Questions tagged [regular-rings]
For questions regarding a commutative Noetherian ring whose localization at each prime ideal is a regular local ring.
81
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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 $...
- 1,506
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1
answer
63
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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
$ht(J)=\mu(J)$, ...
- 105
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1
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59
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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 ...
- 2,377
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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 ...
- 149
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0
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39
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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)^* =...
- 149
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35
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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 ...
- 281
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35
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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*}
...
- 35
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42
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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 ...
- 1,664
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83
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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 ...
- 433
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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?...
- 1,547
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1
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73
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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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52
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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 ...
- 483
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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} = (\...
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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 $R$ ...
- 108
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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]...
- 362
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147
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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. ...
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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 ...
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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....
- 4,800
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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, ...
- 127
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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 ...
- 11
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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 ...
- 1,352
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184
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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,...
- 1,352
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1
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286
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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 $...
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1
answer
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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 ...
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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 ...
- 6,415
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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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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, $I$ ...
- 6,415
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0
answers
79
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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 ...
- 6,415
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0
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204
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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 ...
- 6,415
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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 $...
- 6,415
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0
answers
93
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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{...
- 6,415
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70
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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 ...
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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 $...
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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 ...
- 2,319
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262
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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 ...
- 1,443
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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 \...
- 505
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105
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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....
- 505
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88
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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)...
- 3,665
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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 ...
3
votes
1
answer
206
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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 ...
- 3,665
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21
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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 ...
- 4,364
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106
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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 ...
- 3,665
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0
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163
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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:
...
- 2,990
1
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1
answer
80
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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 ...
- 1,388
4
votes
1
answer
239
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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 $...
- 1,388
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61
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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
(...
- 1,330
1
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0
answers
209
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Regularity of localisation of a polynomial ring.
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 ...
user558035
5
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0
answers
117
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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 ...
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