# Questions tagged [regular-rings]

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21
questions

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### Is the following local map unramified?

Let $(R,m)$ and $(S,n)$ be two local rings, $R \subseteq S$, $R$ regular, $S$ a finitely generated and flat $R$-algebra, and $mS=n$.
In comments to this question it was claimed that in such situation ...

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103 views

### $A \to B$ with $A$ regular imply that $B$ is CM

The answer to this question says the following:
"The general statement is if $A \to B$ is finite and injective, and $A$ is noetherian and regular, then $B$ is CM if and only if $A \to B$ is flat.
...

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174 views

### Local rings $R \subsetneq S$ with $R$ regular and $S$ Cohen-Macaulay, non-regular

Let $R \subseteq S$ be local rings with maximal ideals $m_R$ and $m_S$.
Assume that:
(1) $R$ and $S$ are (Noetherian) integral domains.
(2) $\dim(R)=\dim(S) < \infty$, where $\dim$ is the Krull ...

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438 views

### The ring of global sections of a regular scheme

Let $X$ be a Noetherian regular scheme. Is $\mathcal{O}_X(X)$ a regular ring? For affine schemes this is true, see 02IU on the Stacks project.

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### Geometric regularity for infinitely generated field extensions

Let $k$ be a field. Suppose that for a finite type $k$-algebra $A$, we define two following properties:
$A\otimes_k k'$ is a regular ring for all finitely generated field extensions $k\subset k'$.
$...

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590 views

### On the definition of regular (non-noetherian, commutative) rings

All rings are commutative with unit. A ring $R$ is called regular if it satisfies
(Reg) Every finitely generated ideal of $R$ has finite projective dimension.
Clearly this gives the usual ...

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105 views

### Generating the annihilator ideal up to finite index

Let $\Lambda_2 = \mathbb{Z}_p[[T_1,T_2]]$ be the power series ring over $\mathbb{Z}_p$ in two variables, i.e., $\Lambda_2$ is a regular local ring of dimension 3. Let $M$ be the quotient of an ...

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226 views

### Under which conditions on the homogeneous ideal $ I $, the quotient ring $ \mathbb{C} [X_0, \dots, X_n]/I $ is a regular ring?

If $ I $ is a homogeneous ideal of the ring of homogeneous polynomials $ \mathbb {C} [X_0, \dots, X_n] $ , under which conditions on the homogeneous ideal $ I $, and particularly on $ I_m $, the $m$ -...

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### A semigroup property related to von Neumann regularity

A very common and useful notion in rings is that of von Neumann regular elements: those elements $a\in R$ such that there exists $b\in R$ satisfying $aba=a$. As this is a property defined solely ...

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808 views

### A slick proof (?) of Zariski-Nagata purity in characteristic $p$

I am trying to understand the MathSciNet review written by Mark Kisin of the paper "Almost etale extensions" of Faltings. There Kisin illustrates Faltings' approach to the almost purity theorem with ...

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385 views

### Is the embedding dimension minus the dimension upper semicontinuous?

For a Noetherian local ring $R$ with maximal ideal $\mathfrak{m}$ and residue class field $K$, consider the invariant $$\operatorname{def}(R) := \operatorname{dim}_K(\mathfrak{m}/\mathfrak{m}^2) - \...

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### When a smooth algebra is regular?

Let $A \subseteq B$ be noetherian integral domains, $A$ regular (=every localization at maximal ideal is a regular local ring) and $B$ is a smooth $A$-algebra. For the definition of a smooth algebra, ...

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301 views

### Regularity of a tensor product

Let $A \subseteq B$ and $A \subseteq C$ be commutative noetherian domains.
Assume that $A$ and $C$ are regular rings (=every localization at a maximal ideal is a regular local ring).
Assume that $B$ ...

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197 views

### When every module is a scalar extension?

Let $A \subseteq B$ be commutative noetherian domains.
Of course, if $M$ is an $A$-module, then $M \otimes_A B$ is a $B$-module.
I am curious to know if there exist additional conditions on $A$ and $B$...

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347 views

### Base change of regular schemes [closed]

Let $R$ be a complete DVR with fraction field $K$, $X$ be a regular scheme flat over $R$. Let $L$ be a finite field extension of $K$ and $Q$ be the integral closure of $R$ in $L$. Denote by $Y:=X \...

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402 views

### Reducedness of a ring with prime nilradical

Let $A$ be a regular ring and $\mathfrak q$ be an ideal, such that $\sqrt{\mathfrak q}$ is prime. Further assume that $\mathfrak q$ is locally principal (i.e. $\mathfrak q$ is an irreducible divisor ...

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**1**answer

1k views

### Is every polynomial ring over any field regular?

Is every polynomial ring over any field regular?
For a field that is algebraically closed, it is true as any maximal ideal of $k[x_1,...,x_n]$ corresponds to a point $(t_1,...,t_n)$ in $\mathbb{A}^n$ ...

**12**

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**1**answer

2k views

### geometric interpretation and differences of Gorenstein rings, Complete intersections and regular rings

Let $R$ be a local Noetherian ring.
What is the geometric interpretation of:
1- Gorenstein rings
2- Complete intersections
3- Regular rings?
and how can I realize differences by geometric ...

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729 views

### Does regular field extension preserve regularity?

Let $k$ be an arbitrary field and suppose that $K/k$ is a regular field extension. Let $V$ be regular scheme of finite type over $\text{Spec }k$ (not necessarily smooth). Is it true that $\text{Spec }...

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219 views

### A particular Isomorphism of graded algebras over a regular local ring

In Hartshorne's "Algebraic Geometry", the following statement is a weaker form of Theorem 8.21A (e), which he quotes from Matsumuura's book on commutative algebra:
Proposition. Let $R$ be a regular ...

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398 views

### Tensor product of regular ring (with some conditions)

Basically, my question is whether this answer is correct. Here is the point. Let $R$ be a ring, and let $A$ and $B$ be $R$-algebras. Suppose that $A$ is regular and $B \otimes_R B$ is regular too. ...