Questions tagged [regular-rings]

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33 questions
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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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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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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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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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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-...
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 ...
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?