# 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 ...
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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. ...
1 vote
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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 $ht(J)=\mu(J)$, ...
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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 ...
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1 vote
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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 ...
• 151
1 vote
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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 $R$ ...
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1 vote
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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 ...
• 57
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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 ...
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1 vote
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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, $I$ ...
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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 ...
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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 ...
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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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• 515
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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....
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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 ...
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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: ...
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1 vote
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 ...
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