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Questions tagged [regular-rings]

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Understanding the Zariski tangent space at a closed point of a locally finite type $k$-scheme.

Let $\DeclareMathOperator{\Spec}{Spec} x\in \Spec k[T_1,...,T_n]=\mathbb{A}_k^n$ be a closed point, it is easily seen that $\kappa (x)/k$ must be a finite field extension. Denote the corresponding ...
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A question on regular local rings (of positive characteristic ) of dimension $2$

Let $R$ be a regular local ring of dimension $2$ and of characteristic $p>0$. How to show that for every $f_1,f_2,f_3 \in R$, $\exists 0\ne c\in R$ and $n_0\in \mathbb N$ such that $c(f_1f_2f_3)^{...
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Submodules of modules of finite projective dimension over regular ring

Let $R$ be a regular ring i.e. a commutative Noetherian ring whose localisations at every prime ideal is regular local ring. Then every finitely generated $R$-module has finite projective dimension, ...
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Prove polynomial ring over a discrete valuation ring quotient by powers of maximal ideal is regular?

Let $(R,\mathfrak{m},k)$ be a discrete valuation ring, (of characteristic $p$ if you need). Let $n\geq 1$ be an integer. Is the ring $\frac{R}{\mathfrak{m}^n}[x]$ regular? Note that: Regularity can ...
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Is the trivial ring regular?

In algebraic geometry if $f: X \to Y$ is locally of finite presentation (where $X, Y$ are schemes) then smoothness of $f$ implies that for all $y \in Y$ the "geometric fiber" $\DeclareMathOperator{\...
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Exercise 4.4.1 in Weibel's 'An Introduction to Homological Algebra'.

I can solve this question on the assumption that the $x_i$s are not zero-divisors since $\dim(R/(x)) = \dim(R)-1$ if $x$ is not a zero-divisor. My question is, how do I prove that they are not zero ...
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Noetherian , local, unique factorization domains whose maximal ideal is minimally generated by three elements

Let $(R,\mathfrak m)$ be a local, Noetherian UFD with $\mu(\mathfrak m)=3$. If $J=\mathfrak m$ is the only non-zero ideal of $R$ satisfying $J^2=\mathfrak mJ$, then is $R $ regular ?
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reflexive ideal in regular local ring

Let $I$ be an ideal of a regular local ring $R$ such that $I$ is reflexive as an $R$-module (https://stacks.math.columbia.edu/tag/0AUY) . Then is $I$ a principal ideal (i.e. a free $R$-module i.e. a ...
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On certain local, Noetherian , unique factorization domain whose maximal ideal is minimally generated by $3$ elements

Let $(R,\mathfrak m)$ be a Noetherian , local , UFD with $\mu(\mathfrak m)=3$ ( where $\mu(\mathfrak m):=\min\{|S| : \langle S\rangle=\mathfrak m\}$ ). Also assume that if $J$ is an ideal with $\sqrt ...
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On cancelling $\mathfrak m$-primary ideal of regular local ring $(R,\mathfrak m)$

Let $(R,\mathfrak m)$ be a regular local ring (https://en.wikipedia.org/wiki/Regular_local_ring) . Let $J$ be an $\mathfrak m$-primary ideal such that $J^2=\mathfrak mJ$. Then is it true that $J=\...
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A characterization of regular local ring using primary ideal?

Let $(R,\mathfrak m)$ be a Noetherian local domain such that for every $\mathfrak m$-primary ideal $I$ with $I^2\subsetneq \mathfrak m^2$, we have $I^2\ne \mathfrak mI$. Then is it true that $R$ is ...
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How to find a regular parameter system of the local ring $ \mathbb{C} [X_0, \dots, X_n]_{(P \ )} $?

How to find a regular parameter system of the local ring : $ \mathbb{C} [X_0, \dots, X_n]_{(P \ )} $ with : $ (P \ ) $ a prime ideal of : $ \mathbb{C} [X_0, \dots, X_n] $ ( i.e : $ P \in \mathbb{C} [...
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Projective Nullstellensatz and regular rings

If I am not mistaken, and according to the projective Nullstellensatz, we have: $\mathbb{P}_{\mathbb{C}}^n = \mathrm{Proj} (\mathbb{C} [X_0 , \dots, X_n])$, by the correspondence: $ A \to \mathrm{Proj}...
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Is a complete intersection ring, which is a quotient of a maximal $A$-sequence, Artinian?

