,z]/(xy-z^2)$? I conjecture that the ring is regular at every maximal ideal except $(x,y,z)$. The ...
Anon's user avatar
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Remark in Atiyah, Macdonald: non-singularity $\Rightarrow$ analytic irreducibility

In Atiyah, Macdonald, Introduction to Commutative Algebra, Chapter 11, p. 124, after Proposition 11.24, there is a Remark. It follows from what we have said above that A is also an integral domain. ...
Elías Guisado Villalgordo's user avatar
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Theorem 19.5 from Matsumura's Commutative ring theory, case $A[[ X ]]$.

So on page 157 of Matsumura's "Commutative ring theory", theorem 19.5, to prove "If $A$ is regular then so is $A[[X]]$" it is mentioned that "Now although we cannot say that $...
Jasper's user avatar
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Are injective maps of finite modules over regular ring liftable to termwise injective maps of finite projective resolution?

Let $R$ be a regular noetherian ring and let $M$ be a finitely generated $R$-module. In this situation, we know that $M$ admits a finite resolution in terms of finite projective $R$-modules. Now, let $...
Stabilo's user avatar
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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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Intuitive reason for why $\operatorname{Spec}(k[x,y,z]/(x-yz))$ is smooth at $O$ while $\operatorname{Spec}(k[x,y,z]/(x^2-yz))$ isn't?

Let $k$ be an algebraically closed field and consider the closed subschemes $X=\operatorname{Spec}(k[x,y,z]/(x-yz))$ and $Y=\operatorname{Spec}(k[x,y,z]/(x^2-yz))$ of $\mathbb{A}^3$. Then the gradient ...
imtrying46's user avatar
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1 answer
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Is the ring of regular functions of a simple complex connected linear algebraic group factorial?

The question is basically the subject. I have found references in the case of simply-connected algebraic groups (the field is arbitrary, the group is not necessarily affine): Popov, Picard groups of ...
Grabovsky's user avatar
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Invertible elements in the rings of regular functions of algebraic groups [duplicate]

Let $G$ be a connected simple complex algebraic group and $\mathcal{O}(G)$ be its ring of regular functions. What are the invertible elements of $\mathcal{O}(G)$? Or is it true that $\mathcal{O}(G)^* =...
Grabovsky's user avatar
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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*} ...
willix's user avatar
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48 views

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 ...
Algebear's user avatar
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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 ...
Fabio Neugebauer's user avatar
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119 views

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?...
Z Wu's user avatar
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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.
Lemon's user avatar
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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 ...
Snake Eyes's user avatar
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209 views

Maximal regular sequence coincides with system of parameters

I'm needing help in this question. Let $k$ be a field. Consider the $k$-algebra $R:=k[x,y,z,w]/(z+w,xy+xw)$ and define the ring $A$ the localization of $R$ in its maximal ideal $\mathfrak{m} = (\...
Kevin Vasconcellos's user avatar
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290 views

If the global dimension is equal to the krull dimension of a ring R, then R is regular

Let R be a (non local) noetherian ring, and let $\mathrm{gldim}(R)$ be the global dimension. Recall that R is regular iff for every maximal ideal $m\subset R$ the localization $R_m$ is regular. If $R$ ...
Asvr_esn's user avatar
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$K[t^2,t^3]_{(t^2,t^3)}$ is not regular

I want to show that $K[t^2,t^3]_{(t^2,t^3)}$ is not regular, and to do so I want to show that $dim_F(m/m^2)=2$ for $m=(t^2,t^3).K[z^2,t^3]_{(t^2,t^3)}=(\frac{t^2}{1},\frac{t^3}{1})\subseteq K[t^2,t^3]...
raisinsec's user avatar
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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 ...
raisinsec's user avatar
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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 ...
mariam's user avatar
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1 answer
275 views

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. ...
Squeezelemma's user avatar
1 vote
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226 views

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 ...
jackson's user avatar
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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....
Dylan C. Beck's user avatar
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50 views

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, ...
Singularity's user avatar
2 votes
1 answer
214 views

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 ...
feriado's user avatar
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1 answer
106 views

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 ...
Richard's user avatar
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1 vote
1 answer
228 views

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,...
Richard's user avatar
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2 votes
1 answer
341 views

Completion (as a module) of the integral closure of a Cohen-Macaulay local ring whose completion is reduced

Let $(R,\mathfrak m)$ be a Cohen-Macaulay local ring. Let $\overline R$ be the integral closure of $R$ in the total ring of fractions of $R$. Let $\widehat R$ be the $\mathfrak m$-adic completion of $...
Louis 's user avatar
  • 515
2 votes
1 answer
143 views

Zero is a regular ring or it is not?

In commutative algebra, a regular ring is a commutative Noetherian ring, such that the localization at every prime ideal is a regular local ring: that is, every such localization has the property that ...
Yasin's user avatar
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0 answers
198 views

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 ...
user237522's user avatar
  • 6,695
1 vote
0 answers
121 views

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 ...
user237522's user avatar
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0 votes
1 answer
71 views

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, $I$ ...
user237522's user avatar
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4 votes
1 answer
102 views

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 ...
user237522's user avatar
  • 6,695
0 votes
0 answers
243 views

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 ...
user237522's user avatar
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5 votes
1 answer
171 views

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 $...
user237522's user avatar
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1 vote
0 answers
97 views

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{...
user237522's user avatar
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0 votes
1 answer
71 views

A regular domain is a Krull Domain.

I don't know how to prove that a regular domain is a Krull domain. If we say that $\mathcal{P}=\{p\in \operatorname{Spec}(A)\mid \operatorname{ht}(p)=1 \}$ a Krull domain is a domain s.t.: $A_p$ is a ...
marco's user avatar
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2 votes
0 answers
53 views

Higher partial derivatives of a regular function

Let $X$ be a smooth algebraic variety over a field $k$, and let $x\in X$. Let $(x_1,\dots,x_n)$ be a regular system of parameters at $x$, so that $\Omega^1_{X/k}$ is locally free at $x$ with basis $...
Gaussian's user avatar
  • 473
1 vote
1 answer
214 views

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 ...
theHigherGeometer's user avatar
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1 answer
278 views

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 ...
user102248's user avatar
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1 vote
0 answers
75 views

Depth of tensor product of modules which are locally free on the punctured spectrum of regular local ring

Let $M,N$ be (non-zero) finitely generated modules over a regular local ring $(R, \mathfrak m)$ of dimension $d$ such that $M_P, N_P$ are free (non-zero) $R_P$-modules for every prime ideal $P\ne \...
Louis 's user avatar
  • 515
0 votes
0 answers
116 views

The complete regular ring in Cohen's structure theorem can be chosen to have dimension equal to the embedding dimension of the starting ring?

Let $(R, \mathfrak m)$ be a Noetherian complete local ring. Then by Cohen structure theorem, we have that $R$ is a homomorphic image of a complete regular local ring $(S, \mathfrak n)$ (https://stacks....
Louis 's user avatar
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0 answers
89 views

Notion of simple hypersurface singularity depends on the presentation?

Let $(S, \mathfrak n)$ be a regular local ring. For $0\ne f\in \mathfrak n^2$ define $c(f, S):=\{\text{ideals } I \text{ of } S : f\in I^2\}$ . Now let $(S_1, \mathfrak n_2)$ and $(S_2,\mathfrak n_2)...
user521337's user avatar
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1 vote
1 answer
61 views

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 ...
MathieMcMatherson's user avatar
3 votes
1 answer
220 views

$\operatorname {Ext}$ 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 ...
user521337's user avatar
  • 3,705
1 vote
0 answers
23 views

On an analogy of the highest generating degree and reduction of ideals

Let $R=\mathbb C[x,y]$. Let $\mathfrak m=(x,y)$ . Let $J \subseteq \mathfrak m$ be a homogenous ideal with $\sqrt J=\mathfrak m$ i.e. $\mathfrak m^n \subseteq J$ for some integer $n\ge 1$. Let $a\ge ...
user's user avatar
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2 votes
0 answers
124 views

Embedding dimension of smooth affine variety of dimension $d$ , over an infinite perfect field, is $2d+1$?

Let $k$ be an infinite perfect field. Let $R$ be a finitely generated $k$-algebra of Krull dimension $d$ such that $R$ is regular. Then is it true that there exists a surjective $k$-algebra ...
user521337's user avatar
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0 votes
0 answers
181 views

examples of regular schemes

Let $Y=\operatorname{Spec}\Bbb Z[T]/(T^2+1)$ and $X=\operatorname{Spec}\Bbb Z$. Prove that X and Y are regular. My attempt: Denote $\Bbb Z_{p}$ the localisation of $\Bbb Z$ at $p$. X is regular: ...
Conjecture's user avatar
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1 vote
1 answer
83 views

On the ring $R[X]/(X^q - g)$ being regular

Let $R$ be a Noetherian domain of finite Krull dimension. Let $0\ne g \in R$ be such that $R/gR$ is reduced. let $q$ be a positive integer. Is the following true: $R[X]/(X^q - g)$ is a regular ring ...
uno's user avatar
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4 votes
1 answer
263 views

A local Cohen-Macaulay ring whose dimension is one less than the minimal no. of generators of its maximal ideal

Let $(R, \mathfrak m)$ be a local Cohen-Macaulay ring. If $\dim R=\mu (\mathfrak m)-1$ , then is it true that $R \cong S/(f)$ for some regular local ring $S$ and some (non-invertible) regular element $...
uno's user avatar
  • 1,560
0 votes
1 answer
68 views

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 (...
gete's user avatar
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