Let $A$ be a noetherian regular local ring, $x_1,\dots,x_n$ a regular $A$-sequence and $B = A / (x_1,\dots,x_n)$. Then $B$ is a complete intersection ring by definition. If $(x_1,\dots,x_n)$ is a ...
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Polynomial rings are regular

I want to use the result that the polynomial ring $R:=k[x_1,\cdots,x_n]$ is a regular ring. I can prove that every maximal ideal $\mathfrak m$ in $R$ can be generated by $n$ number of elements so ...
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$k[[x,y]]/I$ is a Gorenstein ring implies that $I$ is generated by 2 elements

I have a problem $k[[x,y]]/I$ is a Gorenstein ring implies that $I$ is generated by 2 elements. I am stuck since I do not have many techniques to prove that an ideal is generated by 2 elements. ...
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Regular local ring if every maximal Cohen-Macaulay module is free

I have a problem like this "Let $R$ be a Cohen-Macaulay local ring, $\dim R=d$. Given that every maximal Cohen-Macaulay $R$-module is free, prove that $R$ is a regular local ring." My lecturer gave ...
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Regularity of $K[X,Y]/(X^p + Y^p - a)$

In his paper "The concept of a simple point of an abstract algebraic variety" Oscar Zariski provides in Example 1 a regular but not smooth variety over a field $K$. In his example $K$ is a non-...
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Regular ring not UFD

I have to prove that the ring $R=K[x,y]/(x^2-y^3+y)$ is not a UFD showing that the prime ideal $(x,y)R$ has height 1 but it's not principal. Do someone know a simple way to prove it? I know there are ...
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Localization of regular local ring by prime ideal

Let $\quad R := k[X_1,\ldots,X_n]/(I)\quad$ be quotient of polynomial ring by some prime ideal, $\mathfrak{p} \subset \mathfrak{m}$ two ideals of $R$, where $\mathfrak{p}$ - prime ideal and $\mathfrak{...
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Why can't $\prod_{i=1}^d a_i^{e_i}$ be in the ideal $(a_1^{e_1 + 1}, \ldots, a_d^{e_d+1})$?

Let $A$ be a regular local ring of dimension $d$ with maximal ideal $\mathfrak{m} = (a_1,\ldots,a_d)$. Let further $e_1,\ldots,e_d$ be natural numbers and set $$a := \prod_{i=1}^d a_i^{e_i}.$$ Now ...
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Regular sequence maps to basis of $\mathfrak{m}/\mathfrak{m}^2$

Let $R$ be a regular local ring with maximal ideal $\mathfrak{m}$. It is known that any elements $r_1, \ldots, r_d$ of $\mathfrak{m}$ which map to a basis for $\mathfrak{m}/\mathfrak{m}^2$ as an $R/\...
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Regular local rings are UFD

For 1-dimensional case, regular local implies PID and hence UFD. That is clear and geometry wise it is basically looking at the germs of smooth functions at a point. For higher dimensional case, what ...
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Why does Hartshorne have hypothesis $(*)$ in II.6?

In Hartshorne's chapter on Weil divisors he fixes the following hypothesis: $(*)$ Every scheme is Noetherian , integral, separated, and regular in codimension 1 I can understand why you would want ...
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An example of Gorenstein ring which is not local regular.

I have to find an example of Gorenstein ring which is not local regular. I take $A=K[|X,Y|]/(XY)$. This is a local ring. As $XY\in (X,Y)^2$ it results that $A$ is not regular. $K[|X,Y|]$ is Cohen-...
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Do flat morphism of schemes induce injection on cotangents?

In the proof that an étale morphism induces an isomorphism on tangents, we use the fact that, if the morphism is unramified, then the induced map on cotangents is surjective. Then we conclude using ...
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Examples of proper, non normal schemes

I'm looking for examples of proper, non-normal schemes over a field $k$, whose global section ring does not coincide with $k$. Looking in the direction of fiber products of simpler objects is often a ...
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Regular rings with F-finite field of fractions

Let $S$ be a regular domain of characteristic $p>0$ with fraction field $K$. Assume that $K$ is $F$-finite, meaning that $K$ is a finite module over $K^p$. Does it follow that $S$ is also $F$-...
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When does an integral group ring have finite global dimension?

Let $G$ be a finite group and $R=\mathbb{Z}[G]$ the integral group ring. If $G$ is such that $R$ is Noetherian (so $G$ polycyclic-by-finite) when does $R$ have finite global dimension? Another way of ...
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How to prove that a ring is Regular?

I have some issues to prove that a certain ring is regular, and therefore to find a "general" method to do that. In order to exemplify, recently I was solving an exercice and one question was about ...
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the localization of R at P is a regular local ring then R is regular local ring

We know the fact that if $R$ is a regular local ring then $R_{P}$ the localization of $R$ at $P$, $P\in\mathrm{Spec}(R)$ is a regular local ring. So, I wonder the converse is true or not? My counter-...
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Normality of localizations in polynomial rings?

Normality of a ring here refers to being equal to it's integral closure in it's field of fractions. The problem is: Let $A=\mathbb{C}[x,y]/(y^2-x^3-x^2)$. Show that $A_m$ is normal for every ...
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Let $(R,m)$ be * local and $R_m$ regular. Is R regular? [closed]

Let $(R,m)$ be *local and $R_m$ regular. Is $R$ regular?